Continuous Transformations in Analysis: With an Introduction to Algebraic Topology

DIE GRUNDLEHREN-DER

MATHEMATISCHEN WISSENSCHAFTEN IN EINZELDARSTELLUNGEN MIT BESONDERER BERUCKSICHTIGUNG DER ANWENDUNGSGEBIETE

HERAUSGEGEBEN VON

R. GRAMMEL .- E. BOPF . H. HOPF . F. RELLICH F. K. SCHMIDT· B. L. VAN DER W AERDEN

BAND LXXV

CONTINUOUS TRANSFORMATIONS IN ANALYSIS

WITH AN INTRODUCTION TO ALGEBRAIC TOPOLOGY

BY

T.RADO AND

P. V. REICHELDERFER

SPRINGER-VERLAG BERLIN· GOTTINGEN . HEIDELBERG

1955

CONTINUOUS TRANSFORMATIONS IN

ANALYSIS

WITH AN INTRODUCTION TO ALGEBRAIC TOPOLOGY

BY

T. RADO RESEARCH PROFESSOR OF MATHEMATICS

IN THE OHIO STATE UNIVERSITY

AND

P. V. REICHELDERFER PROFESSOR OF MATHEMATICS IN THE OHIO STATE UNIVERSITY

MIT 53 TEXTABBILDUNGEN

SPRINGER-VERLAG BERLIN· GOTTINGEN . HEIDELBERG

1955

ALLE RECHTE, INSBESONDERE DAS DER DBERSETZUNG IN FREMDE SPRACHEN,

VORBEHALTEN

OHNE AUSDRDcKLICHE GENEHMIGUNG DES VERLAGES 1ST ES AUCH NICHT GESTATTET, DIESES BUCH ODER TEILE DARAUS

AUF PHOTOMECHANISCHEM WEGE (PHOTOKOPIE, MIKROKOPIE) ZU VERVIELFALTIGEN

ISBN-13 978-3-642-85991-5 e-ISBN-13 978-3-642-85989-2 001: 10.1007/978-3-642-85989-2

Softcover reprint of the hardcover 1st edition 1955

COPYRIGHT 1955 BY SPRINGER-VERLAG OHG. IN BERLIN, GOTTINGEN AND HEIDELBERG

Introduction.

The general objective of this treatise is to give a systematic presentation of some of the topological and measure-theoretical foundations of the theory of real-valued functions of several real variables, with particular emphasis upon a line of thought initiated by BANACH, GEOCZE, LEBESGUE, TONELLI, and VITALI. To indicate a basic feature in this line of thought, let us consider a real-valued continuous function I(u) of the single real variable tt. Such a function may be thought of as defining a continuous translormation T under which x = 1 (u) is the image of u. About thirty years ago, BANACH and VITALI observed that the fundamental concepts of bounded variation, absolute continuity, and derivative admit of fruitful geometrical descriptions in terms of the transformation T: x = 1 (u) associated with the function 1 (u). They further noticed that these geometrical descriptions remain meaningful for a continuous transformation T in Euclidean n-space Rff, where T is given by a system of equations of the form

1-/(1 ff) X-I U, ... ,tt ,.",

and n is an arbitrary positive integer. Accordingly, these geometrical descriptions can be used to define, for continuous transformations in Euclidean n-space Rff, n-dimensional concepts 01 bounded variation and absolute continuity, and to introduce a generalized Jacobian without reference to partial derivatives. These ideas were further developed, generalized, and modified by many mathematicians, and significant applications were made in Calculus of Variations and related fields along the lines initiated by GEOCZE, LEBESGUE, and TONELLI. In their fundamental aspects, these researches constitute a study of continuous transformations in Euclidean n-space Rn in terms of Point Set Theory, Algebraic Topology, and the modern Theory of Functions of Real Variables. The purpose of this volume is to give a systematic account of some of the basic concepts, methods, and results in this general area.

The development of this program involves a study of several individual topics which may deserve mention here. The n-dimensional concepts of bounded variation, absolute continuity and generalized Jacobians, already referred to above, are discussed thoroughly from various points of view in Part IV. The topological index, a concept of

VI Introduction.

general utility in Mathematics, is studied in detail in §§ 11.2, 11.3 for the case of Euclidean n-space, and in §§ VI.1, VI.2 the important case of the plane is given special consideration. The transformation of multiple integrals, a troublesome issue even under the very restrictive assumptions customary in classical areas of Analysis, is discussed in several forms and in extreme generality in various sections of Parts IV, V, VI. Local linear approximations in Euclidean n-space are discussed in § V.2 in terms of the various types of total differentials. Since this treatise has been designed for use by mathematicians primarily interested in Analysis, it did not seem appropriate to assume that the reader is familiar with Algebraic Topology to the extent necessary for our purposes. Accordingly, a self-contained exposition of the required background material in this field is given in §§ 1.4, 1.5, 1.6, 1.7, 11.1. This exposition covers, in a very detailed manner, the basic algebraic apparatus needed in general cohomology theory, as well as some of the fundamental topics in the topology of Euclidean n-space R"'. Thus this exposition may serve as a first introduction to Algebraic Topology, especially since a definite effort has been made to prepare the reader for the study of the excellent comprehensive treatises in this field by ALEXANDROFF-HoPF, LEFSCHETZ, ElLENBERG-STEENROD (see the Bibliography).

While our main concern is the discussion of a general theory applicable in Euclidean n-space, we included a detailed study of the special cases n = 1 and n = 2. The case n = 2 is considered in Part VI, with due emphasis upon the special features which are present in this case and are lost in the general case n> 2. The case n = 1, discussed in § V.1, furnishes a concrete picture of the general abstract concepts involved, and the study of this case yields instructive geometrical interpretations for many of the basic ideas and theorems in the modem theory of real-valued functions of a single real variable.

The method of presentation in this volume is based upon the assumption that the reader has progressed beyond the stage of basic training in the theory of Functions of Real Variables, Point Set Theory, and Group Theory. The background material in these fields is merely summarized in §§ 1.1, 1.2, 1.3, 111.1. For the convenience of the reader, an Index of the terms and symbols used in the book has been added. The Bibliography contains only a list of treatises directly related to our subject. References to literature concerning individual topics are given in footnotes throughout the text, with no attempt at completeness.

The drawings for the diagrams were prepared by our colleague F. W. NIEDENFUHR. We take pleasure in thanking him for his help.

Columbus, Ohio.

January, 1955.

T.RADO

P. V. REICHELDERFER.

Table of Contents. Introduction.

Part I. Background in Topology.

§ 1. 1. Survey of general topology § 1.2. Survey of Euclidean spaces § 1.3. Survey of Abelian groups. § 104. MAYER complexes § 1.5. Formal complexes . . . . § 1.6. General cohomology theory § I. 7. Cohomology groups in Euclidean spaces

Part II. Topological study of continuous transformations in nn. § ILL Orientation in Rn . . . . . . . . . . . § 11.2. The topological index . . . . . . . . . . . . . § II.3. Multiplicity functions and index functions . . . .

Part III. Background in Analysis. § IIL1. Survey of functions of real variables. . . . . . § III.2. Functions of open intervals in Rn . . . . . . .

Part IV. Bounded variation and absolute continuity in nn.

1 19 26 30 45 63 98

110 120 145

190 201

§ IV.1. Measurability questions . . . . . . . . . . . . . . . . . 212 § IV.2. Bounded variation and absolute continuity with respect to a base-

function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 § IV.3. Bounded variation and absolute continuity with respect to a multi-

plicity function . . . . . . . . . . . . . . . . . . . . . . . .. 232 § IVA. Essential bounded variation and absolute continuity . . . 249 § IV.5. Bounded variation and absolute continuity in the BANACH sense 277

Part V. Differentiable transformations in nn. § V.1. Continuous transformations in Rl ......... . § V.2. Local approximations in Rn . . . . . . . . . . . . § V.3. Special classes of differentiable transformations in Rn.

Part VI. Continuous transformations in n 2

§ VL1. The topological index in R2 . . . . . . . . . . . . § VI.2. Special features of continuous transformations in R2 . § VL3. Special classes of differentiable transformations in R2

Index .... Bibliography. . . . . .

292 319 349

377 402 415

439 442

Part 1. Background in topology.

§ 1.1. Survey of general topology.

1.1.1. Sets and mappings. If X is a set, then x E X means that x is an element of X, while y Ef X means that y is not an element of X. A set which has a single element x is denoted by (x). On logical grounds, it is necessary to distinguish between an object x and the set (x) consisting of the single object x. However, as a matter of notational convenience we shall use frequently the same symbol for an object x and the set consisting of the single object x. It is also convenient to use the concept of the empty set which has no element. The empty set will be denoted by the symbol 0.

A set may be finite or infinite. A set X is termed countable if there exists a one-to-one correspondence between X and a set consisting of some or all of the positive integers 1, 2, ... , or if X is the empty set. Thus the elements of a non-empty countable set X can be arranged into a (finite or infinite) sequence Xl' x2 , ...•

If S (x) designates a certain statement relating to the object x, then we write

E = {xIS(x)}

to state that E is the set of those objects x for which the statement S (x) holds. For example, if A and B are two given sets, then their union A U B consists of those objects x which belong to at least one of the sets A and B, and their intersection A n B consists of those objects x which belong to both A and B. These definitions may be stated in the form of the following equations.

A UB = {xlxEA or xEB},

A n B = {x I x E A and x E B} .

The concepts of union and intersection apply to any number of sets. If F is a family of sets, then

U X = {x I x E X for some X E F} , XEF

n X = {x I x E X for every X E F}. Xc: F •

Two sets A, B are said to be disjoint if An B = 0. Rado and Reichelderfer, Continuous Transformation~.

2 Part 1. Background in topology.

If every element of a set A belongs to a set B, then A is termed a subset of B, and we write A(B or equivalently B)A. If A is not a subset of B, then we write A<j:B. As a matter of convenience, the empty set is considered a subset of every set A.

If A is a subset of X, then X and A are said to constitute a pair (X, A). If X is any set, then (X, 0) is a pair. If (X, A) and (Y, B) are two pairs such that X(Y and A(B, then we write (X, A) ((Y, B).

If S (X, then the complement CxS of S with respect to X is the set of those elements of X which do not belong to S. In symbols

CxS={xlxEX, xEfS}.

If S is also a subset of some other set Y, then C y 5 and CxS are different sets. If we consider, in a certain situation, only subsets S of a fixed set X, then we write C S instead of Cx S as a matter of convemence.

The difference A - B of two sets A and B is defined by the formula

A - B = {xlxEA, xEfB}.

Consider a fixed set X and its subsets (thus C will stand for complement with respect to X). The symbols U, n, c, -, introduced above, give rise to a number of identities involving subsets of X. Some examples follow.

CCA =A, A-B=AnCB.

AUA =A, AnA =A, AUB=BUA, AnB=BnA.

AU (B U C) = (A U B) U C, An (B n C) = (A n B) n C.

An (B UC) = (A n B) U (A n C), AU (Bn C) = (A U B) n (A U C).

C(A UB) = CAnCB, C(AnB) = CA UCB.

Similar identities hold in situations involving an arbitrary number of subsets of X. For example, if F is any family of subsets of X, then

CU5 = ncs, SE F,

cns = UCS, SE F.

Furthermore, there exist certain logical relationships between the symbols already introduced. For example, A ( B if and only if A U B = B. Similarly, A (B if and only if An B = A.

If S is a subset of X, then the function c(x, 5) defined by the formula

{ 1 if xES,

c(x, 5) = 'f C5 o 1 X E ,

§ 1.1. Survey of general topology. 3

is termed the characteristic lunction of 5 in X. For any two subsets A, B of X we have the identity

c (x, A UB) + c (x, A n B) = c (x, A) + c (x, B).

If Xl' X 2 are two sets, then their Cartesian product Xl X X 2 is the set of all ordered pairs (Xl' x2) such that xlEXl , x2 EX2 • For example, if (X,A) is a pair of sets, and I:O:;;'u;:;;;;1 is the unit interval on the real number-line, then one has the Cartesian products XxI, A xl, Xx u, A x u (where u is any element of I), and the pairs

(XxI, A xl), (Xxu, A xu),

as well as the inclusion

(Xxu, A xu) «(XxI, A xl,.

Given two sets X and Y, a mapping (or translormation) I:X -+ Y from X into Y is a rule which assigns to each xE X a unique element y = I xE Y as its image.

Given a mapping I: X -+ Y and a subset A =1= 0 of X, we denote by II A the mapping from A into Y which agrees with I on A. Thus

(f I A) x = I x for x EA.

The mapping I I A is referred to as I cut down to A or also as the restriction of I to A.

Given a mapping I:X -~ Y and a subset A of X, the image I A of A under I is the set of the images I x of the elements of A. If B is a subset of Y, then 1-1 B is the set of those elements x of X for which I x E B. In particular, if Yo is an element of Y, then 1-1 Yo is the set of those elements x of X for which I x = Yo'

Given a mapping I: X -+ Y, I is said to be onto if I X = Y, and I is said to be one-to-one if Xl =1= X 2 implies that I Xl =1= I x2 • If I is both onto and one-to-one, then I is termed bi-unique. If I is bi-unique, then for each yE Y the set I-ly consists of a single point of X. Thus in this case we have at our disposal the mapping 1-1: Y -+X, the inverse of I.

The following set of identities, relating to an arbitrary mapping I:X -+ Y, will be constantly used. If F is any family of subsets of X, then

IUA=UIA, AEF, InA(nIA, AEF.

If AI' A 2 are subsets of X such that Al ( A 2' then I Al (j A 2' If ([> is any family of subsets of Y, then

1-1 U B = U 1-1 B , 1-1 n B = n 1-1 B , B E ([> •

1*


If B1 , B2 are any two subsets of Y, then

1-1 (B2 - B1) = 1-1 B2 - 1-1 B1 .

If A, B are subsets of X and Y respectively, then

Consider now two mappings I:X~Y, g:Y~Z. Then the product mapping g I: X ~ Z is defined by the formula

(g f) x = g (f x), x EX.

Note that a product of mappings is to be read Irom the right to the lelt.

Given a mapping I: X ~ Y, consider a subset A of X and a subset B of Y. If I A (B, then I is said to be a mapping from the pair (X, A) into the pair (Y, B), and one writes

I: (X, A) ~ (Y, B).

If I is a mapping from (X, A) into (Y, B) and g is a mapping from (Y, B) into (Z, C) then clearly gl is a mapping from (X, A) into (Z, C).

If X is any set, then the identity mapping i:X ~X on X is defined by ix= x for x EX. If A is a subset of X, then the mapping iIA:A ~X is termed the inclusion mapPing from A into X.

Consider now two pairs (X, A), (Y, B) such that (X, A) ((Y, B), and let j be the identity mapping on Y. Then the mapping

j I X: (X, A) ~ (Y, B)

is termed the inclusion mapping from (X, A) into (Y, B).

If one has to consider several pairs of sets and certain mappings from these pairs, it is helpful to set up a mapping diagram. We give two examples.

Let (X, A) be a pair of sets, and let I:O:;;;'u:;;;'1 be the unit interval on the real number-line. The following diagram will be referred to later

Fig. 1.

on as the lirst homotopy diagram.

In this diagram, s and t denote two real numbers in I. The mappings is, it are the inclusion mappings corresponding to the In

clusions

(Xxs, A xs) ((XxI, A xl), (Xxt, A xt) ((XxI, A xl).

§ 1.1. Survey of general topology.

The mappings hs' hi are defined by the formulas

hsx=(x,s), htx=(x,t), xEX.

The mappings fs' If are defined by the formulas

fs x = (x, s), It x = (x, t), x EX.

5

The mapping g is defined by the formula g(x, u) = x, (x, u)E (X xI). Clearly hs and hi are bi-unique, fs and ft are one-to-one, and g is onto. Furthermore

where i is the identity mapping on X.

The diagram in Fig. 2 will be referred to as the second homotopy diagram.

In this diagram, the mappingsfo and fl have the same meaning as

g fs = g ft = i,

/[X'~A'll~ (X,A) =========::::(Y,E»

mo Fig. 2.

in the first homotopy diagram (corresponding to s = 0, t = 1). The mapping m is an arbitrary mapping from (XxI, AxI) into (Y,B), and the mappings mo and m1 are defined by the formulas

mox=m(x,O), ml x=m(x,1), xEX.

Clearly mo = m fo, ml = m fl'

1.1.2. Preliminary comments on real-valued functions. Let X =l= 0 be an arbitrary set. A real-valued function f (x) on X arises if with every element x of X there is associated a unique real number I(x). The characteristic function c (x, S) of a set S (X (see 1.1.1) is an example. It is convenient to extend the concept of a real-valued function by permitting such a function to assume the values 00 and - 00

also. If a real-valued function f (x) on X does not assume infinite values, then f (x) is termed finite-valued.

Given a mapping T:X-'?-Y (see 1.1.1) and a set SeX, the crude multiplicity function N (y, T, S) is defined, for y E Y, as the number of elements of the set S n T-l y. Thus N (y, T, S) may be equal to 00. The finite values of N (y, T, S) are non-negative integers.

While the use of 00 and - 00 is a matter of convenience, it is also true that one has to provide certain obvious but at times cumbersone explanations in this connection. The following agreements will be used.

00+ 00= 00, - 00+ (- (0) = - 00, a' 00= 00 if 0< a:S;: 00,

a'oo=-oo if -oo:S;:a<O, a+oo=oo if -oo<a<oo.

However, no meaning is assigned to 00 - 00 and 0' 00.


Let 1 (x) be a real-valued function on a set X, such that the finite values of 1 are non-negative and 1 does not assume the value - 00

(however, 1 may assume the value 00). We shall then say that 1 is nonnegative on X, and we shall write 1 (x) ~ 0, x EX. Given a real-valued, non-negative function I(x) on X and a set E (X, we shall have to consider later on the sum 01 the values of 1 (x) on E, to be denoted by L (E, I). Thus

L(E,/)=LI(x). xEE

The following comments will serve to clarify the intended meaning of the symbol L (E, f)·

(i) If f (x) = 00 for some x E E, then L (E, f) = 00.

(ii) If f (x) == 0 on E, then L (E, f) = 0. (iii) If f(x) is finite-valued on E and vanishes on E except for a finite

number of distinct elements Xl' ... , x,u of E, then

L (E,f) = f(x l ) + ... + f(xM )·

(iv) If f (x) is finite-valued on E and vanishes on E except for an infinite sequence of distinct elements Xl' ... , Xk , ..• of E, then

00

k=l

if the series on the right converges, and L (E, f) = 00 if this series diverges.

(v) If the set of those elements x E E where f (x) > ° is non-countable, then L (E, I) = 00.

(vi) If E = 0, then L (E, f) = O.

The sum L (E, f) has various obvious properties, some of which will be listed presently.

(a) If f, F are real-valued, non-negative functions on X and f;S;: F on the set E (X, then

L (E, f) ~ L (E, F).

(b) If f is a real-valued, non-negative function on X and c is a positive constant, then

L(E,c/)=CL(E,/), E(X.

(c) If {f;} is a (finite or infinite) sequence of real-valued, non-negative functions on X, then

L (E, L Ii) = L L (E, fi)' E (X. J, 1

(d) If f is a real-valued, non-negative function on X and {Ii} is a sequence of real-valued, non-negative functions on X such that one

§ I.1. Survey of general topology.

has on a certain set E (X the relations

then 115:./2£.···' li-+I,

L: (E, Ii) -+ L: (E, I)·

7

Let us recall a few definitions relating to sets of real numbers. If 5 is a set of real numbers and B is a (finite) real number such that x ::;:;: B for xES, then B is termed an upper bound for 5. A set 5 of real numbers is said to be bounded Irom above if there exists an upper bound for 5. A fundamental property of the real number system is expressed by the statement that if a set 5 of real numbers is bounded from above, then among its upper bounds there exists a smallest one. This smallest upper bound is called the least upper bound of 5 and is denoted by l.u.b.5. Clearly the relation l.u.b.5E5 holds if and only if 5 contains a largest element (which is then equal to l.u.b.5). As a matter of convenience, we agree to set l.u.b. 5 = 00 if 5 is not bounded from above. The greatest lower bound gr.l.b. 5 of a set 5 of real numbers is defined in a similar manner.

Consider now a real-valued, finite-valued function 1 (x) on a set X. Then the values assumed by 1 (x) on X constitute a set V of real numbers. The least upper bound and the greatest lower bound of 1 (x) on X are defined then by the formulas

l.u.b./(x) = l.u.b. V, gr.l.b./(x) = gr.l.b. V. :rEX "'EX

These quantities may be equal to 00 and - 00 respectively.

If I(x) is a real-valued function on X which takes on the value 00,

then we agree to consider 00 as its least upper bound. Similarly, if 1 (x) takes on the value - 00, then we agree to consider - 00 as its greatest lower bound.

If 1 (x) is a real-valued, finite-valued function on a set X and E =f= 0 is a subset of X, then the oscillation OJ (I, E) of 1 on E is defined as the least upper bound of the set of the numbers I/(x2) -/(xl)1 corresponding to all pairs of elements Xl' x2 of E.

If {x,,} is an infinite sequence of real numbers, then the symbols

lim sup x,,, lim inf XII fI.-+OO n-+-oo

denote the limit superior and the limit inlerior of the sequence respectively in the usual sense. Let us recall merely that the relation

lim sup x" = 00 "-+00

holds if and only if for every real number K there exists an integer n such that x" > K.


1.1.3. Topological spaces. Definition 1. A topological space X is a set of elements (to be termed the points of X) in which a family Q of subsets of X (to be termed the open sets of X) is assigned in such a manner that the following holds.

(i) 0EQ, XEQ.

(ii) If °1 , ... ,Om is any finite system of open sets of X, then no" j = 1, ... , m, is also an open set.

(iii) If F is any (finite or infinite) family of open sets of X, then UO, OE F, is also an open set.

Definition 2. Let X be a topological space. A subset F of X is termed closed if CF is open (where C stands for complement relative to X).

Definition 3. Let X be a topological space. If A is a subset of X, then the closure of A (to be denoted by A) is the intersection of all those closed sets of X which contain A.

Definition 4. Let X be a topological space. If A is a subset of X, then the interior of A (to be denoted by int A) is the union of all those open sets of X which are contained in A.

Definition 5. Let X be a topological space. If A is a subset of X, then the frontier of A (to be denoted by fr A) is defined by the formula

fr A =A nCA.

Definition 6. Let X be a topological space. If A is a subset of X and U is a family of subsets of X such that

ACUU, UEU,

then U is termed a covering of A. If U' is a sub-family of U such that U' is also a covering of A, then U' is referred to as a sub-covering. A covering U of A is finite if U is a finite family. A covering U of A is open (closed) if every set UE U is open (closed).

Definition 7. Let X be a topological space, and let U, ~ be two open coverings of X. If for every UE U there exists a VE Q3 such that U( V, then U is termed a refinement of )8.

Definition 8. Let X be a topological space and U an open covering of X. If UE U, then the star of U relative to U (to be denoted by Stu U) is the set defined by the formula

Stu U = U U', U' E U, U' n U =F 0.

Note that this set is open as a consequence of (iii) in definition 1. Evidently the sets Stll U, corresponding to all the sets U E U, constitute an open covering of X. The open covering so derived from U will be denoted by U*.


Definition 9. Let X be a topological space. If U,5.8 are two open coverings of X such that U* is a refinement of ~( then U is termed a star refinement of 5B.

Definition 10. Let X be a topological space,S a subset of X, and A, B subsets of S. Then A and B are said to constitute a partition of 5 (in symbols, 5 = A I B) provided that

AU B = 5, An B = 0, A =l= 0, B =l= 0, An B = 0, An 13 = 0.

Definition 11. Let X be a topological space. A subset 5 of X is disconnected if there exists some partition 5 = A lB. If there exists no partition of 5, then 5 is connected.

Definition 12. Let X be a topological space. If E is a subset of X and F is a connected subset of E, then F is termed a maximal connected subset of E if F is not a proper subset of any connected subset of E. A non-empty maximal connected subset of a set E (X is termed a component of E.

Definition 13. Let X be a topological space. A non-empty, open, connected subset of X is termed a domain in X.

Definition 14. Let X be a topological space. A subset 5 of X is compact if every open covering of 5 contains a finite sub-covering.

Definition 15. Let X be a topological space. A non-empty, compact, connected subset of X is termed a continuum. If a continuum does not reduce to a single point, then it is termed non-degenerate.

Definition 16. A HAUSDORFF space is a topological space X which satisfies the following additional condition (the so-called separation axiom): if Xl' X 2 are any two distinct points of X, then there exist open sets 01, O2 such that x1E01' x2E02' 01n02= 0.

Definition 17. A topological space X is normal if for every pair of disjoint closed sets F1, F2 there exist open sets 01, O2 such that ~ (01, F2C02,01n02=0.

Definition 18. A topological space X is fully normal if every open covering of X admits of a star refinement.

Definition 19. A topological space X is locally connected if for every open set 0 it is true that the components of 0 are open.

Definition 20. Let X be a topological space. A subset E of X is dense in X if It = X.

Definition 21. A topological space is separable if it contains a countable dense subset.

Definition 22. Let X be a topological space. A subspace A of X is a non-empty subset of X with the following assignment of the family


Q A of the open sets of A: a subset of A belongs to Q A if and only if it is of the form 0 n A, where 0 is open in X.

Note that a subset 5 of a subspace A of X may be open in A without being open in X. In a general way, the meaning of several topological concepts depends upon the topological space relative to which such concepts are considered. Given a topological space X and a subspace A of X, as well as a subset 5 (A, to avoid ambiguity we shall write fr A 5, intA 5, CA 5, and so forth, to indicate that the concepts involved are considered relative to the subspace A. On the other hand, it is easy to see that a set 5 (A is compact relative to A if and only if it is compact relative to X, and similarly 5 is connected relative to A if and only if it is connected relative to X.

Definition 23. Let Xl' X 2 be topological spaces. Then their topological product is the Cartesian product Xl X X 2 with the following assignment of open sets for X l XX2 • Let w be the family of those subsets of Xl X X 2 which are of the form 0 1 X O2 , where 0 1 is open in Xl and O2

is open in X 2. ~:\ subset 5 of Xl X X 2 is then, by definition, open in Xl X X 2

if and only if 5 is the union of a family of sets belonging to w [it is immediate that the conditions (i), (ii), (iii) in definition 1 are satisfied]. The topological product of Xl, X 2 is denoted by the same symbol Xl XX2 as the Cartesian product (it will always be clear from the context which one of the two products is meant).

Definition 24. A pair (X, A) of topological spaces consists of a topological space X and a subspace A of X. As a matter of convenience, one uses also on certain occasions the symbol (X, 0) to refer to a single topological space X.

Definition 25. Let 5 be a topological space and (Xo, Ao) a pair of subspaces of 5 such that Xo and Ao are closed in 5. Furthermore, let F be a family of pairs (X, A) of subspaces of 5 such that X and A are closed in 5. Then we shall write F =? (X 0' Ao) as an abbreviation for the following set of conditions.

(i) (Xo, Ao) ( (X, A) for (X, A) E: F.

(ii) If (Xl' Al)EF, (X2 ,A 2)EF, then there exists a pair (X3 ,A3)EF such that

(iii) If 0 and U are any two open sets of 5 such that Xo -::,0, Ao (U, then there exists a pair (X, A)E F such that X (0, A (U.

Definition 26. Let X be a topological space. A sequence {xn} of points of X converges to a point x of X (in symbols, x" _ x or x = lim x n )

if for every open set 0 containing x there exists an integer N = N (0) such that x"EO for n > N.


Clearly, if X is a HAUSDORFF space, then a sequence {xn } cannot converge to two distinct points.

Definition 27. Given a topological space X, let f(x) be a real-valued, finite-valued function defined on a subset E of X. Then f (x) is said to be continuous (relative to E) at a point xoE E if for every real number e > 0 there exists an open set O=O(e) containing Xo such that If(x)f (xo)1 < e for xE E n O. If f (x) is continuous (relative to E) at every point of E, then f(x) is said to be continuous on E (relative to E).

Definition 28. Given a topological space X and a real-valued function f (x) on X, f (x) is said to be lower semi-continuous at a point Xo

of X if for every real number a < f (xo) there exists an open set 0 = 0 (a) such that xoE 0 and f (x) > a for xE o. If f (x) is lower s€mi-continuous at every point of X, then f (x) is said to be lower semi-continuous on X.

The concepts introduced so far give rise to a number of theorems. We shall list presently a series of such theorems (which will be used later on) as exercises for the reader. The proofs may be found in the treatises on Topology listed in the Bibliography, or else the proofs are easy consequences of the definitions involved.

Exercise 1. Let A be a subset of the topological space X. Then the closure .if of A is closed, and A c.if. Furthermore, if A, B are any two subsets of X, then

A U B = AU Ii , A nBC An Ii .

Exercise 2. Let A be a subset of the topological space X. Then a point x EX belongs to .if if and only if every open set 0 containing x intersects A.

Exercise 3. If 0 is an open subset of the topological space X, then frO C CO.

Exercise 4. Let {OJ} be a (finite or infinite) sequence of pairwise disjoint open sets in a topological space x. Set UOj=O. Then IrOjc/rOCCO.

Exercise 5. Let 0 and F be subsets of a topological space X such that 0 is open, F is closed, and IrO C F. Then 0 - F is open.

Exercise 6. Let X and Y be closed subsets of a topological space 5 such that X) Y=l= 0, X - Y=l= O. Then X - Y is open if and only if 11' XC Y.

Exercise 7. In a topological space X let 0 1 , O2 be open sets such that 0 1 (0 2 .

Put O2 - 0 1 = F12. Then

( 2 ) F.2) /1'02 , O2 - F12 = 0 1 , O2 = 01 U F.2' /1'°1 = 0 1 n F12 ·

Exercise 8. Let U and A be subsets of the topological space X such that U( A. Then V C int A if and only if (int A) U (int C U) = X.

Exercise 9. In a topological space the union of a linite system of compact sets is compact.

Exercise 10. Let A be a closed subset of the fully normal topological space X. Then A (as a subspace of X) is fully normal.

Exercise 11. Let E =l= ° be a subset of the topological space X. Then the following statements hold concerning the components of E.

(i) If C1 , C2 are distinct components of E, then C1 n C2 = o. (ii) If F is a non-empty connected subset of E, then F is contained in a (unique)

component of E.


(iii) E is the union of its components. (iv) If C is a component of E and F is a connected set such that C (F (E,

then C=F. (v) If E is closed, then every component of E is closed.

Exercise 12. If the subsets A, B of the topological space X constitute a partition of X (see definition 10), then A and B are both open and closed.

Exercise 13. Let A be a subset of the connected topological space X such that A is both open and closed. Then either A = 0 or A = X.

Exercise 14. In a topological space X, let °1,°2 be non-empty, disjoint open sets. Then 01 and 02 constitute a partition of 01 U02 (see definition 10).

Exercise 15. Let °1 ,°2 ,°3 be open subsets of the topological space X, such that °1 , 02 are non-empty and disjoint, and 0 3 does not intersect both 01 and °2 ,

Then 01 U 02 U 0 3 is disconnected.

Exercise 16. Let E and F be subsets of a topological space X such that E is connected and E (F (E. Then F is connected.

Exercise 17. Let E and r be subsets of a topological space X such that r is connected. If r intersects both E and CE, then it also intersects IrE.

Exercise 18. If E is a non-empty proper subset of the connected topological space X, then fr E =F 0.

Exercise 19. Let X, Y be non-empty, closed subsets of the locally connected HAUSDORFF space 5, such that X) Y and X - Y is non-empty and connected. Thcn X - Y is contained in a unique component 01 of CY. Let 02 be the union of those components of CY which do not contain X - Y (if no such components exist, thenput02=0). SetX*=XU02, y*= YU02. Then the following holds.

(i) X* and y* are closed. (ii) X*=XUY*, y=xny*, S-Y*=01' (iii) 5 - x* is empty or non-empty according as X - Y is open or not open. (iv) 5 - x* is empty or non-empty according as Y does or does not contain Ir X.

Exercise 20. In a HAUSDORFF space X, a single point x (considered as a subset of X) is closed, and C x is open.

Exercise 21. Let X be a HAUSDORFF space which consists of precisely two distinct points a and b. Then the family of open sets of X consists of the sets 0, X, a, b.

Exercise 22. If A is a closed subset of the compact topological space X, then A is compact.

Exercise 23. If F is a compact subset of the HAUSDORFF space X, then F is closed.

Exercise 24. In a HAUSDORFF space X, let {C1} be a sequence of non-empty compact sets such that C1 ) C2 ) •••• Set E = n Cj . Then the following holds.

(i) E is non-empty and compact.

(ii) If ° is an open set containing E, then ° )C j for j sufficiently large.

Exercise 25. Let F and ° be subsets of the normal topological space X such that F is closed, ° is open, and F(O. Then there exists an open set C such that FCC, GC 0.

Exercise 26. Let U and A be subsets of the normal topological space X such that U is open, A is closed, and U CA. Let F be the family of all the pairs of the form (X - V, .4 - V), where V ( U and V is open. Then (see definition 25)

F =} (X - U, A - U).

§ 1. 1. Survey of general topology.

Exercise 27. Let Y be a non-empty, compact subset of the HAUSDORFF space X, and let {Fn} be a sequence of compact subsets of X such that F I ) F2 ) ••• , n Fn = Y. Denote by F the family of the pairs (X, Fn), n = 1, 2, .... Then (see definition 25)

F=9 (X, Y).

Exercise 28. Let 0 be a non-empty open subset of the locally connected, separable topological space X. Then the components of 0 constitute a countable family of pair-wise disjoint domains.

Exercise 29. A compact HAUSDORFF space is normal.

Exercise 30. Let {Gi } be a sequence of continua in the HAUSDORFF space X such that GI ) G2 ) •• '. Then n Gi is a continuum.

Exercise 31. Let X, Y be non-empty, compact subsets of the connected HAUS

DORFF space S, such that X) Y and X - Y = 0 is open and non-empty. Then the following holds.

(i) 0 and 11' 0 are compact. (ii) fl'O +- 0,0 - frO = O. (iii) X = OUY, frO = on Y.

Exercise 32. Let F be a closed subset of the locally connected topological space X. If D is a component of CF, then jr DC F.

Ecercise 33. Let D be a component of an open subset 0 of the locally connected topological space X. Then

IrDCtrOCCO.

Exercise 34. Let Do, DI be non-empty domains in the connected topological space X, such that 150 C Dl and 11' Do is connected. Then DI - Do is connected.

Exercise 35. Let F be a closed subset of the compact, locally connected HAUS

DORFF space X, such that CF is connected. Let 0 be an open set containing F. Then there exists an open set U such that F C U CO and C U is connected.

Exercise 36. If Xl' X 2 are compact HAUSDORFF spaces, then their topological product Xl X X 2 is also a compact HAUSDORFF space.

Exercise 37. Let I (x) be a real-valued, finite-valued, continuous function defined on a connected subset C of the topological space X. Then the following holds.

(i) If I (x) takes on the values a, b on C, then it also takes on every value between a and b.

(ii) If the set V of the values taken on by I (x) is countable, then I (x) is constant on C.

Exercise 38. If f (Xl' "', x,,) is a real-valued, finite-valued, continuous function of the variable points Xl'"'' x" on the compact topological space X, then I(xl , ... , xn) attains a maximum and a minimum on X.

Exercise 39. If fIx) is a real-valued, finite-valued continuous function on the compact topological space X, then there exist two points Xl' x2 such II (x2 ) - t (xIll is equal to the oscillation (J) (I, X) of t (x) on X.

Exercise 40. Let A be a compact subspace of the compact HAUSDORFF space X. Consider the topological products X X I, A X I, where I is the unit interval 0;;;;; u::;;; 1

on the real number-line. Given tEl, let 0, U be open subsets of X X I such that 0) X X t, U) A X t. Then there exists an E> 0 (which depends upon t, 0, U) such that

(X X s, A X s) C (0, U) for s E I, Is - tl <e.


Exercise 41. Let (X, Y) be a pair of compact HAUSDORFF spaces such that o '*' Y '*' X. Let CY be represented as the union of a (finite or infinite) sequence of pair-wise disjoint open sets 01' ... , On' .... Set Y" = COn' Y: = YUO", Fn = n Yk ,

k = 1, ... ,n. Then the following holds.

(i) Y,,, Yn*, F" are non-empty and compact. (ii) Y,,) Y, Y:) Y, Ym ) Yn* if m '*' n, F;) F2) "', Yn )Fn , 0" C V,i, trO"C Y. (iii) y=nFn, X=YnUOn , trOn=y"no". (iv) {(X, F,,)} =} (X, Y) (see definition 25).

Exercise 42. In a HAUSDORFF space S, let (Xl' 1S,), (X2' Y2 ) be two pairs of compact subsets such that (a) (X2' Y2) ((Xl' 1S,) and (b) X;-Yi is connected and open, i = 1, 2. Then either Xl - Y1 and X 2 - Y; are disjoint, or else Xl - 1S, is a subset of X 2 -}~.

Exercise 43. If X is a fully normal topological space, then X is normal.

1.1.4. Metric spaces. Definition 1. A metric space arises if in a set X of elements (to be termed the points of the metric space) there is assigned a distance d (Xl' x2) for every pair of points Xl' x2 of X, such that the following holds. (i) d (Xl' X 2) is real-valued, finite, and nonnegative. (ii) d (Xl' X2) = 0 if and only if Xl = x2. (iii) d (Xl' X2) = d (X2' Xl)'

(iv) If Xl' X2, X3 are any three points of X, then one has the triangle inequality

Definition 2. Let X be a metric space, X a point of X, and r a positive real number. Then the set

Llr(X)={X'iX'EX, d(x,x')<r}

is termed the open spherical neighborhood of X with radius r. As a matter of convenience, we shall also use the symbol Ll (x, r) to refer to this spherical neighborhood.

Definition 3. The natural topology of a metric space X is determined by the following assignment of the family of the open sets of X: A set E (X is termed open if and only if for every point x E E there exists a real number r = r (x) > 0 such that Ll, (x) (E. It is immediate that X is then a HAUSDORFF space.

Definition 4. Let E be a subset of the metric space X. Then the diameter of E, to be denoted by bE, is defined by the formula

bE = l.u.b.d(xl ,x2), x1EE, x2EE.

It is convenient to agree that the diameter of the empty set is equal to zero.

Definition 5. A subset E of the metric space X is bounded if bE < 00.

Definition 6. A metric space X is boundedly compact if every bounded closed subset of X is compact.


Definition 7. Let E be a non-empty subset and x a point of the metric space X. Then the tcart of x from E, to be denoted by e(x, E), is defined by the formula

e(x,E) = gr.l.b. d(x, y), yE E.

Definition 8. Let {xi} be a sequence of points in the metric space X. Then {xi} is termed a Cauchy sequence if for every 13 > 0 there exists an integer n = n (e) such that d (xi' Xk) < 13 if j, k> n.

Definition 9. A metric space X is complete if every CAUCHY sequence in X is convergent.

Definition 10. Let X be a metric space. A family F of subsets of X is termed completely additive if the following holds. (i) Every open set of X belongs to F. (ii) If E E F, then CE E F. (iii) If E 1 , E 2 , •.• is any (finite or infinite) sequence of sets of F, then their union also belongs to F.

Clearly, those subsets of X which belong to every completely additive family of subsets of X constitute a completely additive family. This observation justifies the next definition.

Definition 11. The completely additive family of those subsets of X which belong to every completely additive family in X is termed the Borel class in X. A set which belongs to this class is termed a Borel set in X.

As an immediate consequence of this definition it follows that every open set and every closed set in X is a BOREL set, and that the complement of a BOREL set in X is again a BOREL set. Furthermore, countable unions and countable intersections of BOREL sets are again BOREL sets.

Definition 12. Let X be a metric space. A determining system ~ of closed sets in X arises if with every finite sequence of positive integers 111 , .•. ,nk there is associated a closed subset F (n1' ... ,nk) of X. If p= (n1 , ... , nk> ... ) is an infinite sequence of positive integers, then we put 00

Fv = n F(n1 ,···, 11k ), A(~) = UF." k=l ,.

where in the second formula the union is taken with respect to all infinite sequences p of positive integers. The subsets A (~) of X which are obtained in this manner are termed analytic sets in X.

Definition 13. Let X be a metric space. A determining system ~ of closed subsets F (n1' ... ,nk ) in X is termed regular if always

F(nl' ... , nk)) F(n1' ... , nk , nk-i-l)'

Definition 14. Let f (x) be a real or complex-valued, finite-valued function on the metric space X. Then f(x) is said to be uniformly continuous on X if for every 13 > 0 there exists an 'Y) = 'Y) (e) > 0 such that If (x2) -f (Xl) I < 13 whenever d (Xl' x2) < 1).


We proceed to list some theorems, relating to the various concepts introduced in this section, in the form of exercises (the proofs may be found in the treatises on Topology listed in the Bibliography, or else the proo'fs are easy consequences of the definitions involved).

Exercise 1. If f (x) is a real or complex-valued continuous function on the compact metric space X, then t (x) is uniformly continuous on X.

Exercise 2. Let x be a point and E a non-empty compact subset of the metric space X. Then there exists a point yEE such that d(x,y) =e(x,E) (see definition 7).

Exercise 3. In the metric space X, let E be a non-empty compact set and x a point such that xEtE. Then e(x,E) > O.

Exercise 4. A metric space is fully normal.

Exercise 5. If X is a boundedly compact metric space, then X is complete.

Exercise 6. Let U be an open covering of the compact subset F of the metric space X. Then there exists a number 1} > 0 such that the following holds: if S is any subset of X such that,) S < 1} and S n F =F 0, then S is contained in some set U E U.

Exercise 7. In a complete, separable metric space X the following holds for BOREL sets and analytic sets.

(i) Every BOREL set is analytic. (ii) If ~ is a determining system of closed sets, then there exists a determining

system ~' of closed sets F(n1' ... , 12k) such that A (~') = A (~), ~!' is regular, and for every infinite sequence 111 , 122 , •.• of positive integers one has

1.1.5. Continuous transformations. Definition 1. Let f: X -+ Y be a transformation from a topological space X into a topological space Y. Then f is termed continuous at a point xoE X if for every open set 0)' in Y such that / xoE 0)' there exists an open set Ox in X such that xoE Ox and /0 x (0)'. If / is continuous at every point of X, then / is said to be continuous on X.

De/inition 2. Let /: (X, A) -+ (Y, B) be a transformation, where (X, A) and (Y, B) are pairs of topological spaces. Then / is a transformation from X into Y such that / A (B (see 1.1.1), and the statement that / is continuous is to be interpreted in the sense of definition 1.

De/inition 3. A transformation /:X -+ Y, where X and Yare topological spaces, is termed a homeomorphism from X onto Y if it is biunique, and if furthermore / is continuous on X and f-1 is continuous on Y. If there exists a homeomorphism from the topological space X onto the topological space Y, then X and Yare said to be homeomorphic.

De/inition 4. Given two metric spaces X, Y, a transformation T:X -+ Y, and a non-empty subset E of X, the oscillation w (T, E) of T on E is defined by the formula

w(T,E) = 15 TE.


Definition 5. Given two metric spaces X, Y, two transformations ~ : X -i>- Y, T2 : X -i>- Y, and a non-empty subset E of X, the deviation of T2 from ~ on E is measured by the quantity

e(~, T2 ,E) =l.u.b.d1'(~x, T2 x), xEE,

where d1' denotes distance in Y.

Definition 6. Let A be a subspace of the topological space X and let f:X-i>-A be a continuous transformation from X into A. If fx=x for xE A, then f is termed a retraction from X onto A. If such a retraction exists, then A is termed a retract of X.

Definition 7. Let (X, A) and (Y, B) be pairs of topological spaces, and let mo, m1 be two continuous transformations from (X, A) into (Y, B). Then a continuous transformation

m: (X X I, A X I) -i>- (Y, B) ,

where I is the unit interval 0 :;;;;;'u:;;;;;'1 on the real number-line, is termed a homotopy connecting mo and m1 provided that

mo x = m (x, 0), m1 x = m (x, 1), x EX.

If such a homotopy exists, then the continuous transformations mo, m1

are termed homotopic.

Definition 8. Let A be a subspace of the topological space X. Then A is termed a deformation retract of X if there exists a continuous transformation t: (X, A) -i>- (X, A) such that (a) f is a retraction from X onto A and (b) f is homotopic to the identity transformation i: (X, A) -i>- (X, A).

We shall now list, in the form of exercises, a series of theorems relating to the preceding concepts. The proofs may be found in the treatises on Topology listed in the Bibliography or else the proofs are easy consequences of the definitions involved.

Exercise 1. Let i: X -+ X be the identity transformation in the topological space X. Then i is continuous on X.

Exercise 2. Let A be a subspace of the topological space X. Then all the transformations occurring in the first homotopy diagram (see 1.1.1) are continuous, and the transformations denoted there by Its and Itt are homeomorphisms onto.

Exercise .J. Let T:X --;.. Y be a continuous transformation from the topological space X into the topological space Y. If F is a closed subset of Y, then T-l F is closed in X.

Exercise 4. Given a topological space X and a HAUSDORFF space Y, let T: X --;.. Y be a continuous transformation from X into Y. Then for every point y E Y the set T-l y is closed in X.

Exercise 5. Let T:X -+ Y be a continuous transformation from the topological space X into the topological space Y. If U1' is an open covering of Y, then the sets T-I U1" corresponding to the sets Uy E U y, constitute an open covering of X.

Rado and Reichelderfer) Continuous Transformations. 2


Exercise 6. Let X, Y be topological spaces. Consider an open subset O;t of X and a continuous transformation T:O;t-+Y. Assume that T is one-to-one and carries open sets in O;t into open sets in Y. Then T is a homeomorphism from O;t onto TO;t.

Exercise 7. Given two HAUSDORFF spaces X and Y, consider a continuous transformation T: E -+ Y from a compact subset E of X into Y. Assume that T is one-to-one. Then T is a homeomorphism from E onto T E.

Exercise 8. Given two topological spaces X and Y, consider a continuous transformation T: X -+ Y. If 5 is a compact subset of X, then T 5 is a compact subset of Y. Similarly, if 5 is a connected subset of X, then T S is a connected subset of Y.

Exercise 9. Given a continuous transformation T:X -+Y from the HAUSDORFF

space X into the HAl:SDORFF space Y, let E be a subset of X such that E is compact. Then TE=TE.

Exercise 10. Given two topological spaces X and Y, let Jz: X -+ Y be a homeomorphism from X onto Y. If 5 is any subset of X, then JzjrS=/rh5, h5=h5, hCS= Cft5, hint 5 = int hS.

Exercise 11. Let Y be a metric space and X a compact metric space. If T:X -+ Y is a continuous transformation from X into Y, then T is uniformly continuous on X in the following sense: for every 6 > 0 there exists an 1] = 1] (e) > 0 such that if the distance (in X) of two points Xl' x2 is less than 1], then the distance (in Y) of the image points T Xl' T x 2 is less than c.

Exercise 12. Let:r;" T2 be two continuous transformations from the metric space X into the metric space Y, and let E be a non-empty compact subset of X. Then there exists a point Xo EE such that (see definition 5)

where dy stands for distance in Y.

Occasionally there arise situations where one has to consider discontinuous transformations in connection with topological spaces. We shall now discuss in some detail an important example.

Modification theorem. Let there be given a fully normal topological space X, an open covering U of X, and a non-empty closed subset Y of X. Then there exists an open set 0) Y, a star refinement ~~ of U, and a (generally discontinuous) transformation f:X -+X, such that fOe Y and fVe 5t"i[j V for every VE ~R

Proof. In view of 1.1.3, definition 18 we can select a star refinement ~\ of U. Denote by 0' the union of all those sets V E m which intersect Y. Then clearly YeO' and 0' is open. By 1.1.3, exercises 43 and 25 there follows the existence of an open set 0 such that

YeO, Oeo'. (1 )

We now define a (generally discontinuous) transformation f: X-+X in the following steps.

f x = x if x E C 0' . (2)

§ 1.2. Survey of Euclidean spaces. 19

If xEO', then by the definition of 0' we can select a set V(x) E ~ such that xE V(x), V(x) n Y =l= 0. Accordingly, we can choose a point in V(x)nYas the image Ix of x. Then

xEV(x)E~~, IxEV(x)nY for xEO'.

We assert that )8, 0, I satisfy the requirements of the theorem. Indeed, (3) implies that 1 x EY if xEO', and hence 10'(Y. In view of (1) it follows that 10 (Y. There remains to show that

IV(StfJ] V if VE~. (4)

Select a set V E ~ and consider a point x E v. If xECO', then 1 x =

xEV(StfJ]V. If xEO', then in view of (3) we have IxEV(x)(St'l:;V, since V(x) contains the point x E V and hence vn V(x) =l= 0. Thus 1 x E StfJ] V in either case. Since x was an arbitrary point of V, (4) follows and the theorem is proved.

§ 1.2. Survey of Euclidean spaces.

1.2.1. Preliminaries. For each positive integer n, Euclidean n-space R" is the metric space defined as follows. The points of R" are ·n-term sequences (Xl, ... , x") of real numbers. Of course, the superscripts are indices and not exponents. If

x = (Xl, ... , x"), y = (yI, ... , y")

are two points of R", then their distance d (x, y) is defined by the formula

d(x, y) = [.f (xi - yi)21~. 1~1

Clearly the conditions required in the definition of a metric space are satisfied (see 1.1.4, definition 1). As a matter of convenience, one also considers a zero-dimensional Euclidean space RO which consists of the real number zero alone.

If x = (xl, ... , x") is a point of R", n~ 1, then Xl, ... , x" are termed the coordinates of x. It is convenient to use vector notation in R" to condense formulas. A vector v in R" has n components v!, ... , v", and one writes v = (VI, ... , vn) if it is desired to display the components of v. Thus points of Rn and vectors in Rn are identified by assigning an n-term sequence of real numbers. In the sequel, the terms point in R" and vector in Rn will be used interchangeably. The reader is assumed to be familiar with the fundamental properties of vectors, and we merely state here certain agreements concerning terminology. If

2*


are two vectors in Rn, then their sum and difference are given by the formulas

The scalar product VI • V2 is given by the formula n

VI • v2 = L v{ v~ . i~l

If VI = v2 = v, then one writes v2 instead of V • v. Thus the length i I V II of a vector V is given by the formula II V II = (V2)~.

If v = (vI, ... , vn) is a vector and c is a real number, then cv is the vector given by the formula cv = (cvI, ... , cvn).

If x = (Xl, ... , xn), y = (yl, ... , yn) are two poipts in Rn, then in vector notation one has d (x, y) = I i x - y II. Thus, in particular, II x I! is the distance of the point x from the origin (0, ... ,0).

We shall now list, in the form of exercises, various elementary theorems about R". The proofs may be found, for instance, in the ALEXANDROFF-HoPF treatise (see the Bibliography), or else the proofs are easy consequences of the definitions involved.

Exercise 1. Since R n is a metric space, for each point xE R" and each real number r > 0 one has the open spherical neighborhood ,1r (x), in the sense of 1.1.4, definition 2. In the present case, ,1r (x) and its closure 2fr (x) are given by the formulas

,1r (x) = {x' I x' ERn,

LTr (%) = {x' 1%' ERn,

!IX-%'II <r},

I i x - %' II ~ r}.

Furthermore, the following holds. (i) ,1r (%), 2fr (%) and (i f n? 2) fr ,1r(x) are connected. (ii) ,1r(%) =int 2fr(%)' (iii) %EL/,(%) - fr!1r(%). (iv) 2fy (x) and jriJr (%) are compact. (v) fr ,1, (%) = fr ZIr (%).

Put Exercise 2. In R n, n :; 1, take a point x and a real number r such that 0 < r < 1.

Ar(%) = {X' 1%' '" R",

By(x) = {x'I%'ERn,

11% - x'il ~ -~.-},

1'~ Ilx-.'f'11 ~-~-}. }'

Then the following holds.

(i) Ar(%) By (%), Ar (%) - Br (%) = ,1r (%). (ii) Ar (x) and By (x) are compact. (iii) Ay (x) - Br (%) is a domain (see 1.1.3, definition 13).

Exercise 3. If %0 is a point of R", E is a subset of R", and ,1,-(xo) nE =l= 0 for every r > 0, then %0 '" E.

E%ercise 4. R n is separable, connected, locally connected, and boundedly compact (see 1.1.3, definitions 21,11,19, and 1.1.4, definition 6).

E%ercise 5. A subset E of R" is termed isolated if for every point .'f E E there exists an open set 0 (x) containing % such that E no (%) = %. An isolated set in R" is countable.

§ I.2. Survey of Euclidean spaces. 21

Exercise 6. Let F be a non-empty compact subset of RI!, n ~ 1. Then CF has precisely one unbounded component if n > 1, and precisely two unbounded components if n = 1.

Exercise 7. If E is a non-empty compact subset of R" and T is a continuous transformation from E into R", then T is bounded in the sense that there exists a (finite) positive constant M such that II T x II < M for x E E.

Exercise 8. In Rl, a continuum (see 1.1.3, definition 15) is a set consisting of all those real numbers x which satisfy the condition a;:;;; x;;;;; b, where a, b are any two real numbers such that a;:;;; b. The most general bounded domain in Rl is a set consisting of all those real numbers x which satisfy the condition a < x < b, where a, b are any two real numbers such that a < b. The frontier of the domain is then the set consisting of a and b.

Exercise 9. Let there be given in R" a finite system of vectors vi = (vf' ... , vj). j = 1 •... , m. Then these vectors are said to be linearly dependent if there exist real numbers c1 • ... , cm such that at least one of c1 • .. " cm is different from zero and c1 VI + ... + cm V", = o. Otherwise VI' ••. , V", are said to be linearly independent. The following holds. (i) A system of n vectors Vi = (vf' .... vi). j = L .... n. in R" is linearly independent if and only if the determinant of the matrix (v}) is different

from zero. (ii) If the vectors VI' •.. , V" in R n are linearly independent, then every vector V in R n can be represented in the form V = C1 VI + ... + c" V". where c1 ' ... , c" are uniquely determined real numbers.

Exercise 10. Let there be given n vectors Vi = (v} • ... , vi). i = 1, ...• n. in R". such that II vi.! ~ = 1. j = 1 •... , n, and vJ • vk = 0 for j '*' k. Then the determinant of the matrix (vi) has the value ± 1.

Exercise 11. Let b" be a vector in R". n > 1. such that!' b" I' = 1. Then one can select vectors b1 ••..• b .. _ 1 in R" such that Ilbill=1. i=1 •...• 11. and bi·bi=O if i '*' j.

Exercise 12. If VI' V2 are vectors in R", then I VI' v2 1 ;:;;; ! VI : i; V2 ,

1.2.2. Elementary figures in Rn. The n-cell En in R", n;;;:' 1, is defined by the formula

£" = {xlxE R", Ilx:1 S; 1}.

The O-cell EO in RO is defined as EO = RO (thus EO Gonsists of a single point).

The n-sphere 5" in R"+1, n'20, is defined by the formula

5n = {xlxE R,,+l, ilxil = 1}.

Thus 5° consists of precisely two points (namely, + 1 and -1).

For n;;;:' 1, we have in Rn+l the following configuration (which will be referred to as the basic diagram in Rn). Starting with the (n + 1)-cell En+1 in Rn+1, we consider the following sets as constituting the basic diagram (in describing these sets, x = (Xl, ... , x"+ I ) denotes a generic point of Rn+1).

£,,+1 = {xlllxll:S: 1}, 5n = {x III x II = 1}.

E'i-= {xlllxli = 1, x"+l;;;:, a}, 5,,-1 = {xlllxli = 1, X,,+l = o}.

E~ = {x III x II = 1, X'H·I;;;;;: O},

22 Part I. Background in topology.

If ai' bi , i = 1, ... , n, are real numbers such that ai< bi , i = 1, ... , n, then the set of those points x = (Xl, ... , x") in Rn which satisfy the conditions aj;;S; xi;:;;;: bj , i = 1, ... , n, is called an n-interval in R", or briefly an interval in Rn. If bj - aj = h > 0 for i = 1, ... ,n, then the n-interval is called an oriented n-cube of side-length h. Thus the points x = (Xl, ... , nn) of an oriented n-cube Q (R" may be characterized by means of inequalities of the form

aj:S;; xi ;;;;;: a7 + h, i = 1, ... , n,

where h> 0 is the side-length of Q. For fixed i, the set of those points of Q for which xi = aj is termed an (n - 1) -dimensional face of Q, and the set of those points of Q for which xi = ai + h is also termed an (n -1 )-dimensional face of Q. Thus the number of (n - 1 )-dimensional faces of Q is equal to 2 IZ. The point

( h h

al + 2 ' ... , a" + 2)

is called the center of Q, and the point (aI' ... , an) is called the initial vertex of Q.

In Rn a segment with end-points Xl, x2 is defined as the set of those points x which are (in vector notation) of the form

x=tx2 +(1--t)xl , O:;:;;t;:;;;:1.

A non-empty subset E of Rn is termed convex if Xl E E, x2 E E imply that the segment with end-points Xl' x2 is contained in E.

Let there be given in R" a point Xo = (x~, ... , x~) and n linearly independent vectors Vi = (vI, ... , v7), i = 1, ... , n. Then the parallelotope P with initial vertex Xo and edge-vectors VI' ... , vn is the set of those points x = (Xl, ... , x') which can be represented in the form

n .j_J-j'" i·_ x-xo -.L../1iV;, ]-1, ... ,n,

'~l

where /11' ... ,/1" are real numbers such that O;:;;;:/1,:S;;1, i=1, ... , 'i1.

In vector notation, the points x of P may be represented in the form

" x = Xo +L: /1; Vi' 0;;;;: /1i :;:;; 1, i = 1, ... , 11. '~l

The point n

Xo +! L: Vi i=l

is termed the center of the parallelotope. Consider in Rn';-l, n ~ 1, the n-sphere S". Denote by R" the set

of those points x = (Xl, ... , xn+l) of R,,+l for which X,,+l = 0 [thus

a generic point x of R" is of the form (xl, ... , x", 0)]. Denote by N

§ 1.2. Survey of Euclidean spaces. 23

the point (0, ... ,0,1) (the north pole of sn). If x = (Xl, ... , x", 0) is a point of Rn , then the line through x and N is easily seen to intersect sn in precisely one point-x distinct from N. The transformation I: Rn~sn - N defined by I x = x is easily seen to be a homeomorphism from R" onto S" - N. The inverse 1-1 of I, which is a homeomorphism from sn - N onto R", is termed the stereographic projection from S" - N onto Rn. On denoting by h the transformation from R" onto Rn defined by

h (xl, ... , x", 0) = (Xl, ... , xn ),

clearly hl- 1 is a homeomorphism from S"-N onto R". This homeomorsphism hl- 1 is frequently referred to as the stereographic projection from sn - N onto R".

We shall list presently a series of elementary theorems in the form of exercises. The proofs may be found in the ALEXANDROFF-HoPF treatise (see the Bibliography), or else the proofs follow readily in view of the definitions involved.

Exercise 1. The following holds for the basic diagram in R"+l (see above).

(i) All the sets occurring in the diagram are compact. (ii) 5"-1 is homeomorphic to the (n - 1)-sphere Sn-1

(iii) E~, E'.:. are homeomorphic to the n-cell Ell.

(iv) S" = E~ U E'.:., 5"-1 = E~ n E'!...

Exercise 2. Let Q be an oriented n-cube in R", 11 :2 1, of side-length Ii> o. Let Xo = (x~, ... , xrrl be the initial vertex of Q. Then Q consists of those points x = (Xl, ... , x") which can be represented in the form

xi = x~ + f.lj It , j = 1, ... , 11,

where f.lI' ... , f.ln are real numbers such that 0;;;;; f.lj ;;;; 1, j = 1, "', n.

Exercise 3. Let I be an n-interval in RH. Then I is convex, compact, and connected, and int I is a convex domain.

Exercise 4. In R", n ;:;;; 1, let P be the parallelotope with center y and edgevectors VI' "', v". The P consists of those points x E R n which can be represented in the form

n

X = Y + L}.jVi' i~l

where AI' "', }'n are real numbers such that

-LS;;}'i;;;;;~' i=l, ... ,n.

Exercise 5. The n-cell E" is convex, compact, connected, and int E" is a convex domain.

Exercise 6. The n-sphere S", 11 :2 1, is a non-degenerate continuum. Furthermore, S" is locally connected and separable.

Exercise 7. In R", n ~ 1, the following holds for the open spherical neighborhoods Ll,(x) (see 1.1.4, definition 2). (i) Ll,(x) is a convex domain. (ii) Ar(x) is a convex continuum.


Exercise 8. If 0 is an open set in Rn, n Z I, then there exists a sequence 11 , 12 , •••

of n-intervals such that If (0, i = I, 2, '" , and 0 = U int If'

Exercise 9. In R", n:;:: I, let F be a compact subset of an open set O. Then there exists a finite system of oriented n-cubes Q1' ... , Qm such that

Exercise 10. For n;;;: 1, the stereographic projection (see above) can be used to transfer configurations from R" to S", and vice versa. The following statements are useful in this respect.

(i) If n ;;;; 2, and F =F 0 is a compact subset of sn such that S" - F is non-empty and connected, then F is homeomorphic to a non-empty compact subset F* of R n

such that R" - F* is connected. (ii) In R", n :? 1, let Y be a non-empty compact set. Then there exists a non

empty compact subset y* of S" such that (a) y* is homeomorphic to Y, and (b) on denoting by q the number of components of SI! - y* and by k the number of bounded components of R" - Y, one has q = k + 1-

Exercise 11. In R", n'i 2, denote by Qn the unit cube [the set of those points x = (xl, ... , x") for which 0 ::;;;; xi :;;;: I, i = I, '" , n]. If F is a non-empty compact set in R" such that R" - F is connected, then there exists a non-empty compact subset F* of int Qn such that (a) F* is homeomorphic to F, and (b) Q" - F* is connected.

Exercise 12. Let Q be an oriented n-cube in R", 11:: 1. Then Q is homeomorphic to the n-cell En, and IrQ is homeomorphic to the (n - I)-sphere sn-l.

Exercise 13. In R", n? 1. the origin (0, ... ,0)· is a deformation retract of the n-cell En.

Exercise 14. Let F be a non-empty compact subset of the one-sphere 51, such that 51 - F is non-empty and connected. Then either F consists of a single point, or else F is homeomorphic to the one-cell E1.

1.2.3. Subdivisions in RH. Definition 1. Let nand m be positive integers. Then the subdivision .1 (n, m) of RIO consists of those oriented n-cubes in R" which are determined by inequalities of the form

!!.L<;;.xi:::;;' ki+~ . 1 2m - - 2'" ' I = , ... , n,

where kl' ... , k" are arbitrary integers.

Definition 2. Let Q": 0 S xi S 1, j = 1, ... , n, be the unit It-cube in RI!, n:£ 1, and let m be a positive integer. Then the subdivision .1* (n, m) of Q" consists of those cubes of the subdivision .1 (n, m) of R" which are contained in QI!.

Definition 3. Let PI' ... , Pj' .. , be the sequence of positive primes (thus PI = 2, P2 = 3, ... ), and let n be a positive integer. Then the subdivision Dpi of R" consists of those oriented n-cubes in RI! which are determined by inequalities of the form

'!i.-..-- .i ~ k i + 1 . 1 Pi';;;;" - Pi ' t = , ... , n,

§ I.2. Survey of Euclidean spaces.

where kI' ... , k" are arbitrary integers. Given an oriented n-cube

Q: ai -;;;, xi 5,;. a; + h, i = 1, ... , n, h > 0,

25

in R", the subdivision Dpi of Q consists of all those oriented n-cubes which are determined by inequalities of the form

k· . k· + 1 . ai + --'- h -;;;, x' -;;;, ai + ~, ~- h, Z = 1, ... , n, Pj Pj

where kI' ... , k" are integers such that o-;;;,k;-;;;'Pj-1, i=1, ... , n.

Definition 4. If K is a collection of cubes of the subdivision Ll* (n, tn) (see definition 2), then IK I denotes the union of the cubes qEK.

Definition 5. Two cubes q', q" of the subdivision Ll* (n, m) (see definition 2) are strongly adjacent if q' =!= q" and q'n q" is a common (n - 1 )-dimensional face of q' and q".

Definition 6. A finite sequence ql' ... , qs of s?.,; 2 cubes of the subdivision Ll* (n, m) is termed a chain if qj and qj-i-l are strongly adjacent, j=1, ... , s-1 (see definitions 2 and 5). Thus qj=!= qj+I , but one may have qj = qk if I j - k I :2 2.

Definition 7. A collection K of cubes of the subdivision Ll* (n, m) is strongly connected if (a) K is empty, or (b) K consists of a single cube, or (c) for every pair q', q" of distinct cubes of K there exists a chain ql' ... , qs in K such that qi = q', qs = q" (see definitions 2 and 6).

We shall state presently, in the form of exercises, a series of elementary theorems relating to the preceding concepts. The proofs may be found in the ALEXANDROFF-HoPF treatise (see the Bibliography), or else the proofs are easily deduced from the definitions involved.

Exercise 1. Let Xo be a point of R", n :2; 1, such that no coordinate of Xo is equal to zero. Then there exists an integer jo such that for j > jo the point Xo is contained in the interior of a (unique) cube of the subdivision DPi of R".

Exercise 2. Given an oriented n-cube Q (R", 12 ~ 1, and a point Xo E int Q. there exists an integer jo such that for j > jo the point Xo is contained in the interior of a (unique) cube of the subdivision DPi of Q.

Exercise 3. Let r be a finite collection of s "" 2 distinct cubes of the subdivision il (1'1, In) of R", n 1· Then the cubes of r can be aITanged into a sequence qi' .... q, in such a manner that on setting

s-J .

Y = qs n (j~l qi)'

Y is a proper subset of Irqs'

Exercise 4. Let Fl' Fs be disjoint compact subsets of the unit cube Q" in R". n ? 1. Then there exists an integer Ino such that for 111 > 1110 no cube of the subdivision Ll* (n, In) of Q" intersects both Fl and F2 .

Exercise 5. Let 0 and F be subsets of R", n :;;;: 1, such that 0 is open, F is compact, and F (0. Denote by F", the union of all those cubes of the subdivision Ll (n, In) of R" which intersect F. Then there exists an integer Ino such that F", (0

for m > mo'


Exercise 6. Consider the subdivision Lf* (n, m) of the unit cube Q" in Rn. Let q', qO be a pair of strongly adjacent cubes of Lf* (n, mi. Then there exists a point Xo

such that the folIowing holds.

(a) Xo Efrq' n fr qO. (b) If q is a cube of Lf* (n, m) such that Xo Eq, then either q = q' or q = qO.

Exercise 7. Let r be a continuum in the unit cube Qn of R", n ;;;;; 1. Denote by K the colIection of all those cubes of the subdivision Lf* (n, m) of Q" which intersect r. Then K is strongly connected.

Exercise 8. Let Ks be a strongly connected collection of s cubes of Lf* (n, m), where s ;::; 2, and let q* be a cube of Ks. Then the cubes of Ks can be arranged into a sequence ql' ... , qs in such a manner that the folIowing holds. (i) q'-l and qs are strongly adjacent. (ii) The collection K,_l consisting of the cubes ql' ... , q'-l is stlOngly connected, and q*EKs _ 1 .

Exercise 9. Let D be a bounded domain in R", n ;;;;; 1. Then there exists in R" a sequence of domains {Di } such that the folIowing holds.

(i) Di is the interior of the union of a finite number of cubes of the subdivision Lf (n, mil of R", where m1 < m 2 < ....

(ii) fr Di is the union of a finite number of (n - 1)-dimensional faces of cubes of Lf (n, mi).

(iii) 751 (Di +1 , j = 1. 2, ... .

(iv) D=UDi , j=1,2, ... .

§ 1.3. Survey of Abelian groups.

1.3.1. Abelian groups, factor groups, direct sums. In the present § 1.3. we collect; for convenient reference, the definitions and facts concerning Abelian groups which will be needed later on. Proofs and further details may be found in the ALExANDRoFF-HoPF treatise listed in the Bibliography.

Abelian groups will be written additively (that is, the group operation will be denoted by the symbol +). An Abelian group G contains at least one element, namely o. If G consists of a zero-element alone, then G. is termed trivial, and one writes G = o. The following special groups will frequently occur in the sequel.

The additive group of integers will play an important role, and will be denoted by I. For each positive integer n, an Abelian group I" is defined as follows. The elements of In are n-term sequences (kl' ... , k,,) of integers. Addition in In is defined by the formula

til' ... ,j,,) + (hI' ... , "II) = (il + kI' ... ,j" + kll) .

In particular, one has 11 = I. As a matter of convenience, one sets 10 = O. A further important group lID (where w denotes the cardinal of the set of positive integers) is defined as follows. The elements of lID are those infinite sequences {kj } of integers which contain at most a finite number of non-zero terms. Addition in lID is defined by the formula

{kj} + {if} = {kj + ij}.

§ 1.3. Survey of Abelian groups. 27

An Abelian group G is termed infinite cyclic if it contains an element go such that (a) every element of G can be written in the form ngo, where n is some integer, and (b) one has ngo = 0 if and only if the integer n is equal to zero. An element go with these properties is termed a generator of the infinite cyclic group G. If go is a generator of the infinite cyclic group G, then - go is also a generator, and G possesses no further generators. The additive group I of integers is the prototype of infinite cyclic groups, its generators being + 1 and - 1.

-Let K be a subgroup of the Abelian group G. If g is an element of G, then the set of all those elements g' of G for which g' - gE K is called a coset relative to K, and is denoted by [g, K], or merely by [gJ if the subgroup K is thought of as fixed. For given K, the cosets relative to K constitute a group, the factor group GIK, in which addition is defined by the formula

I t is easy to see that the sum so defined is independent of the choice of gl' g2 in the cosets [gl], [g2J. The factor group GIK is again an Abelian group.

Let GI , ... , Gm be a finite system of subgroups of the Abelian group G, such that each element gEG can be represented in one and only one way in the form

g = gl + ... + gm' gl E GI , ... , gm EG",.

Then G is said to be the direct sum of its subgroups GI , ... , Gm , and one writes

G = GI + ... + G",.

Let {Gj } be an infinite sequence of subgroups of the Abelian group G, such that the following holds: for every element g =f= 0 of G there exists a unique (finite) system of integers kl' ... , km such that 0 < kl < ... < kIll and

g=gk,+···+gkm ' gkiEGki , gki =f= 0, i=1, ... ,m.

Then G is said to be the weak direct sum of the subgroups GI , G2 , ••••

We shall now state, in the form of exercises, a few facts needed in the sequel. The proofs may either be found in the ALExANDRoFF-HoPF treatise (see the Bibliography) or can be readily deduced from the definitions involved.

Exercise 1. Let [g] be an element of the factor group G/K. Then [g] = 0 if and only if g E K.

Exercise 2. If 0 denotes the subgroup of G consisting of the zero-element of G alone, then G/O = G.

Exercise 3. G/K = 0 if and only if K = G.


Exercise 4. If go is a generator of the infinite cyclic group G and n1 , n2 are integers, then n1 go = n 2 go if and only if n1 = n 2 •

Exercise 5. Let n? 2 be an integer. For each integer i = 1, ... , n, denote by Ii the subgroup of I" consisting of those elements (kl' ... , knl of I" for which k i =0 for i oF i. Then If is infinite cyclic, and I" is the direct sum of I~, ... , I~.

Exercise 6. For each positive integer i. denote by Ii" the subgroup of those elements {ki} of IW for which k, = 0 if i oF i. Then Ii" is infinite cyclic, and IW is the weak direct sum of 11'. I~ , ....

1.3.2. Homomorphisms, isomorphisms. Let h:GC~G2 be a mapping from the Abelian group GI into the Abelian group G2 • If

h (g~ + g~') = h g~ + h g~'

for every pair of elements g~, g~' of Gl , then h is termed a homomorphism from GI into G2. If hGI = G2, then h is said to be a homomorphism from GI onto Gz. The set of those elements gl EGI for which hgl = 0 is termed the nucleus of h. If hGl = 0 (that is, if every element gl of GI

is carried by h into the zero-element of G2), then h is called a zerohomomorphism. If either GI = 0 or G2 = 0, then clearly there exists only one homomorphism h:GC +G2 , and this unique h is a zero-homomorphism. In these cases, one refers to this unique h as the trivial zero-homomorphism. If G is any Abelian group, then the mapping h:G-+G defined by hg=g for gEG is clearly a homomorphism, termed the identity homomorphism in G. One writes h = 1 to state that h is the identity homomorphism.

If the nucleus of the homomorphism h: GI -+ G2 reduces to the zeroelement of Gl , then clearly h is one-to-one. If this is the case, then h is called an isomorphism from GI into G2 , and if furthermore hGI = G2, then h is called an isomorphism from GI onto G2, and one writes h: Gl ~ G2. Two Abelian groups GI , G2 are termed isomorphic if there exists an isomorphism from Gl onto G2 •

Let G be an infinite cyclic group and h: G ~ G an isomorphism from G onto G. If go is a generator of G, then it is easy to see that either hgo = go or hg 0 = - go. In the first case, h is termed even, and in the second case h is termed odd. It is immediate that these properties of h do not depend upon the choice of the generator go.

Let h:GI"-~G2 be a homomorphism from the Abelian group GI

into the Abelian group G2, and let K I , K2 be subgroups of GI , G2

respectively such that hKI (K2 . For each element gl of Gl denote by [gIJI the coset in GI , relative to K I , which contains gl' and let [g2J2 have an analogous meaning relative to G2 , K 2 . Consider an element [gIl of the factor group GIIKl . Then [hgrJ2 is an element of the factor group G21K2 , and it is easy to see that this element depends only upon the coset [gIJI and is independent of the particular element gl selected

§ 1.3. Survey of Abelian groups. 29

in that coset. Accordingly, we can define a mapping

h. : G1/K1 -+ G2/K2

by the formula h.[gIJl = [hg1J2' and it is immediate that h. is a homomorphism. The homomorphism h. so obtained is said to be induced by the homomorphism h:G1-+GS (which is assumed to satisfy the condition hKl ( K 2).

The proofs of the statements included in the following exercises are either contained in theALExANDRoFF-HoPF treatise (see the Bibliography) or can be readily deduced from the definitions involved.

Exercise 1. If h: Gc + Ga is a homomorphism from the Abelian group G1 into the Abelian group Ga. then the nucleus of h is a subgroup of G1 • and hG1 is a subgroup of Ga.

Exercise 2. Given three Abelian groups G1 • Ga. Ga. let h1 :G1 -+Ga• h2:G2-+Ga be homomorphisms such that h2 is an isomorphism into. Then the homomorphism h = h2 hI has the same nucleus as hI'

Exercise 3. Given three Abelian groups G1 • G2• Ga. let h1 :G1 -+G2• h2 :G2 -+Ga be homomorphisms such that h2hl is onto. Then h2 is also onto.

Exercise 4. Consider a homomorphism h: 1--+ I. where I is the additive group of integers. If h is onto. then it is an isomorphism onto.

Exercise 5. Given two Abelian groups G1 • G2 • let hI: G1 -+ G2 • h2 : G2 -+ G1

be homomorphisms such that h2hl is the identity homomorphism in G1 • Then one has the relation G2 = II -+- N2 (see 1.3.1). where N2 is the nucleus of h2 and II = hI G1 .

Exercise 6. Consider two groups I'. IS. where r is equal to some non-negative integer or to w. and similarly s is equal to some non-negative integer or to w (see 1.3.1). Then I' and IS are isomorphic if and only if r = s.

Exercise 7. Let G be an Abelian group which is the direct sum of its subgroups G1 • .... G",. If these subgroups are infinite cyclic. then G is isomorphic to 1m (see 1.3.1).

Exercise 8. If {Gi } is an infinite sequence of infinite cyclic subgroups of an Abelian group G. and G is the weak direct sum of these subgroups. then G is isomorphic to I"' (see 1.3.1).

1.3.3. Exactness 1• Let G, H, K be Abelian groups, and let g:G-+H, h: H -+ K be homomorphisms. We shall use the diagram

G~~H~~K

to refer to this sort of situation, and we shall say that we are given a three-term sequence of groups and connecting homomorphisms. The three-term sequence is said to be exact if the nucleus of h coincides with gG.

More generally, a diagram of the form

••• hl_'~ G. hi-.~ G . .l:L~ G. h;+I~ ••• 1-1 1 1+ 1

1 For historical comments and further details. see the ElLENBERG-STEENROD treatise listed in the Bibliography.


will be used to refer to a situation where we are given a (finite or infinite) sequence of Abelian groups and connecting homomorphisms. Such a sequence is termed exact if every three-term sequence

G hj_, G hj G i-l-~ i -~ i+1

contained in it is exact. The ElLENBERG-STEENROD treatise (listed in the Bibliography) contains a comprehensive study of exact sequences. We shall consider here merely a few elementary facts relating to such sequences.

Lemma 1. If the three-term sequence

G ~G.l2.G I ~ 2 ~ 3

is exact and G3 = 0, then hI is onto.

Indeed, the assumption G3 = 0 implies that the nucleus of h2 coincides with G2 • Since the sequence is exact, it follows that G2 = hI GI , and thus hI is onto. The proofs of the following lemmas are equally obvious.

Lemma 2. If the three-tenn sequence

G hi G.l2. G 1--- 2 ,. 3

IS exact and GI = 0, then h2 is an isomorphism into.

Lemma 3. If the three-term sequence

G ", G h, G 1->- 2--- 73

IS exact and G1 = 0, G3 = 0, then G2 = O.

Lemma 4. If the four-term sequence

G ~G.l2.G~G 1 ,.. 2 ;lo- 3 )- 4

IS exact and G1 = 0, G4 = 0, then h2 is an isomorphism onto.

§ 104. MAYER complexes l .

I.4.1. Basic definitions. A MAYER complex

M = M(CP, oP) (1 )

is a mathematical system consisting of the following objects. (i) For every integer p (positive, negative, or zero) there is given an Abelian

1 For historical comments and further details concerning this fundamental concept, see the ElLENBERG-STEENROD treatise listed in the Bibliography. We restrict ourselves to a detailed discussion (designed for a reader unfamiliar with this type of algebraic approach) of those topics which are actually needed in the sequel.

§ I.4. MAYER complexes. 31

group 0. (ii) For every integer p, there is given a homomorphism

(2) such that

bP+l bP = o. (3)

Explicitly, (3) means that if cP is any element of CP, then bP+1 bP cP is the zero-element of CPH.

Generically, an element of 0 will be termed a p-cochain of the Mayer complex M, and 0 will be termed the group of p-cochains of lvI. If cPECP, then the (P+l)-cochain bPcP is termed the coboundary of cPo A p-cochain whose coboundary is zero is termed a p-cocycle. Thus the p-cocycles constitute precisely the nucleus (see 1.3.2) of the homomorphism bP, and hence the p-cocycles form a subgroup of CP which will be denoted by ZP. A p-cochain is termed a p-coboundary if it is the coboundary of some (P -1 )-cochain. Thus the p-coboundaries constitute precisely the image of 0-1 in 0 under the homomorphism bP-l, and hence (see 1.3.2, exercise 1) the p-coboundaries form a subgroup of cP which will be denoted by BP. Thus cP E BP if and only if there exists some cP- 1 E 0-1 such that cP'= bP- 1 cp- 1• By (3) it follows that bPcP = bP(bP- 1 cP- 1) = O. Hence every p-coboundary is a p-cocycle. In summary:

(4) The factor group

(5)

is termed the p-th cohomology group of the Mayer complex M. As explained in 1.3.1, the elements of HP are cosets in ZP with respect to the subgroup BP. If zP is an element of ZP, then the coset (relative to BP) containing zP will be denoted by [zPJ and will be termed the cohomology class of zp. Thus the generic element of HP is a cohomology class [zPJ, where zP is a p-cocycle of M. By 1.3.1, exercise 1, the relation [zPJ = 0 holds if and only if zP EBP. In other words, the cohomology class [zPJ is the zero element of HP if and only if the p-cocycle zP is a p-coboundary. If zf, z~ are two p-cocycles, then the relation [zfJ = [z~J holds if and only if the difference z~ - zf is a p-coboundary.

If clarity requires explicit reference to the complex M, then notations like b~, c~, O(M), ZP(M), BP(M), HP(M), [z~lM will be used as needed.

1.4.2. Co chain mappings. Let there be given two Mayer com-pie xes

K=(O(K),b~), L=(CP(L),bt). (1 )


Suppose that for every integer p there is also given a homomorphism

"p CP(K) ----CP(L)

}..P:cP(K)-+cP(L). (2)

Consider the diagram in Fig. 3.

CP·'(K)--"--:---~ CPH(L) "P+I

Starting with a p-cochain c~E CP(K), we can apply to c~ the homomorphisms band ,t in two different ways, obtaining the (p + 1)-cochains bi,tP c~ and ,tP+1 b~ c~ of L. If

Fig. 3. (3 )

for every integer p and for every p-cochain c~, then the homomorphisms (2) are said to constitute a cochain mapping from K to L. Briefly, we have a cochain mapping if the vector law holds in the preceding diagram.

Suppose that the homomorphisms ,tP constitute a cochain mapping. We assert then that

,tP ZP (K) (ZP (L) ,

,tP BP (K) (BP (L).

(4)

(5)

That is, under a cochain mapping cocycles are carried into cocycles and coboundaries are carried into coboundaries. Indeed, consider any cocycle z~EZP(K). Then, in view of (3),

M_ X" z~ = ,tP +1 b~ z~ = 0,

since b~z~=O. Thus).P z~EZP(L), and (4) is proved. Similarly, consider any coboundary b~ = bV1 C~-l. Then

,tP blK = ,tP blK-1 C~-l = M.-1 ,tP-l C~-l E BP(L) ,

and (5) is also proved. In view of (4), (5) and 1.4.1 (5), there arises (see 1.3.2) for each p a homomorphism from HP(K) into HP(L), to be denoted by

,t~ :HP(K) -? HP(L). (6)

The discussion in 1.3.2, applied to the present situation, yields the following explicit description of the homomorphism ,t~. The genenc element of HP (K) is of the form (see 1.4.1)

(7)

Then ,tP zlK is a p-cocycle of L by (4), and hence ,tP z~ determines a cohomology class [,tP Z~JL in zP (L) relative to BP (L). We have then explicitly

(8)

§ 1.4. MAYER complexes. 33

The homomorphisms A~ are said to be induced by the cochain mapping (2).

In addition to the complexes (1) and the cochain mapping (2) let there be given a third MAYER complex

(9) and a cochain mapping

fhP: cP (L) --+ CP (M). (10)

We have then the homomorphisms

fhP ;,.P : cP (K) --+ cP (M) . (11)

On taking any p-cochain ciKEcP(K), we obtain [since (2) and (10) are cochain mappings] the relation

b~ fhP )',P ciK = fhP+l bt AP ciK = fhP+l AP+1 biK ciK,

showing that the homomorphisms (11) constitute a cochain mapping from K to M. Briefly, the product oj two cochain mappings is again a cochain mapping.

The cochain mappings (2), (10), (11) induce homomorphisms A~, fh~' (fhP AP)* for the cohomology groups of the complexes K, L, M. There arises the following diagram.

Fig. 4.

We proceed to verify the relation

(fhP AP) * = fh~ A~ , (12)

which will be referred to as the vector law jor induced homomorphisms. Consider any element

In view of (8) we obtain

(fhP AP)* h~ = [fhP AP Z~JM = fh~ [AP Z~JL = fh~ A~ [Z~JK = fh~ A! h~,

and (12) is proved.

1.4.3. Standard triples of MAYER complexes. Let there be given three MAYER complexes

L = (CP(L), bf), K = (CP(K), b~), F = (cP(F), C>t;), (1) Rado and Reichelderfer, Continuous Transformations. 3


and cochain mappings

r/: cP(L) ~ CP(K) , :JlP:cP(K) ~ cP(F). (2)

It will be convenient to write (2) in the concise fonn

cP (L) ~ cP (K) ~~ cP (F).

The complexes L, K, F, jointly with the cochain mappings (2), will be said to constitute a standard triple of MAYER complexes if the following conditions hold for every integer p.

(i) The homomorphism r/ is an isomorphism into. That is, if ct E cP (L) and 1'l ct = 0, then ct = o.

(ii) The homomorphism :JlP is onto. That is, if ctE CP (F), then there exists at least one p-cochain c~EcP(K) such that :JlPc~=ct.

(iii) The three-tenn sequence (3) is exact (see 1.3.3). That is, if c~EcP(K), then :JlPc~=O if and only if there exists a cochain ctECP(L) such that c~ = 17P cf.

Observe that a standard triple consists of three MAYER complexes and of two cochain mappings, as indicated in (3), with the properties (i), (ii) , (iii).

Given a standard triple as in (1) to (3), the cochain mappings in (3) induce homomorphisms (see 1.4.2)

HP (L) 'i~ ~ HP (K) ": , HP (F). (4)

Next, it will be shown how the properties (i), (ii), (iii) may be used to introduce, for each integer p, a homomorphism

r5~ :HP (F) -~ HP+l (L), (5)

termed the coboundary homomorphism of the standard triple. A definition for r5~ is suggested by an inspection of the following diagram.

Fig. 5.

The generic element hf,EHP(F) is of the fonn (see 1.4.1)

ht = [ztJF' ztEZP(F). (6)


Since 'JT,P is a homomorphism onto by (ii), there exists a cochain c~ E CP (K) such that

(7)

Then <5~ c~ is a coboundary in BP+1 (K); since one seeks an element of HP+1 (L) to assign as the image of h~ in (6) under <5!, one is led to ask whether there is a cocycle zf+1EZP+1(L) such that

(8)

Note that if such a co cycle zf+1 exists, then it is unique by (i). We shall find that zf+1 does exist, and we shall also find that on assigning [Zf+lJL as the image of [Z~JF there results a homomorphism from HP (F) into HP+1 (L). To facilitate the discussion, we introduce a binary relation

zf+1m:z~, zf+1EZP+1(L), z~EZP(F), (9)

as an abbreviation of the statement that for the co cycles z~, zf+1 there exists a cochain c~E cP (K) such that (7) and (8) hold. The following diagram indicates the groups and homomorphisms that will be used in the course of the discussion.

c p +Z (ll '1 p.2

, C P•2 (K)

~~.' ~P+' K

CP+I (Ll rIP"

~ CP"(K) n P"

.. CP"CF)

sP L

sP K

sP F

c P ell rIP

~ c P (Kl n P

~ cP (Fl

p.' SP-' SK F

CP-' lK) n P- 1

.. cP-'(Fl Fig. 6.

Lemma 1. Given z~EZP (F), there exists at least one cocycle zf+1fZP+l (L) such that (9) holds.

3*


Proof. Since 'j(l is onto, there exists a cochain c~E cP (K) such that

(10) We have then

(11)

since z$ is a cocycle. In view of the exactness property (iii) above, (11) implies the existence of a cochain Cf+l E CP+1 (L) such that

( 12)

In view of (9), (10), (12), the lemma will be proved if we show that

cf+1 E ZP+1 (L). ( 13)

Now by (12) and 1.4.1 (3) we have

'YjP+2 bf+1 cf+1 = bf/l 'YjP+1 cf+1 = b~+1 b~ c~ = 0.

Since 'YjP+2 is an isomorphism into, it follows that bf+l ci+1 = 0, and ( 13) is verified.

The proofs of the following lemmas 2, 3, 4, 5, 6 are quite similar and are left to the reader.

Lemma 2. If Zf+l'1(Z$ and zf+1 '1( zt, then

(zf+1 ± zf+1) '1( (z$ ± zt). Lemma 3. If zf+1E BP+1 (L) and z$E BP (F), then (9) holds.

Lemma 4. If zf+1 '1(z$, and zt is a p-cocycle of F such that z$ - z$ E BP (F), then zf+1 '1( zt.

Lemma 5. If zt+1 '1(z$ and zf+1 is a (p + 1 )-cocycle of L such that Zf+l - zf+1 E BP+1 (L), then zf+1 '1(z$.

Lemma 6. If Zf+l'1(O, then zf+lE BP+l (L).

We are now ready to define the homomorphism b~ [see (5)J. The generic element h$ of HP (F) is of the form (see 1.4.1)

h$=[Z$JF' z$EZP(F). (14)

By lemma 1 there exists a cocycle Zf+lEZP+l (L) such that

( 15) We define

(16)

To justify this definition, we have to show that b~h$ is independent of the particular choice of the cocycles z$, Z!+l [subject to the conditions (14), (15)J. SO let zt, zf+1 be any other pair of cocycles such that

( 17)


We have to show that (18)

Now zt-z~EBP(F) by (14) and (17). and hence in view of (15) we conclude by lemma 4 that

zt+l ~z$.

From (17) and (19) we obtain, by lemma 2.

(zt+l - zt+l) ~ O.

(19)

(20)

By lemma 6 (applied to the cocycle zt+l- zt+l) it follows from (20) that zt+l- zt+l E BP+l (L), and (18) is proved (see 1.4.1).

It is thus shown that b~ is a single-valued mapping from HP (F) into HP+l(L). If h$, h$ are any two elements of HP(F), then lemma 2 and the defining formulas (14), (15), (16) yield directly the relation

b~ (h~ + h~) = b~ h$ + b~ h~. showing that b~ is a homomorphism.

Lemma 7. Given two cocycles z$EZP(F) and zt+lEZP+l(L), the relation

holds if and only if (21)

(22)

Prool. By the definition of c5~. there exists a cocycle zf+l such that

zt+l ~z$.

c5~ [Z~JF = [Zt+1JL'

(23)

(24)

Suppose now that (21) holds. Then from (21) and (24) it follows that

zt+l - zf+l E BP+l (L). (25)

In view of lemma 5, (23) and (25) imply (22). Conversely, if (22) holds. then (21) follows directly from the definition of c5~.

Jointly with the homomorphisms nt and nt in (4), the homomorphism c5~ gives rise to the following sequence (infinite in both directions):

6,,-1 "I" n" 6" 1 11,,+1 ... ~ HP (L) ~ HP (K) ~- .. HP (F) ~ HP+ (L) ~ . .. (26)

which is called the cohomology sequence of the standard triple.

Theorem. The cohomology sequence of a standard triple is exact (see 1.3.3).

Prool. Let us use the symbols I (n~), I (nt).I (~) to denote the images of the groups HP(L), HP(K), HP(F) under the homomorphisms

38 Pari 1. Background in topology.

1]~, n~, b~ respectively. Similarly, let us denote by N (n~), N (b~), N(1]~+1) the nuclei of the homomorphisms n~, b~, 1]~+1.

Exactness at HP(K). Let h'K be an element of HP(K) such that

h'KE I (1]~). (27)

Then there exists an element [zfJLEHP(L) such that

h'K = rJ~ [ZfJL = [rJP ZfJK' I t follows that

n~ h'K = n! [rJP ZfJK = [nP rJP ZfJF = 0,

since nP rJP zf = 0 by the exactness property (iii) above. Thus h'KE N (n~). Since h'K was an arbitrary element satisfying (27), we conclude that

I(rJ~)(N(n!).

Consider now an element h'K= [z'KJKEHP(K), such that

h'K = [Z'KJK E N (n~).

(28)

(29)

Then n~ h'K = 0, and hence nP z'K E BP (F). Hence there exists a cochain ct-1 E CP-l (F) such that

(30)

Since nP- 1 is onto, there exists a cochain c'K-lE CP-l (K) such that

(31)

From (30) and (31) we conclude that

nP (z'K - b'K-1 c'K-1 ) = nP z'K - nP b'K-1 c'K-1

= nP z'K - bt-1 nP- 1 c'K-1 = nP z'K - bt-1 ct-1 = O.

By the exactness property (iii) above it follows that z'K - b'K-1 c'K-1

is the image, under rJP, of an element of CP (L). We denote this element by zf, anticipating the fact (to be verified in a moment) that this element is a cocycle. We have

z'K - b'K-1 c'K-1 = rJP z£. (32)

To verify that zf is a cocycle, we calculate [using (32)J:

rJP+l bt zt = b'K rJP zt = b'K z'K - b'K t5'K- 1 c'K-1 = 0 - 0 = 0,

since z'K is a cocycle. As rJP+l is an isomorphism into, it follows that bf z£ = 0, and thus z£ is a cocycle. Now since b'K-1 c'K-1 E BP (K), we have

[z'K - b'K- 1 C'K-1JK = [Z'KJK = h'K.

From (32) we conclude now that

rJ~ [Z£JL = [ll ZfJK = [z'K - b'K-1 C'K-1]K = h'K.


Thus J4EI(17~)' Since hft. was an arbitrary element satisfying (29), it follows that N(n~) (1(17:), By (28) we conclude that N(n!) =1(17:), and exactness at HI> (K) is proved.

Exactness at HI>+1 (L). Take an element ht+1 = [Zf+1JL of HI>+1 (L) such that ht+1EI(<5~). Then there exists an element ht= [ztlFEHP(F) such that ht+1 = <5~ ht. By lemma 7 we have then the relation zt+1\llz~, and hence [see (7), (8), (9)J there exists a cochain cft.E CI> (K) such that

171>+1 zt+1 = 15ft. cft. ' nl> cft. = z~ . It follows that

17~+1 h£+1 = [171>+1 Zt+1JK = [15ft. Cft.JK = 0,

since <5ft.cft.E BP+1(K). Thus ht+1EN(17:+1), and it is proved that

(33)

Consider next an element ht+1 = [Zt+1JL of HP+l (L) such that h£+1E N (17:+1). Then

17:+1 ht+1 = [if+1 Zt+1JK = 0,

and hence there exists a cochain cft. E cP (K) such that

17P+l zt+1 = 15ft. cfc. (34) We calculate

<5~ nl> cfc = nP+1 15ft. cft. = nP+I 17P+l zf+1 = 0,

by the exactness property (iii) above. Thus nl> cfc is a cocycle of F:

nl>cft.=ztEZI>(F). (35)

From (34) and (35) it follows [see (7), (8)J that zt+1\llzt. Hence, by lemma 7, ht+1 = <5~ [ZtJF' and it is thus proved that

(36)

From (33) and (36) it follows that 1 (<5t) = N (17:+1), and exactness at HI>+I (L) is proved.

Exactness at HI> (F). Take an element htE HI> (F) such that htEI(n:). Then there exists an element [zft.JKE HI> (K) such that

ht = n: [Zft.JK = [nl> Zft.JF.

Now <5~ht is [see (14), (15), (16)] of the form

<5t ht = [zt+1JL' zt+1 \llnl> zfc.

Accordingly, there exists a cochain cft.E CI> (K) such that

171>+1 zt+1 = 15ft. eft., nl> cft. = nl> zft..

(37)

(38)

40 PartI. Background in topology.

Thus :rr;P (c~ - z~) = o. By the exactness property (iii) above, there follows the existence of a cochain c£ E CP (L) such that

c~ - z~ = r/ cf. (39)

From (38), (39) we infer that

'YjP+1 (Z£+1 - (l£ C£) = 'YjP+1 zf+1 - 'YjP+1 (If c£

= (l~ c~ - (l~ 'YjP cf = (l~ c~ - (l~ c~ - (l~ z~) = 0,

since z~ is a cocyc1e. Since 'YjP+l is an isomorphism into, it follows that Zf+1_(lfcf=0 and hence zf+1EBP+1(L). In view of (37) it follows that (l~ ht = 0, and thus htE N (l~). This shows that

I (n!) ( N (l!). (40)

Consider now an element [ztJFEHP(F) such that [ztJFEN(l!). Now [see (14), (15), (16)J

(41)

and there exists a cochain c~ E CP (K) such that

(42)

Since (l![ZtJF= 0 by assumption, we conc1ude from (41) thatzf+1EBP+1(L). Hence there exists a cochain cf E cP (L) such that

zf+1 = (If cf·

From (42) and (43) we obtain

(l~ (c~ - 'YjP cf) = (l~ c~ - 'YjP+1 (If c£ = (l~ c~ - 'YjP+l z£+l = o.

Thus c~ - 'YjP c£ is a cocyc1e of K:

c~ - 'YjP cf = zj; E ZP (K) .

In view of (44), (42) we have then

n~ [Z~JK = [nP Z~JF = [nP c~ - n P 'YjP CfJF = [nP C~JF = [ZtJF'

(43)

(44)

since nP 'YjP cf = 0 by the exactness property (iii) above. Thus [zt JF E I (n~), and hence

N (l~) ( I (n~). (45)

From (40) and (45) it follows that I(n~)=N(l!), and the proof is complete.

1.4.4. Mappings for standard triples. To avoid prohibitive notational complications in the sequel, we make the following agreements. Given two MAYER complexes K. L and a cochain mapping from K


to L as in 1.4.2, we shall write

K~L or A:K~L

as an abbreviation for the explicit formulas (1), (2), (3) in 1.4.2. Similarly, the formulas 1.4.3 (1), (2), (3), relating to a standard triple, will be abbreviated by writing

Let there be given two standard triples

Tl : Ll ..!!4 Kl ~ Fl

T2:L2..:E....".K2~F2

and three cochain mappings I k t

Ll ---+ L2, Kl ---+ K 2' Fl ---+ F2·

For each integer p we have then the following diagram.

CPCL,) 17; • CPCKJ

rr; • CPCF,)

,P k P fP

CP(Lz) • CPCKz) • CPCF;) 17~ n P

l Fig. 7.

(1 )

(2)

(3)

(4)

The cochain mappings I, k, f will be said to constitute a mapping from the standard triple (1) to the standard triple (2) provided that the vector law holds in each box of the preceding diagram (for every integer Pl. Explicitly, this condition means that

kP'f}f = 'f}g lP, fPnf = ng kP.

Let there be given a third standard triple

and cochain mappings

L2~L3' K2~K3' F2~F3·

By 1.4.2 we have then also the cochain mappings


The following statement is an immediate consequence of the preceding definitions.

Theorem 1 (product law lor triple mappings). If I, k, I constitute a mapping from TI to T2 and A, x, cp constitute a mapping from T2 to 7;, then Al, xk, cpI constitute a mapping from TI to 7;.

Returning to the standard triples 7;., T2 in (1), (2), suppose that the cochain mappings (3) constitute a mapping from 7;. to T2 • There arises the following diagram.

.. · .. ---HP(L, ) r'1 ~" ~HI>(K, ) n;" • HP(F,) b~ .. • HP .. (L,)--.. · ..

,p kP fP P" (6) .. .. .. I"

.. · .. -HP(Lz} '1f ..

;0 HP{Kz)

n~* • HP(~ )

b~* p+,( ) H L z -----0" .. _.

Fig. 8.

This diagram consists of the cohomology sequences (see 1.4.3) of the standard triples (1) and (2), connected by homomorphisms (indicated by vertical arrows) which are induced (see 1.4.2) by the cochain mappings (3).

Theorem 2 (vector law lor cohomology sequences). The vector law holds in each box of diagram (6). Explicitly: for every integer p we have the relations

k~ 1Jf* = 1J~* l~,

I~ nf* = n~* k~,

1~+1 <5f* = <5~* I~.

(7)

(8)

(9)

Prool. In view of (5), the relations (7) and (8) follow directly from the vector law for induced homomorphisms (see 1.4.2). To prove (9), take any element

(10)

By the definition of the homomorph:sm bf* [see 1.4.3 (14), (15), (16), (9)J, we have then a cocyc1e

zt:-l E ZP+1 (L I )

and a cochain cP E cP(K ) K, I

such that

<5f*h~, = [Zi;-lk, (11 )

§ 1.4. MAYER complexes.

By (11) and the relations (5), applied for p and p + 1, we obtain

'YlP+1Ip+! ZP+l = kP+! 'YlP+! zP+1 = kP+1 tl cP = tJP kP cP . '/2 L, '/1 L, K, K, K, K,

Since I, k, I are cochain mappings, we have (see 1.4.2).

43

(12)

(13)

IP+! zt;-1 E ZP+l (L2) , IP z~, E ZP (F2) , kP c~, E cP (K2) •

Accordingly, applying 1.4.3 (9) to the standard triple (2), the relations (12) and (13) mean that

IP+I zP+1 ~ IP zP . L., F,

Hence, by lemma 7 in 1.4.3,

tJP [IP zp] = [IP+l ZP+l] . 2* F, F, L, L,

Observe now that [jP Z~,JF. = I~ [zt,k = I~ ht, '

[IP+! zP+!] = IP+l [ZP+l] = IHI tJP hP L, L, * L, L, * 1* F,'

by (10) and (11). From (14), (15), (16) we conclude that

tJ~* I~ ht, = 1~+1 tJf* ht" and (9) is proved.

1.4.5. Homotopy. Let there be given two MAYER complexes

L= (CP(L),o£), K= (CP(K),o~),

and two cochain mappings

(14)

( 15)

(16)

(1 )

By the agreement stated in 1.4.4, 11 and 12 are abbreviations for homomorphisms

It:CP(L)-+CP(K), It:CP(L)-+CP(K).

satisfying the condition 1.4.2 (3), The cochain mappings 11.12 induce (see 1.4.2) homomorphisms

If* :HP(L) -+ HP(K), I~* :HP(L) -+ HP(K).

Our present objective is to state a relationship between the cochain mappings 11 and 12 which implies that If * = I~ * for every integer p.

The cochain mappings 11' 12 are termed homotopic, In symbols II -= 12 , provided that there exists homomorphisms

DP:cP(L) -+ CP-l (K), P = 0, ± 1, ± 2, ... , (2)


such that for every cochain ct E cP (L) the following relation holds:

!5~-1 DP ct + DP+1 !5f cf = I~ cf - If ct· (3)

The homomorphisms DP are then said to constitute a homotopy operator for the pair of cochain mappings 11 ,/2 , and (3) is called the homotopy identity.

Theorem. If 11 = l2' then If * = I~ * for every integer p. Proof. Take any element

hf = [z£k E HP(L).

Since z£ is a co cycle and hence <5£ z£ = 0, the substitution ct = zf in (3)

yields IP zP - IP zP = <5P- 1 DP zP E BP (K) 2L lL K L .

Hence (see 1.4.1, 1.4.2)

and consequently

If* [zfk = [If ZfJK = [le ZfJK = 19* [zfk· Since [ZfJL was an arbitrary element of HP (L), the theorem is proved.

1.4.6. Subcomplexes. Let there be given two MAYER complexes

L = (CP(L), <5f), K = (CP(K), <5~). (1 )

Then L is termed a subcomplex of K, in symbols L (K, if the following two conditions hold for every integer p.

(i) CP(L) is a subgroup of cP(K). (ii) <5£=<5~lcP(L) (see 1.1.1). Lemma 1. Suppose that L(K. For each integer p, let

r/:CP(L) _cP(K)

denote the identity homomorphism (that is, r/ cP = cP for every cochain cPE cP (L). Then the homomorphisms r/ constitute a cochain mapping from L to K.

Proof. If cP is any element of cP(L), then by condition (ii) above we have

<5~ r/ cP = <5~ cP = <5£ cP = r/+1 <5£ cp•

Thus the condition 1.4.2 (3) is satisfied, and the lemma is proved.

Lemma 2. Given a MAYER complex K= (cP(K), <5~), let there be assigned for each integer p a subgroup Gp of cP (K), such that <5~Gp( GP+1

for every p. Then the subgroups Gp and the homomorphisms <5~ I Gp

constitute a sub complex of K. This is a direct consequence of the definition of a subcomplex.

§ 1.5. Formal complexes.

Lemma 3. Let there be given two MAYER complexes

and a cochain mapping (see 1.4.4 for the notation)

;':KI -+K2 •

Furthermore, let

LI = (cP (LI), t5f.), L2 = (CP (L2), t5t,)

45

(2)

be subcomplexes of K I , K2 respectively, such that ;'PCP(LI) (CP(L2 )

for every p. Then the homomorphisms (see 1.1.1)

p.P = ;,p I cP (LI) : cP (LI) -+ CP (L 2)

constitute a cochain mapping from LI to L2.

Proof. Take any cochain cPE cP (LI)' Since (2) is a cochain mapping, we obtain

t5t p.P cP = t5t ;,p cP = ;'PH t5t cP = p.P+I t5t cp.

Thus the condition 1.4.2 (3) is satisfied, and the lemma is proved.

§ 1.5. Formal complexes l .

1.5.1. p-functions. Let X =1= 0 be an arbitrary set and p"?:. 0 an integer. A p-function cP (xo, ... , xp) for X is an integral-valued function of p + 1 variable points xo, ... , xp of X. Thus, for example, a O-function CO (xo) is an integral-valued function defined for all points xoEX, a i-function c1(xo, Xl) is an integral-valued function defined for every choice of the points xo, xIEX, and so forth. The sum of two p-functions cP and dP, to be denoted by cP + dP, is the p-function defined by the formula

(cP+ dP) (xo, ... , xp) =cP(xo, ... , .'rp) +dP(xo, ... , xp). (1)

Since cP and dP are integral-valued, clearly cP + dP is also integralvalued. It is obvious that for fixed P"?:. 0 the p-functions for X form an Abelian group with addition defined by (1). This Abelian group will be denoted by q (X). The subscript F stands for formal, and in turn the term formal is used in the present § 1.5 to emphasize the fact that the concepts to be discussed here relate to un-topologized sets, and hence are purely formal in character.

1 These complexes are frequently referred to as abstract simplicial complexes. For historical comments and further details concerning this subject, see the treatises by ALEXANDROFF-HoPF, LEFSCHETZ, and ElLENBERG-STEENROD listed in the Bibliography. We restrict ourselves to topics actually needed in the sequel. The presentation is designed for a reader unfamiliar with this subject.


Given a O-function cOE C~(X) and any two points XU' xIEX, the expression CO (Xl) - CO (xo) may be thought of as representing the increment of CO over the pair of points Xo, Xl. The same expression may be thought of as a (manifestly integral-valued) function of the variable points XU' xIEX, and hence as a i-function for X. This i-function will be denoted by o~co. Thus o~co is the i-function for X defined by the formula

(2)

Similarly, given clE C} (X), the increment of clover a triple of points Xu, Xl' X2 is a 2-function for X, to be denoted by o1:CI and given by the formula

In generalizing this process, the following notation will be convenient. Given an array of points Xu' ... , xmEX, the symbol xo,···, Xi' ... , xm

will denote the array xu' ... , Xi-I' Xi+l' •.• , Xm obtained by deleting the point Xi. Given then a p-function cPE q (X), the symbol o~cP will designate the (P + i)-function defined by the formula

P+l (o!- cP) (xo, ... , xpH) = L (- 1)i cP (xo, ... , xi' ... , Xp+l)' (4)

;=0

For P = 0 and P = 1, the general formula (4) reduces to the special formulas (2) and (3) respectively.

If cPE q (X), then o~cP is clearly an element of q+l (X). Accordingly, the symbol o!- may be interpreted as a mapping

o!-: q (X) -+ q+l (X) , P ~ 0, (5)

which assigns to the generic element cP of q (X) the element o~cP of q+l (X) as its image.

Lemma 1. o~ is a homomorphism (see 1.3.2).

Proal. The assertion means that on choosing any two elements cP, dPE q (X), one has the relation

(6)

Observe that the expressions appearing in (6) are (P + 1)-functions. The equality of two functions means that they have the same value for every choice of the variable arguments. Accordingly, (6) means that on choosing any array xu' ... , xP+1 of P + 2 points in X, one has [see (1)J

(o~(cP+dP)) (xo, ... , xpH) = (o!-cP) (xo, •.• , xpH) + (o!-dP) (xo, .. , xpH). (7)

§ I.5. Formal complexes. 47

Now (7) can he verified by direct calculation [using (1) and (4)J as follows.

P+l (b~(eP + dP)) (xo, ... , xP+l) = L (- 1}i (eP + dP) (x o, ... , xi' ... , XH1)

i=o P+I PH

= L (- 1)i eP (xo, ... , xi' ... , XP+l) + L (- 1}i dP (xo, ... , xi' ... XP+l) ;=0 i=o

= (b~ eP) (xo, ... , XP+l) + (b~ dP) (xo, ... , XP+l) ,

and (7) is proved.

Remark 1. The simple proof of lemma 1 has been carried out in full detail as an illustration of the steps involved in proving an identity between p-functions. The detailed discussion of similar identities in the sequel has the purpose of providing exercises for the reader who is studying this subject for the first time.

Remark 2. If p-functions relative to two sets X, Yare to be considered simultaneously, then notations like c~, et, bt, and so forth will be used to avoid ambiguity. However, consistent use of fully descriptive notations would soon lead to prohibitive complications, and hence we shall use the simplest ad hoc notations consistent with clarity.

While the homomorphism b~ has the effect of raising the dimension, another basic homomorphism to be introduced presently operates in the opposite direction. Let a point v, to be termed the vertex, be selected in X. A homomorphism

I'/: q(X) -+ q-l (X) , p;;;;: 1,

is then defined as follows: if cPE q(X), then I'/ cP is the (P -i}-function defined by the formula

(I',/cP) (x o, ... , xp _ 1) =(-1)P cP(xo, ... , xp _ 1' v), P?1. (8)

For p = 1 and p = 2, for example, one has

(F./c1) (xo) = - c1(xo, v), (I',,2 e2) (xo, Xl) = c2 (xo' Xl' v).

Since cP is integral-valued, clearly F,/ cP is also integral-valued. It is also immediate that if cP, dP are any two elements of ct (X), then I'"P (eP + dP) = I'"P cP + I'"P dP• It is customary to refer to I'"P as the cone homomorphism with vertex v.

Lemma 2. If P;;;;'1 and cf, c~ are elements of q(X), then

cf = c~ if and only if

I'''pef= ['''peg

for every choice of the vertex vEX.

(9)

(10)


Proal. Clearly (9) implies (10). Suppose, conversely, that (10) holds for every vEX. Let the points Xo,"" xpEX be arbitrarily assigned. On choosing v = xp ' (8) and (10) yield

cf(xo, ... , xp) = (-1)p(r~cf) (xo, ... , xp- 1)

= (- 1)P (r~c~) (xo, ... , xp- 1) = c~ (xo, ... , xp) ,

and (9) is verified.

Remark 3. This lemma enables us to reduce the verification of the equality of two p-functions cf, c~ to the verification of the equality of two (p -1 )-functions rt cf, J;;p c~. This remark will be utilized in proofs based upon induction with respect to p.

Lemma 3. If cPE C~ (X), then

(11)

for every choice of the vertex vEX.

Proal. On choosing any array X o, ... , xp of P + 1 points in X. (4) and (8) yield

p

i=o p (12) (O~-l T"P cP) (xo .... , xp) = L: (- 1)i (T/ cP) (xo, ... , xi' ... , xp) I

= (- 1)PL: (-1)icP(xo, ... , xi' ... , X P' v), j=o

Adding (12) and (13), we obtain

(O~-l r/ cP + T"P+l o~ cP) (xo, .... xp) = cP (xo ..... xp),

Since the points xo, ... , xp were arbitrarily chosen, (11) follows.

Remark 4. (11) is termed the cone identity. It enables us to express the combination ro in terms of the combination or, and vice versa.

Lemma 4. If cPEq(X), then

O~+l o~ cP = 0, P ~ 0. ( 14)

Proof. The assertion means that

(O~+l o~ cP) (xo, ... , XP+2) = ° for every choice of the points xo,"" xp+2EX. For p = 0, direct calculation yields

(o}o1-cO) (xo, Xl' X2) = (o1-cO) (Xl' X2) - (o1-cO) (Xo, X2) + (01 CO) (XO' Xl)

= (CO(X2) - CO (Xl)) - ((CO (X2) - CO (XO)) + (CO (Xl) - CO (XO)) = 0,

§ 1.5. Formal complexes. 49

and (14) is verified for p = 0. Proceeding by induction, assume that (14) is known to hold for a certain integer p;;;:. 0. We have to show that, as a consequence of (14),

(15 )

for cP+IEC~+l(X). By lemma 2, (15) will be proved if we show that

( 16)

for every choice of the vertex vEX. Two successive applications of the cone identity (11) yield

F"P+3 b~+2 b~+l cP+I = b1P CP+I - b~+I I'vP+2 b~+1 CP+I

= b~+I cP+I - b~+I (cP+I - ~ I',;p+I CP+I) = b~+1 b~ F"P+I CP+I = 0,

in view of the induction assumption (14). Thus the inductive proof of the lemma is complete.

1.5.2. The formal complex MF(X). Given a set X =p 0, the corresponding formal complex

(1 )

is a MAYER complex (see §I.4) defined as follows. For p;;:;,O, the groups C~(X) and the homomorphisms b~ are set up as explained in 1.5.1. For p < 0, one agrees to set C~ (X) = 0 (that is, ct (X) consists of a zero-element alone for p < 0). Accordingly, for p < ° the homomorphism b~ from ct(X) into Ct;+1(X) is a zero-homomorphism (see 1.3.2). To show that these definitions yield a MAYER complex, we have to verify that

b~+1 b~ cP = 0 for every p, (2)

where cPECt(X). If P<O, then cP=O and (2) is obvious. If p;;;:.O, then (2) follows by 1.5.1, lemma 4.

The cochains, cocycles, coboundaries, and cohomology groups of the formal complex MF (X) will be termed the cochains, cocycles, coboundaries and cohomology groups respectively of X. In analogy with the notation ct (X) for the group of p-cochains, the symbols Z$ (X) and Bt; (X) will designate the groups of p-cocycles and p-coboundaries of X respectively, while the cohomology groups will be denoted by H$(X). One sees that for p;;;:.O the p-cochains of X are merely the p-functions of X, while for p < 0 the only p-cochain of X is the zero-element of the trivial group C$(X).

The introduction of the trivial cochain groups ct; (X) = 0 for p < 0 is a matter of convenience. Agreement with the general theory of MAYER complexes is accomplished in this manner, and various basic

Rado and Reichelderfer, Continuous Transformations. 4


theorems can be stated in a uniform manner. A significant exception arises, however, in connection with the cone homomorphism (see 1.5.1). This point will be discussed presently in some detail.

Definition 1. The cone homomorphism

Ij': q (X) -+ q-l (X),

with vertex vEX, is defined as in 1.5.1 (8) for P?:.1. As ct-1(X) =0 for p;;;;: 0, in accordance with 1.3.2 the homomorphism r/ is a zerohomomorphism if p;;;;: o.

Thus definition 1 extends, in a trivial manner, the cone homomorphism considered in 1.5.1 to the case p ;;;;:0. There arises the question as to the status of the cone identity 1.5.1 (11) in this new situation. In considering this question, the following definition will be useful.

Definition 2. The unit O-cocycle of X, to be denoted by C~, is the O-function which has the constant value 1 on X.

Remark 1. To justify this definition, we have to show that C~EZ~ (X). It is obvious, however, that a O-function CO is a cocycle if and only if it has a constant value on X. Thus Z'j,. (X) consists of those O-functions CO

that reduce to constants on X. This result implies that Z'j,. (X) is isomorphic to the additive group I of integers. Indeed, choose a point xEX. Then the mapping m :Z~(X)-+I, defined by mzo=ZO(x) for zOEZ~(X), is obviously an isomorphism from -4 (X) onto I.

Lemma 1. If cP E q (X), then (see definition 2)

O~-l rj' cP + rj'+1 o~ cP = cP if P oF 0,

0Xl I:0 CO + I'.l o~ CO = CO - CO (v) C~.

(4)

(5)

Proof. If P<O, then cP=O and (4) is obvious. If P?:.1, then (4) is merely a re-statement of 1.5.1 (11). Turning now to (5), observe that I:0 CO = 0 by definition 1, and hence (5) reduces to

On choosing an arbitrary point xEX, one obtains

(I:l o~ CO) (x) = - (o~ CO) (x, v) = CO (x) - CO (v) = (CO - CO (v) C~) (x),


(6)

Remark 2. Comparison of (4) and (5) reveals that the cone identity (4) generally fails to hold if P = O. From (5) we infer the precise statement: for p = 0 the cone identity (4) holds for those and only those O-cochains CO

that have the value zero at the vertex v.

§ 1. 5. Formal complexes. 51

Lemma 2. The cohomology groups Ht (X) of a set X =f= $3 are as follows:·

Ht(X) = ° if P =f= 0,

H~(X) R:i I,

where I is the additive group of integers.

(7)

(8)

Prool. Since Ht (X) =zt (X)/ Bt (X), (7) is equivalent (see 1.3.1, exercise 3) to the relation zt(X) = Bt(X), which in tum is equivalent to the relation

zt (X) (Bt (X) for P =f= 0, (9)

since Bt (X) is a subgroup of zt (X). Now if zP is any element of zt(X), then b~zP=o, and (4) yields (see § 1.4)

zP = b~-l r: zP E Bt(X) .

As zP was an arbitrary element of zt(X), (9) follows. Turning now to (8), observe first that B~(X) = bx:1 Cil(X) = ° since Cil(X) = 0. Thus (8) is equivalent (see 1.3.1, exercise 2) to the relation Z~(X) !::::; I, and this last relation has already been established in remark 1.

1.5.3. Induced cochain mappings. Given two sets X =f= 0 and Y =f= 0, consider a mapping (see 1.1.1)

I:X-+Y. (1)

Then I gives rise, for every integer p, to a homomorphism

1$: q(Y) -+ q(X) (2)

in the following manner. If p 2. ° and c~E q(Y), then the image 1$ c~ of c~ is the p-function for X defined by the formula

(/$ c~) (xo, ... , xp) = c~(/ xo, ... , I xp), p;;;;. 0. (3)

If P<O, then both of the groups q(X), q(Y) are trivial, and 1$ is the trivial zero-homomorphism, in accordance with 1.3.2. It is immediate that

It(c~ + d~) = It c~ + 1$ d~

for any two elements c~, d~ of q(y), and thus It is a homomorphism. Lemma 1. The homomorphisms It constitute a cochain mapping

from MF(Y) to MF(X),

Prool. The assertion means (see 1.4.2) that

(4)

for c~Eq(Y). If P<O, then c~=O and (4) is obvious. If P~O, then on choosing an arbitrary array of P + 2 points xo, ... , xpH EX, direct

4*


calculation yields P+l

(b~ It ct) (xo, ... , xP+l) = L (- 1)i (It ct) (xo, ... , xi' ... , XP+1) i=o

PH ~

= L (- 1)i ct(f xo, ... , I xi' ... , I XP+l) i=o

= (bt ct) (I xo, . .. J I Xp+l) = (It+! bt ct) (Xo, ... , XP+l) ,

and (4) is verified. The cochain mapping considered in lemma 1 is termed the cochain

mapping induced by the mapping I. Let there be given three sets X =1= 0, Y =1= 0, Z =1= 0, and mappings

I:X-+Y, g:Y-+Z.

Then we have also the product mapping (see 1.1.1)

g/:X-+Z.

The mappings (5), (6), (7) induce cochain mappings

It:q(Y) -+q(X), gt:q(Z) -+q(Y),

(g I)t: q (Z) --7 q (X).

The mappings (5) to (10) give rise to the following diagram.

(5) (6)

(7)

(8) (9)

(10)

Lemma 2 (the vector law for induced cochain mappings). Under the conditions just stated, we have the relation

(g I)~ = Itgt·

Prool. Explicitly, (11) means that

(gf)tc~=ltgtc~ for c~Eq(Z).

(11)

( 12)

If P < 0, then c~ = ° and (12).is obvious. If P?::. 0, then on choosing an arbitrary array of p + 1 points Xo, ... , xp EX, direct calculation yields:

[(g I)tc~] (xo, ... , xp) = c~(g I xo, ... , g I xp)

= (gt c~) (I xo, ... , I xp) = (It gt c~) (xo, ... , xp) , and (12) is proved.


Remark 1. Given a set X =F 0, let i:X --+X denote the identity mapping in X (that is, ix= x for every point xEX). Then the corresponding induced homomorphism i$: q (X) --+ q (X) is the identity homomorphism in q (X). Explicitly:

i$cP = cP for cPE ct(X). (13)

If P< 0, then cP = ° and (13) is obvious. If p;;:;;; 0, then

(itcP) (xo,"" xp) = cP(i xo, ... , i xp) = cP(xo, ... , xp)

for arbitrary points xo, ... , xpEX, and (13) follows.

Remark 2. Suppose that X =F 0 is a subset of Y. Then the identity mapping i in X is also a mapping from X into Y, and hence we have induced homomorphisms

it: q (Y) --+ q (X) . (14)

Observe that the symbol it has altogether different meanings in (13) and (14). As we noted earlier, completely descriptive notations in this theory would lead to prohibitive complications, and accordingly we shall have to use (as in the present case) identical symbols with different meanings in different situations. Returning to (14), the following comments are instructive. Take P;;:;;; 0, and consider a p-function ct for Y. On choosing an array of p + 1 points xo, ... , xpEX, the inclusion X (Y implies that ct(xo, ... , xp) is a well-defined integer. In other words, on restricting the arguments of ct to the subset X of Y, we obtain a p-function for X which we denote by ct IX (representing the function ct cut down to the subset X). We have then the relation

( 15)

Indeed, for arbitrary choice of the points Xo,"" xpEX, we obtain

(i$ct) (xo, ... , xp) = ct(i xo, ... , i xp)

= ct(xo, ... , xp) = (ctIX) (xo, .... ' xp).

Remark 3. Consider the following diagram. f fP

c~ (X) .... , _....;F ___ c;!YJ x------.... ,y

q v

z-----~· w C~(Z)-· ----c:(W) u u~

Fig. to.

In this diagram, X, Y, Z, W denote non-empty sets, f, g, u, v denote mappings as indicated by the arrows, and f$, g$, ut, v$ designate


the corresponding induced homomorphisms. Then v I and ug are mappings from X into W which induce homomorphisms (v f)t and (ug)t respectively from q(W) into q(X). Suppose now that

vi = ug. (16) Then we assert that

Itvt=gtut· Indeed, in view of (16) we conclude from lemma 2 that

It vt = (vl)t = (ug)t = gt ut, and (17) is verified.

(17)

Remark 4. Given the mapping I:X ~Y as in (1), choose a point vEX. Then we have (see 1.5.2) the cone homomorphism

I;P: q (X) ~q-l(X).

Similarly, the image point IvEY gives rise to the cone homomorphism

.z;~:q(Y) ~q-l(Y).

There arises the following diagram (in which the homomorphisms It and It-1 are induced by the mapping I).

C~(X) .. f;

C~(Y)

r P y r:.

C~-'(X) • C~-'(y) f P -'

F Fig. I!.

We shall now verify the formula

~P IP = IP-l I',P v F F lv' (18)

which states that the vector law holds III the preceding diagram. Explicitly, (18) means that

I'/ltc~=lt-l.z;~C~ for c~Eq(Y). (19)

If p:;;;.O, then (19) reduces to 0=0 (since q-l(X) =0 for p-;;:;'O). If p~ 1, then on choosing an arbitrary array of p points Xo,"" xp_1EX, we obtain [(see 1.5.1 (8)J:

(I'! It c~) (xo, ... , xp_ 1) = (- 1)P (It c~) (xo, ... , xp_ 1 , v)

= (- 1)P c~(f X O, •• , I xp- 1 , tv) = (.z;~ c~) (f xo, ... , I xp--1 )

= (ft-1 .z;~ c~) (xo, ... , xp- 1),

and (19) follows.

§ 1. 5. Formal complexes. 55

1.5.4. The homotopy operator DP. Let there be given two sets X =f= 0. Y =f= 0. and two mappings

(1 )

from X into Y. We have then the corresponding induced homo-morphisms

(2)

In terms of I and g the homotopy operator DP is defined as a homomorphism

in the following manner. If p~O. then q-l(X) =0. and DP is the trivial zero-homomorphism. in accordance with 1.3.2. If P 21. then the image DPety of an element etyE q(Y) is the (P -i)-function for X defined by the formula

P-l (DP ety) (xo •...• xp_ l) = L (-1)1 ety(fxo • ...• I xi' g xi' ...• gxP_l)' (4)

;=0

For example. for p = 1. 2. 3 the detailed expressions for DP are as follows:

(Dl e~) (xo) = e~(f xo. g xo).

(D2 e~) (xo• Xl) = e~(f xo. g xo• g Xl) - e~(f xo• I Xl. g xJ. (D3 e~) (Xo• Xl. X 2) = et(f xo• g Xo• g Xl' g X 2) -

- et (f xo• I Xl' g Xl. g X2) + et (f xo• I Xl. I X2 • g X2) •

Clearly. if ety. dty are any two elements of q(y). then

DP (ety + dty) = DP ety + DP dty.

and thus DP is a homomorphism. Observe that the defining formula (4) is not symmetric in I and g.

The homotopy operator DP is related to the cone homomorphism [see 1.5.1 (8). 1.5.2J in the following manner. Take an arbitrary vertex vEX and an integer p 21. Then we have the cone homomorphism

r!:q(X) -+ q-l (X).

The images Iv. gv of v give rise, in Y. to the cone homomorphisms

r:"+I;q+I(Y) -+q(y). IJ~;q(Y) -+q-l(y).

We have then the following two identities (the straightforward verifications are left to the reader).

F.P DP+I eP+I = - DP TNl eNl - fP- l r,P TNI eP+I P'> 1 (5) v Y gv Y F Iv gv y. <=.


where ct+!E C~+l (Y). In terms of the auxiliary expression

LIP ct = !5~-1 DP ct + DP+! !5t ct - g~ ct + It ct, ct E q (Y) , (6)

we have the identity

r!+l LlP+l ct+1 = LIP r/v+l ct+! for ct+1 E q+l (Y). (7)

The fundamental property of the homotopy operator DP is expressed by the homotopy identity (see 1.4.5)

!5~-1 DP ct + DP+l !5t ct = gt ct - It ct, ct E q (Y). (8)

The proof to be presented here is based on induction with respect to p. If P < 0, then ct = 0 and (8) is trivial. If p = 0, then (8) reduces to

Dl !5'{, c'{, = g~ c'{, - I~ c'{, .

On choosing an arbitrary point xoEX, direct calculation yields

(Dl!5'{, c'{,) (xo) = (!5'{, c'{,) (f xo, g xo)

= c'{, (g xo) - c'{, (f xo) = (g~ c'{, - I~ c'{,) (Xo) ,

(9)

and (9) follows. A similar explicit calculation, left to the reader, shows that (8) holds for p = 1. Proceeding by induction, we assume that (8) is known to hold for a certain integer p~ 1. In terms of LIP, defined in (6), the induction assumption appears in the form

LlPct=o for ctEq(Y),

and we have to prove, on the basis of (10), that

LIP+! ct+ 1 = 0 for ct+1 E q+l (Y).

(10)

(11)

Take any element ct+!E C~+1 (Y) and choose a point vEX. On setting

cP = rp+1 CHI Y gv Y'

we obtain from (10) and (7) the relation

FP+l LlP+l CHI = LIP THI CHI = LIP cP = 0 v Y gv Y Y' ( 12)

Since the vertex vEX was arbitrary, (12) implies (11) by 1.5.1, lemma 2, and the inductive proof of (8) is complete.

1.5.5. The formal complex MF (X, A). Given a set X =1=0 and a subset A of X, take an integer p;;;:: o. Then an element cP of the group C~ (X) is said to vanish on A if cP (xo, ... , xp) = 0 whenever xo, ... , xp EA. Clearly, those elements cPE C~(X) that vanish on A form a subgroup of q(X) which will be denoted by q(X, A). For p <0, we have


q(X)=o, and in this case we set q(X,A)=q(X)=O. We assert

that c5iq(X, A) (q+1(X, A). (1)

If P < 0, then q (X, A) = ° and (1) is obvious. So assume that P?; 0, and let cP be any element of C~(X, A). On choosing an arbitrary array of points ao, ... , ap+1 E A, we obtain

P+l (c5i cP) (ao, ... , ap +1) = L (- 1)i cP (ao, ... , aj , ... , ap +1) = 0,

i~O

since ao,"" aj' ... , ap+ 1 E A and cP vanishes on A. Thus c5i cP E C~+1 (X, A), and (1) follows. Accordingly (see 1.4.6), the groups C~(X, A) and the homomorphisms c5i constitute a subcomplex

MF(X, A) = (q(X, A), c5i) (2)

of the complex MF(X). The groups of p-cocycles and p-coboundaries of the complex (2) will be denoted by Z~ (X, A) , B~ (X, A) and will be termed the groups of p-cocycles and p-coboundaries respectively of the pair (X, A). Similarly, the cohomology groups of the complex (2) will be denoted by H~ (X, A) and will be termed the cohomology groups of the pair (X, A). The MAYER complex MF(X, A) will be referred to as the formal complex of the pair (X, A). Let us recall that the concept of a pair (X, A) implies that A (X (see 1.1.1).

Remark 1. Actually, we should write c5i I q (X, A) in (2), in conformity with 1.1.1, since c5i is now operating from the subgroup C~ (X, A) of q (X). However, since this fact is clear from the context, we continue to use the symbol c5i for the sake of simplicity. Let us note that clearly Z~(X, A) and Bf;(X, A) are subgroups of Z~(X) and B~ (X) respectively. For p = 0, it follows that Z~ (X, A) = ° if A =\= 0. Indeed, if zOEZ~(X, A), then ZO is constant on X by 1.5.2, remark 1, and since ZO vanishes on A =\= 0, it follows that ZO vanishes identically on X.

Remark 2. If A = 0, then every element cP of C~(X) vacuously satisfies the condition of vanishing on A. Accordingly, we have

q(X, 0) = q(X), Z~(X, 0) =Z~(X), Bf;(X, 0) = B~(X),

Ht(X, 0) =Ht(X), MF(X, 0) =MF(X),

Lemma 1. If (X, A) is a pair of sets such that A =\= 0, then

Ht(X, A) = ° for every p.

Proof. In view of 1.3.1, exercise 3, (3) is equivalent to the relation Z~(X, A) =B~(X, A), which in tum is equivalent to the relation

Z~(X, A) (B~(X, A), (4)


since Bt(X, A) (zt(X, A). If P=O, then Z~(X, A) =0 since A =1= 0 (see remark 1) and (4) is obvious. IfP<O, thenZt(X,A)=Oand (4) is again trivial. Consider finally the case P:?; 1. Observe that zPE zt(X, A) implies that zPEZt(X), and hence 1.5.2 (4) yields

(5)

Since A =1= 0, we can select the vertex v as a point of A. We assert then that

Tj'zPE q-l(X, A). (6)

Indeed, on choosing an arbitrary array of points ao, ... , ap_1EA, we obtain

(Tj' zP) (ao, ... , ap_1) = (- 1)P zP (ao , ... , ap_1, v) = 0,

since ao, ... , ap- 1 ' vE A and zP vanishes on A. Thus Tj'zP vanishes on A, and (6) is verified. From (5) and (6) it follows that zPEBt(X, A), and (4) is proved.

Remark 3. Observe that (3) fails to hold if p = ° and A = 0, since then [see remark 2 and 1.5.2 (8)J H~(X, O) =H~(X) ~ I.

Let there be given two sets X =1= 0, Y =1= 0, and a mapping

I:X-+-Y.

We have then the induced homomorphisms (see 1.5.3)

It:q(Y) -+-q(X) ,

which satisfy the relation

(7)

(8)

(9)

Let A and B be subsets of X and Y respectively, such that (see 1.1.1)

I: (X, A) -+-(Y, B). We assert then that

It q(Y, B) (q(X, A).

(10)

(11)

If P<O, then q(Y, B)=O and (11) is obvious. So assume that p;;;:,O, and consider any element cPE ct(Y, B). On choosing an arbitrary array of points ao, ... , apE A, we obtain

(It cP) (ao, ... , ap) = cP (I ao, ... , lap) = ° , since lao, ... , lapEB by (10) and cP vanishes on B. Thus ItcP vanishes on A. Hence ItcPE q(X, A), and (11) follows. Setting (see 1.1.1)

f~F = It I q (Y, B), (12)

we obtain in view of (11) homomorphisms

I~F:q(Y, B) -+-q(X, A), (13 )


which by (9) satisfy the relations

J:P .tP fP+I J:P uX/*F = *F Uy. (14)

Thus the homomorphisms f~F constitute a co chain mapping (see 1.4.2) from MF(Y, E) to MF(X, A) which is said to be induced by the mapping (10).

If (X, A), (Y, E), (Z, C) are three pairs of sets, and if there are given mappings f:(X, A)~(Y, E), g:(Y, E)~(Z, C), then lemma 2 in 1.5.3 yields [see (12)J the vector law

(gf)~F=f~Fg~F' (15)

Remark 4. If A = 0, E = 0 in the situation considered in (7) to (14), then clearly f~F reduces to ft. If A=0,E=l=0, then (11) reduces to ft q(Y, E) (q(X, 0), an obvious relation, since q(Y, E) ( q(Y) and hence

Similarly, the reader will readily see that the vector law (15) remains meaningful and valid in the special cases when A = 0, E = 0, C = 0, or A = 0, E= 0, C=l= 0, or A = 0, E=I= 0, C=l= 0.

Consider now two pairs of sets (X, A), (Y, E) and two mappings

f:(X, A) ~ (Y, E), g : (X, A) ~(Y, E).

The relations (16) imply that

f:X~Y, g:X~Y,

fA (E, gA (E.

( 16)

(17)

(18)

Corresponding to the mappings (17) we have (see 1.5.4) the homotopy operator

(19) We assert that

DPq(Y, E) (q-l(X, A). (20)

If P 5:.0, then DP is a zero-homomorphism, and (20) is obvious. So assume that p;;;:; 1, and consider any element c of ct(Y, E). On choosing an arbitrary array of points ao, ... , ap- I E A, we obtain

p-l (DPc) (ao, ... , ap-I) = L (-1)i c(/ao, ... , f ai , gai' ... , gap_I) = 0,

i=O

since fao,"" fai, gai' ... , gap- I E E by (18) and c vanishes on E. Thus DP c vanishes on A. Hence DPcECt-1(X, A), and (20) follows. On setting (see 1.1.1)

D~ = DP I q(Y, E) , (21)


we obtain in view of (20) homomorphisms

D~: q(y, B) -7 q-l (X, A), (22)

for which the homotopy identity in 1.5.4 yields directly [see (12)] the identity

!5~-1 D~ c + D~+1!5? c = g~F C - tt:wF C, c E q(Y, B) . (23)

Remark 5. The reader will readily see that (20) to (23) remain trivially meaningful and valid in the special cases when A = 0, B = 0 orA=0,B=I=0.

1.5.6. Excision. Given a pair (X, A) and a set U(A, the pair (X - U, A - U) is thought of as being derived from the pair (X, A) by the process of excising the set U. If i is the identity mapping in X - U (that is, ix=x for xEX - U), then clearly

i: (X - U, A - U) -7 (X, A)

in the sense of 1.1.1. This mapping is termed an excision mapping. It induces (see 1.5.5) homomorphisms

(1)

which constitute a cochain mapping. To avoid trivialities, we assume that

A =1=0, U=I=0, X-A=I=0. (2)

The case when simultaneously

A - U = 0 and p = 1 (3)

will be termed the exceptional case. The following diagram will be helpful to the reader in connection with the lemmas to be discussed presently.

~ ~-z po. Q A bx

C;-z (X.A)-----~ Cr' (X.A) ------~ C; (X,A)

• p-' I#F

C~-2 (X-U.A-U) ~N C;-'(X-U.A-U) x-u

p-' ' C; (X·U. A-U) ~x·u

Fig. 12.

Lemma 1. Assuming (2), we have

i~Fq(X, A) = q(X - U, A - U). (4)

Proof. We have to show that the homomorphism (1) is onto. If p < 0, then both of the groups occurring in (1) are trivial, and (4) is

§ 1.5. Formal complexes.

obvious. So assume that p:;;;;; 0, and assign any element

cP E q (X - U, A - U).

Define a p-fundion dP on X as follows:

dP ( ) _ { cP (xo, ... , xp) if xo, ... , xp - ° th . o eIWlse.

We assert that

61

(5)

(6)

(7)

To verify (7), we have to show that dP vanishes on A. Consider any array of points ao, ... , apE A. If at least one of these points lies in U, then dP (ao, ... , ap) = 0 by (6). On the other hand, if ao, ... , apE A - U, then (since A- U(X-U)

(8) by (6) and

cP(ao,···,ap)=o, (9)

since cP vanishes on A - U by (5). From (8) and (9) it follows that dP (ao, ... , ap) = ° in this case also, and the lemma is proved since clearly i~F dP = cPo

Lemma 2. Assuming (2) and excluding the case when simultaneously A - U = 0 and p = 0, we have

i~FZ~(X, A) =Z~(X - U, A - U). (10)

Proal. Since the homomorphisms (1) constitute a cochain mapping, by 1.4.2 we have the inclusion

i~FZ~(X, A) (Z~(X - U, A - U).

Hence (10) will be proved if we can show that

i~FZ~(X, A) )Z~(X - U, A - U). (11)

Since the groups involved in (11) are trivial if P<O, we can assume that p;;:;; 0. Taking first p;;:;; 1, we assert that

Z~(X-U,A-U)=B~(X-U,A-U) for p~1. (12)

Indeed, if A - U =l= 0, then (12) follows from 1.5.5, lemma 1. On the other hand, if A - U = 0, then by 1.5.5, remark 2 and 1.5.2, lemma 2,

Z~ (X - U, A - U) = Z~ (X - U) = B~ (X - U) = Bf, (X - U, A - U),

since p ~ 1. Thus (12) is verified. Assign now any p-cocycle

zPEZ~(X - U, A - U). (13 )


In view of (12), there exists a (p -1)-cochain

cP-1E q-l(X - U, A - U), such that

By lemma 1 there exists a (p -1)-cochain

ct-1 E q (X, A), such that

Let us put

Then zfEBt(X, A) (zt(X, A) by (16), and [see (18), (17), (15)]

(14)

( 15)

(16)

( 17)

(18)

As zP was an arbitrary element of zt(X - u, A - U), the inclusion (11) is verified for the case p ~ 1. Finally, if p = 0, then A - U =f= 0 by assumption, and hence Z~(X - U, A - U) =0 (see 1.5.5, remark 1). Thus (11) is trivial in this case, and the proof is complete.

Lemma 3. Let there be given a p-cocycle

(19) and a (p -1)-cochain

cP-1E q-l(X - U, A - U), (20) such that

(21)

Assuming (2) and excluding the exceptional case 0), there exists then a (p - 1 )-cochain

dP- 1 E q-1 (X, A), such that

6~-1 dP- 1 = zP, it;,;J dP- 1 = cp- 1 .

(22)

(23)

Proof. Note first that Z$(X, A) = B$(X, A) by 1.5.5, lemma 1, since A =f= 0. Thus (19) implies the existence of a (p -1 )-cochain

ct-1 E q-l (X, A), (24) such that

6~-1 ct-1 = zp. (25)

On setting C~-l = i~-t cP- 1 , we have

cP- 1 = i P- 1 cP- 1 E 0-1 (X - U A - U) 2 #F 1 F ' , (26)

§ 1.6. General cohomology theory. 63

and furthermore [see (25)]

5!.p-l cP-l - 5!.P-l ;p-l cP-l - iP 5!.p-l cP-l - iP zP UX-U 2 - UX-U"""F 1 - ""FUx 1 - ""F • (27)

From (21) and (27) it follows that

lJ~:=_lU (cP- l - C~-l) = 0, and hence

cP- l _ c~-lEZ~-l(X - U, A - U). (28)

Since the exceptional case (3) has been excluded, we cannot have simultaneously p -1 = 0 and A - U = 0. Accordingly, we can apply lemma 2 (with P replaced by p -1), and we conclude thus from (28) the existence of a (p - 1 )-cocycle

zf-1 E Z~-l (X, A), such that

it"""J zf-l = cP- 1 - C~-l . (29)

Consider now the (p - 1 )-cochain

dP- 1 = Cf-l + zf-1 E q-l (X, A) . (30)

In view of (30), (26), (25), (29) we obtain

b~-l dP- l = 6~-l Cf-l = zP, i~ff dP- l = C~-l + (cP- l -- C~-l) = cP- l ,

and (23) is proved.

§ 1.6. General cohomology theoryl.

1.6.1. The cohomology groups HP(X, A). Given a pair (X, A) of topological spaces, where A is a subspace of X, we set up a corresponding MAYER complex M (X, A) as follows. For each integer p, we first define a subgroup cP(X, A) of the formal cochain group q(X, A) by the following agreements.

1 Cohomology theory is in a certain sense dual to homology theory. For a comprehensive study of the algebraic apparatus and geometrical motivation of singular homology theory see the ALEXANDROFF-HoPF treatise listed in the Bibliography. There exist several versions of cohomology theory (see the treatises by LEFSCHETZ and ElLENBERG-STEENROD. listed in the Bibliography. for historical comments and a comprehensive discussion). The idea of defining cohomology groups in terms of p-functions is credited to ALEXANDER and KOLMOGOROFF. For a detailed discussion of a version of this approach (credited to A. D. WALLACE). see E. H. SPANIER. Cohomology theory for general spaces [Ann. of Math. 49. 407-427 (1948)]. For the version used here. see T. RADO. On general cohomology theory [Proc. Amer. Math. Soc. 4. 244-246 (1953)].


(i) For p < O. the group q (X. A) is trivial. and hence we have no choice but to set CP (X. A) = o.

(ii) For p ;;;;: O. the group cP (X. A) is defined as consisting of those elements cP of C~(X. A) that vanish locally on X. Explicitly. this conditions means that for every point xEX there exists an open set U = U (x. cPl. which depends upon both x and cPo such that cP(xo • ...• xp) =0 whenever xo • ...• xpE U. Clearly. these elements cP form a subgroup of q(X. A). Explicitly. a p-function cPo p;;;;.O. belongs to cP (X. A) if and only if (oc) cP vanishes locally on X and ({3) cP vanishes on A in the sense that cP(xo • ...• xp)=O whenever xo • ... , xp lie in A.

We observe that cPE cP(X, A) implies that o~cPE cP+1(X, A). For p < 0, the assertion is trivial, since then cP = O. For p;;;:: 0, we have surely o~cPE q+1 (X, A), and there remains to show that o~cP vanishes locally on X. Now since cPE cP (X, A), there exists an open covering U of X such that cP vanishes on each set UE U. For xo, ... , xpH E U E U, in the formula

P+1 (o~ cP) (xo, ... , Xp+1) = L (- 1)i cP (xo, ... , xi' ... , xpH)

i=O

every term of the summation vanishes, since xo, ... , Xi' ... , xp+1E U. Thus o~cP vanishes on each set UEU. Hence the groups CP(X, A), jointly with the homomorphisms~, constitute a MAYER complex

M(X, A) = {cP(X, A), c5~},

which is a subcomplex of the formal complex MF(X, A). Applying to this situation the general concepts relating to MAYER complexes, we have the groups ZP (X, A), BP (X, A) of the p-cocycles and p-coboundaries of the complex M (X, A). The cohomology groups

HP(X, A) = ZP(X, A)/BP(X, A)

will be referred to as the cohomology groups of the pair (X, A) 1.

While the formal complex MF(X, A) did not involve topological concepts, the complex M (X, A) depends upon the topology of X and A through the use of locally vanishing p-cochains cP [see condition (ii)

1 For the case when A is closed in X (which is the only one relevant for applications in this volume) the cohomology group HP (X, A) so defined is isomorphic to the cohomology group of dimension p -1 occurring in alternative versions. This shift in dimension could be avoided, of course, by a mere change of notation. However, since we apply the general theory primarily to subsets of Euclidean spaces, there seems to be a certain advantage in operating with the dimensions directly obtained from the definition. A further aspect of the version adopted here is that it yields directly the reduced theory. Thus it is unnecessary to introduce separately the so-called reduced cohomology groups (see the ElLENBERG-STEENROD treatise listed in the Bibliography).


above]. In point of notation, observe that we use the subscript F whenever the formal complex is considered.

The definition of the complex M(X, A) remains meaningful if A = 0. In this case, we write M (X), CP (X), ... , HP (X) instead of M(X, 0), cP(X, 0), ... , HP(X, 0). Then cP(X) for example, for p ~ 0, consists of those elements cP of the formal cochain group ct (X) that vanish locally on X.

Clearly, CP(X, A) is a subgroup of cP(X), and hence M(X, A) is a subcomplex of M (X), since the same homomorphisms !5~ are used throughout.

To illustrate and to clarify the meaning of these general concepts, a series of remarks will be discussed presently.

Remark 1. For p < 0, cP (X, A) = 0 by definition, and thus clearly HP (X, A) = 0 for p < O. In particular HP (X) = 0 for p < O.

Remark 2. In fact, CO (X, A) =0 always. Indeed, consider any element cOE Co (X, A). For any point yEX, we must have an open set U containing y such that CO(x) =0 whenever xE U. Since yE U, it follows that CO (y) = O. As y was an arbitrary point of X, it follows that CO = O. Thus CO (X, A) consists of a zero-element alone, and hence CO (X, A) =0. Accordingly, HO(X, A) =0 always. In particular, HO(X) =0 always.

In view of the preceding two remarks, one sees that HP(X, A) =0, HP(X) =0 for p:s::O.

Remark 3. HP(X, X) =0 for every p. For p;;;;;,O, this follows from the preceding remarks. For p;;;;: 1, an element cP of cP(X, X) must vanish, by definition, on X, and hence cP = O. Thus cP (X, X) consists of a zero-element alone, and the assertion HP (X, X) = 0 follows.

Remark 4. If y is any point of X, then HP (X, y) = HP (X), for every p. Indeed, one has M(X, y) =M(X). To see this, note that for p:s:: 0 one has CP (X) = 0 = CP (X, y) by Remarks 1 and 2. For p ~ 1, consider any p-cochain cPEcP(X). Since cP vanishes locally on X, we have an open set U such that yE U and cP(xo,"" xp) =0 whenever xo, ... , xpE U. In particular, it follows that cP(y, ... , y)=O. Thus cP vanishes on y (considered as a subspace of X). This means that cPE CP (X, y), and hence CP (X) (cP (X, y). Since clearly cP (X, y) ( cP (X), it follows that cP (X) = cP (X, y). As the same homomorphisms o~ are used in setting up the complexes M (X) and M (X, y), it follows that M(X, y) =M(X), and hence HP(X, y) =HP(X).

Remark 5. If X consists of a single point x, then HP (X) = 0 for every p. Indeed, by the Remarks 3 and 4 we have then

HP(X) =HP(X, x) = HP(X, X) = o. Rado and Reichelderfer, Continuous Transformations.


Remark 6. If cPEcP(X,A), and p;;;;;O, then cP(x, ... ,x)=O for every point xEX. This follows directly from the fact that cP vanishes locally on X.

Remark 7. Always JI1(X, A) R;P(X, A). Indeed, by Remark 2 we have CO (X, A) =0, and hence BI(X,A)=b~CO(X,A)=O. Thus JI1 (X, A) =Zl (X, A)/O R;ZI (X, A) (see 1.3.1, exercise 2). In particular, always JI1(X) R;P(X).

Remark 8. Let X be a HAUSDORFF space consisting of just two distinct points a and b. Then JI1(X) R; I (the additive group of integers).

Proof. In view of Remark 7, we have to show that P(X) R; I in this case. To accomplish this, we introduce a homomorphism h:P (X)-;.I by setting hz=z(a, b), where z is a generic notation for an element of P(X). We first show that h is onto. To see this, let there be assigned any integer n. Consider the (formal) zero-cochain c~E C~(X) defined by c~(a) =0, c~(b) =n. Then we assert that

(1)

Indeed, since bb=O, surely zEZ}(X). To verify (1), we have to show yet that z vanishes locally on X. In the present case, this merely means that z(a, a) =0, z(b, b) =0, and these relations are immediate. Indeed,

z(a, a) = (b~ c~) (a, a) = c~(a) - c~(a) = 0,

z(b, b) = (b~ c~) (b, b) = c~(b) - c~(b) = 0.

Once (1) is established, we conclude that

hz = z(a, b) = (b~ c~) (a, b) = c~(b) - c~(a) = n - ° = n.

Thus h is onto. To show that h is an isomorphism into, we have to verify that if zEP(X) and z(a, b) =0, then z(xo, Xl) =0 for every choice of xo, xlEX. The only choices are now (a, a), (b, b), (a, b), (b, a). Since z vanishes locally on X, we have z(a, a) =0, z(b, b) =0, while z (a, b) = ° by assumption. Finally, since z is a 1-cocycle, we have

0= (biz) (a, b, a) = z(b, a) -- z(a, a) + z(a, b) = z(b, a) - ° + 0.

Thus z(b, a) =0 also. Summing up: h is an isomorphism onto, and consequently HI (X) R; I.

Remark 9. A central objective of cohomology theory is to establish relations between geometrical properties of spaces on the one hand and algebraic properties of their cohomology groups on the other hand. In the way of illustration, we prove the following theorem: a topological space X is connected if and only if HI (X) = 0.

§ I.6. General cohomology theory. 67

It is convenient to divide the proof of this theorem into three parts.

(i) Let ZEZl(X). Suppose A and B are subsets of X having a point x in common such that z vanishes on each of the sets A, B. For points aE A, bEB, one finds z(a, b) = (bl-z) (a, b, x) = O. Thus z clearly vanishes on AUB.

(ii) Again consider zEzt(X). For a fixed point xEX let U be the class of all open sets containing x on which z vanishes. This class is not empty, since z vanishes locally on X. Denote by U the union of those sets in U. Clearly z vanishes on U by (i), and thus U is the largest open set containing x on which z vanishes. Let y be any point in U. Since z vanishes locally on X there is an open set V containing y on which z vanishes. Because V meets U (see 1.1.3, exercise 2) it follows by (i) that z vanishes on the open set V U U. But U is the largest open set containing x on which z vanishes. Thus yE V (U, and hence U is closed. Summarizing, with each element zEzt(X) and each point xEX there is associated an open and closed set U containing x on which z vanishes.

(iii) The theorem may now be proved as follows. In view of remark 7 it suffices to show that X is connected if and only if Zl (X) = 0. First, suppose that X is connected. Then the only non-empty set in X which is both open and closed is the space X itself (see 1.1.3, exercise 13). From (ii) it is thus evident that every zEzt (X) must vanish on X. That is, Zl(X) =0. Next, assume that X is not connected. Then X = A UB, where A and B are non-empty disjoint open (and closed) sets (see 1.1.3, exercise 12). Define

c (x) = {O ~f x E A, 1 If xEB,

z(x, y) = (b~c) (x, y) = c(y) - c(x) for x, YEX.

Obviously z vanishes on each of the sets A, B, and thus it vanishes locally on X. On the other hand, for aEA, bEB, z(a, b) = 1. Since bl-z = 0, one finds in z a non-trivial element of Zl (X). This establishes the theorem.

1.6.2. Induced cochain mappings. Let (X, A), (Y, B) be two pairs of topological spaces, where A is a subspace of X and B is a subspace of Y, and let (see 1.1.5, definition 2) t:(X, A)--+(Y, B) be a continuous mapping. By 1.5.5, t induces then a cochain mapping

Clearly (1 )

5*


In view of 1.4.6, lemma 3, we infer from (1) that the homomorphisms

f~ = f~FI cP(Y, E) (2)

constitute a cochain mapping

f~:CP(y, E) ~ CP(X, A).

By 1.4.2, we obtain corresponding induced homomorphisms

f~*: HP(Y, E) ~HP(X, A)

for the cohomology groups of the complexes M(Y, E), M(X, A). If zPEZP(Y, E), then the homomorphism f~* is given explicitly by the

formula P [P] [fP P] f # * Z YB = # Z XA,

where the square brackets, with subscripts Y E, X A denote the cohomology classes [relative to the complexes M(Y, E) and M(X, A) respectively], that contain the indicated p-cocycles.

If (X, A), (Y, E), (Z, C) are three pairs of topological spaces, where A, E, Care subspaces of X, Y, Z respectively, and if f: (X, A) ~(Y, E),

HP (y, B) g.: (Y, E) ~ (~, C) are con-

~ tmuous mappmgs, then gf:

;/

(X, A)~(Z, C) is also a con-f P qP tinuous mapping. Accordingly, #" ~* we have the induced homo-

morphisms indicated by the

HP(X..A)--.------- HP{Z.C) diagram in Fig. 13· (Clf) P We assert that the vector law

"J ~* Fig. 13.

holds in this diagram. In view of 1.4.2, it is sufficient to verify that in the diagram

cP(Y,BI

/~ CF(X,A) C"(Z.,C)

(qf): Fig. 14.

the vector law (g f)~ = f~ g~ (3)

holds. In view of (2), the relation (3) is an immediate consequence of the vector law established in 1.5.5.

The preceding definitions and statements remain obviously meaningful and valid if one or more of the subspaces A, E, C reduces to the


empty set O. Whenever an empty subspace occurs, we agree to drop the subscript # to simplify notations. In particular, if A = 0, B = 0, C = 0, then we have the situation indicated by the following diagrams.

/y~ /C\ ;/~\ X • 'Z. ePOC) • P c"(Z) HP(X)' HP(ZI _ .1 W,

Fig. 15.

The vector law holds throughout. The reader should set up the diagrams corresponding to the cases A = 0, B* 0, C* ° and A = 0, B = 0, C*0.

Remark 1. An important special case arises in connection with identity maps. Let A be a subspace of the topological space X, and let i:X -+X be the identity map defined by i x = x for every xEX. Then i: (X, A) -+ (X, A) is a continuous mapping and hence we have the induced homomorphisms

i~:cP(X,A)-+cP(X,A), } (4) i~* :HP(X, A) -+ HP (X, A).

In view of (2), it follows from 1.5.3, remark 1, that i~cP=cP for every cPECP(X, A), and hence obviously i~*hP=hP for every hPEHP(X, A). Thus we have the theorem: the identity map i: (X, A)-+(X, A) induces the identity homomorphism in the cohomology groups HP(X, A).

Remark 2. As an immediate consequence, we obtain the topological invariance of the cohomology groups in the following precise form. Let (X, A), (Y, B) be two pairs of topological spaces. The pairs (X, A), (Y, B) are termed homeomorphic if there exists a homeomorphism I:X-+Y such that IA=B. Then f:(X,A)-+(Y, B). Given such a homeomorphism I, we assert that the induced homomorphism

I~* :HP(Y, B) -+HP(X, A)

is an isomorphism onto, for every p. To prove this, let

g: (Y, B) -+(X, A)

be the inverse of I (since I is a homeomorphism, g = t-1 is single-valued and continuous, and gB=A). Then gl=i is the identity map in X, and Ig=j is the identity map in Y. Using the vector law and remark 1, we infer that


The first relation shows that I~* is a homomorphism onto HP(X, A). The second relation shows that I~ * v = 0 implies that v = 0, and thus I~ * is also an isomorphism into.

As a matter of fact, it is obvious from the basic definitions that even the cochain groups CP (X, A) are topologically invariant in a similar sense.

Remark 3. Given a continuous mapping

I: (X, A) -+ (Y, E), (5) suppose that

IX(E. (6)

Then we have also the relation

I: (X, A) -+ (E, E) , (7)

as well as the inclusion mapping

i: (E, E) -+ (Y, E) . (8)

The mappings (5), (7), (8) yield the following diagram.

The homomorphisms 1'# *' I~ ,i~ are induced by the mappings (5), (7), (8) respectively. If v is any element of HP(Y, E), then by the vector law

1,#* v = I~ i~ v. (9)

Fig. 16. On the other hand, i~ v = 0 since HP (E, E) is trivial by 1.6.1, remark 3. From (9) it follows

therefore that I~ * v = o. Since vE HP (Y, E) was arbitrary, we obtain the following conclusion: if the continuous mapping I: (X, A) -+ (Y, E) satisfies the condition IX (E, then the corresponding induced homomorphism I~ * is a zero homomorphism [that is, 1'# * HP (Y, E) = 0].

1.6.3. Excision 1. Let A and U be subspaces of the topological space X, such that U (A. If i:X - U -+X - U is the identity mapping on X - U, then clearly

i: (X - U, A - U) -+ (X, A). (1 )

Since the pair (X - U, A - U) is obtained by excising the set U simultaneously from X and A, the mapping (1) is termed, in the

1 The discussion of excision would be considerably simpler if we were operating with the unreduced theory. However, from the point of view of actual applications, the advantage of the unreduced theory is perhaps more apparent than real, due to the necessity of introducing separately the reduced cohomology groups (see the ElLENBERG-STEENROD treatise listed in the Bibliography).


present context, an excision mapping (see 1.5.6). The corresponding homomorphism

i~*:HP(X, A) -+HP(X - U, A - U) (2)

is termed the excision homomorphism. A definitive study of the excision homomorphism will be made in 1.6.14. However, some special lemmas to be used in that connection will be established presently, to provide some immediate applications of the information contained in 1;6.1, 1.6.2. The condition U (A will be replaced, in these special lemmas, by the stronger requirement

U (intA. (3)

By 1.1.3, exercise 8, condition (3) is equivalent to

x = (int A) U (int C U). (4)

To avoid trivialities, we assume that

A=*0, U=*0, X-A=*0. (5)

The case when simultaneously

A - U = ° and p = 1 (6)

will playa special role, and will be termed the exceptional case (see 1.5.6).

Lemma 1. Under the assumptions (3) and (5), the excision homomorphism i~ * is onto.

Proal. In view of the basic definitions, the assertion is a direct consequence of the relation

(7)

which will be verified presently. For p 5:.0, the groups involved in (7) are trivial (see 1.6.1, remarks 1 and 2) and hence we can assume that p;;:;; 1. Assign (see 1.6.1) any p-cocycle

zEZP(X - U, A - U) (zt(X - u, A - U). (8)

By 1.5.6, lemma 2, we have then a p-cocycle

(9) such that

(10)

and clearly (7) is proved if we can show that Zl vanishes locally on X. So let x be a point of X. If xE int A, then on choosing V = int A, we have xEV and Zl(XO' ... , xp) = 0 if X o, ... , xpEV, since V (A and Zl


vanishes on A by (9). If x does not lie in int A, then

xE int C U by (4). By (10) we have

(11)

Zl(XO""'xp)=z(xo,""xp) for xo, ... ,xpECU. (12)

By (8), Z vanishes locally on C U, and hence there exists an open set 0 of X such that xEO and

(13 )

On choosing V = 0 n int C U, we obtain an open set of X which contains x [see (11)]. Furthermore, if xv' ... , xpEV, then by (12) and (13) we see that Zl (xo, ... , xp) = O. Thus Zl vanishes locally on X.

Lemma 2. Assuming (3) and (5), and excluding the exceptional case (6), the excision homomorphism i~ * is an isomorphism into.

Proof. Since the groups involved in (2) are trivial for P <';;;'0 (see 1.6.1, remarks 1 and 2) we can assume that p?:;,,1. In view of the basic definitions, the assertion to be proved is equivalent to the following: if we are given a co cycle

such that i~ Z E BP (X - U, A - U),

then we have also zEBP(X,A).

Now (15) means that there exists a cochain C such that

C E 0-1 (X - U, A - U), b~-=-lU C = i~ z.

(14)

(15 )

(16)

( 17)

By 1.5.6, lemma 3, there follows the existence of a cochain C1 such that

C1 E q-1 (X, A), (18)

(19)

(20)

Thus (16) will be established if we can show that c1EO-1(X, A), and in view of (18) there remains to show that C1 vanishes locally on X. Now c1 vanishes on A by (18), and c1(XO' ... , xp_1) =c(xo, ... , xp- 1) for Xv' ... , xp_1ECU by (19). Also, C vanishes locally on CU by (17). From these facts, jointly with (4), it follows that c1 vanishes locally on X, by the same argument as the one used in the proof of lemma 1.

Combining the preceding two lemmas, we obtain directly the following statement.


Lemma 3. If U ( int A and A - U =l= 0, then the eXCISIOn homomorphism i~ * is an isomorphism onto for every p.

The case when U (int A and simultaneously A - U = 0 gives rise to the following comments. In this case, we have A = U and hence

if = U (int A ( A .

It follows that if =A = int A. Since if is closed and int A is open, it follows that A is both open and closed. Hence, on setting B = CA, the set B is also both open and closed, and B=l= 0, A =l= 0 by (5). Thus we have now the following situation:

X=AUB, A=l=0, B=l=0, AnB=0, (21)

A open, B open. (22)

Accordingly (see 1.1.3, definition 11) the topological space X is disconnected and the sets A, B constitute a partition of X. The identity mapping (1) becomes now

i:B~(X, A), (23)

while the excision homomorphism reduces to

(24)

For this case, the lemmas 1 and 2 yield the following information.

Lemma 4. The excision homomorphism (24) is onto for every p, and for p =l= 1 it is an isomorphism onto.

Remark 1. Let us observe that for p = 1 the second statement in lemma 4 breaks down. Indeed, let X be a HAUSDORFF space consisting of two distinct points a and b. Then we can identify the sets A and B with the points a and b respectively, obtaining for p = 1 the homomorphism

By 1.6.1, remarks 4,8,5, we haveHl(X,a)=Hl(X)~I,Hl(b)=O, and hence in this case the excision homomorphism is certainly not an isomorphism onto.

Remark 2. Let us add that, in the situation considered in remark 1, we have HP (X) = 0 for p =l= 1. Indeed, using lemma 4, jointly with the remarks 4 and 5 in 1.6.1, we conclude that for p =l= 1 we have

Combining this result with remark 8 in 1.6.1, we obtain the following theorem: If X is a HAUSDORFF space consisting of precisely two points, then HP (X) = 0 for p =l= 1 and HI (X) ~ I, the additive group of integers.


1.6.4. Exactness. Let X be a topological space, and let A, B be closed subsets of X such that

X)A)B. (1 )

Then A and B, as subspaces of X, give rise to the three pairs (X, A), (X, B), (A, B). In view of (1) we have then the inclusion mappings

i: (X, B) ----?-(X, A),

i:(A, B) ----?-(X, B),

(2)

(3)

which in turn give rise to cochain mappings that we shall denote by

iP:CP(X, A) ----?-CP(X, B),

iP:CP(X, B) ----?-cP(A, B).

Thus, for each p, we have the diagram

(4)

(5)

(6)

We shall verify presently that the complexes M(X, A), M(X, B), M (A, B), jointly with the cochain mappings indicated in (6), constitute a standard triple, in the sense of 1.4.3. Since the groups occurring in (6) are trivial for p;;;:O by 1.6.1, remarks 1 and 2, we have to discuss (6) only for P ~ 1.

(a) The homomorphism iP in (6) is onto. Indeed, assign c E CP (A, B). Define a p-cochain d on X by the agreement

d(xo, ... , x) = {C(xo, ... , .xp) if xo, ... , xpE A,} (7) P 0 otherwIse.

Since c vanishes on B, clearly d also vanishes on B. Now observe that c vanishes locally on A. Hence, there exists an open covering UA of A such that c vanishes on each set UA E UA • Since A is a subspace of X, each set UA E UA can be written in the form UA = UxnA, where Ux is open in X. Noting that CA is open, we define an open covering Q3x of X as comprised of CA and of all the sets Ux just referred to. We assert that d vanishes on each set VxE Q3x' Indeed, if Vx=CA and xo, ... ,xpEVx , then d(xo,""xp)=O by (7). IfVx is one of the sets Ux, then VxnAEUA • If xo, ... , xpEVx , then either xo, ... , xpEA, and then xo, ... , xpEVxnAEUA , and hence d(xo, ... , xp) = c (xo, ... , xp) = 0, since c vanishes on each set of UA ; or else, the points xo, ... , xp do not all lie in A, and then d(xo, ... , xp)=O by (7). Thus d vanishes on B and vanishes locally on X, and hence dE CP (X, B).


Finally, for xo, ... , xpE A, we have by (7)

W d) (xo, ... , xp) = d (j xo, ... , j xp) = d (xo, ... , xp) = c (xo, ... , xp).

Thus jP d = c, proving that jP is onto.

(b) Since E(A, it is clear that cP(X, A) is a subgroup of cP(X, E). For c E CP(X, A) and xo,"" xpEX, we have Wc) (xo, ... , xp) = c(ixo, ... , ixp)=c(xo, ... , xp), and thus iPc=c. Hence iPc=O implies that c = 0, proving that iP is an isomorphism into.

(c) If cEcP(X, A) and xo, ... , xpEA, then

WiPc) (xo, ... , xp) =c(ijxo, ... ,ijxp) =c(xo, ... , xp) =0,

since c vanishes on A. Thus jPiPc=O, proving that iPCP(X, A) IS

contained in the nucleus of jP.

(d) Let c E CP(X, E) be such that jPc=O. For xo, ... , xpEA we have then

0= Wc) (xo, ... , xp) =c(jxo, ... ,jxp) =c(xo, ... , xp).

Thus c vanishes on A, and [since cE CP(X, E)] it vanishes locally on X. Thus cE CP(X, A). But then c=iPc, as we observed under (b), proving that the nucleus of jP is comprised in iPcP(X, A). In view of (c) it follows that iPcP(X, A) coincides with the nucleus of jP. Thus we have exactness in (6).

The facts established in (a) to (d) mean that the diagram (6) represents a standard triple. Accordingly, by the scheme described in 1.4.3, we obtain homomorphisms

O~-l: HP-l (A, E) -+ HP (X, A),

which jointly with the induced homomorphisms

i~ :HP (X, A) -+ HP (X, E), i~ :HP(X, E) -+HP(A, E),

yield the cohomology sequence

(8)

'P-l 6P-1 .p .p c5P

•.. ~ HP-l(A, E) -*-,. HP(X, A) ~ HP(X, E)~" HP(A, E) -* ,. ... , (9)

corresponding to the complexes M(X, A), M(X, E), M(A, E). The sequence (9) will be termed the cohomology sequence 0/ the triple (X, A, E). By 1.4.3 this cohomology sequence possesses the fundamental property of exactness.

Remark 1. In establishing this fact, we assumed that X)A)E and that A and E are closed subsets of X. The assumption that A is closed was used in showing that jP in (6) is onto. On the other hand, the assumption that E is closed was not used at all. The generalization implied by this observation is irrelevant for our purposes.


Remark 2. It is instructive to visualize the explicit description of the homomorphism b!-1 occurring in the cohomology sequence (9). For P:;;;;'1, the group HP-1(A, B) is trivial by 1.6.1, remark 2, and hence it is sufficient to consider the case p ~ 2. The generic element of HP-1(A, B) is of the form [ZJAB' where ZEZP-1(A, B), and [ZJAB indicates the cohomology class containing z, relative to the complex M (A, B). Define a (p -1)-cochain d on X by the agreement

d( ) _{Z(Xo"",XP_l) if xo,···,xp_1EA, } ( ) xo, ... , xP_ 1 -. 10 o otheIWlse.

By the argument employed under (a) above (in showing that jP is onto), one sees that dE 0-1 (X, B). Let us denote this cochain d by ~ z, where the symbol ~ is interpreted as the operator effecting the natural extension from A to X, described explicitly by (10). One verifies readily that on setting

one has the relation zlEZP(X, A). Inspection of the general scheme in 1.4.3 reveals that the homomorphism bt- 1 in (9) is now described explicitly by the formula

bt-1 [zJAB = [b~-ldzJXA'

where []x A means the cohomology class relative to the complex M(X, A).

Remark 3. An important special case arises if B = 0. The cohomology sequence (9) reduces then to

... it-" HP-1(A) ot-" HP(X,A) i~) HP(X).J.l.>-HP(A)~ ••• , (11)

and is referred to as the cohomology sequence of the pair (X, A). Thus the sequence (11) is exact. Let us recall that A is assumed to be a closed subset of the topological space X.

1.6.5. Examples. In a topological space S, let A and B be nonempty, closed sets such that A n B = 0. Then the following holds.

(a) If HP-1 (A) = 0 and HP (A U B) = 0 for a certain integer p, then HP (B) = 0 for the same integer p.

(b) If HP(A) =0 and HP(B) =0 for a certain integer P=t=1, then HP (A U B) = 0 for the same integer p.

Proof 0/ (aJ. On setting X =A UB, the subspace X is a closed subset of S, and hence A and B are closed relative to X also. Since A n B = 0, it follows that A and B are also open relative to X. By 1.6.3, lemma 4,


the excision homomorphism

(1 )

is therefore onto. In the cohomology sequence of the pair (A UE, A), the groups Hp-I (A), HP (A UE, A), HP (A UE) are consecutive terms. The first and the third of these groups are trivial by assumption. Hence, by 1.3.3, lemma 3, we have HP(A UE, A) =0. Since the homomorphism (1) is onto, it follows that HP (E) = o.

Proal 01 (b). Since p =\= 1, the homomorphism (1) is now an isomorphism into, by 1.6.3, lemma 4. Since HP (E) = ° by assumption, it follows that

HP(AUE,A)=O. (2)

In the cohomology sequence of the pair (A UE, A), the groups HP(A UE, A), HP(A UE), HP(A) are consecutive terms. The first group is trivial by (2), while the third group is trivial by assumption. By 1.3.3, lemma 3 it follows that HP(A UE) =0.

Remark. Let 5 be a HAUSDORFF space consisting of just two points a and b. We can then choose A =a, E= b, and forp= 1 we have HI (a) =0, HI (b) =0, HI(aUb) =\=0 by 1.6.1, remarks 5 and 8. Thus the exclusion of the case p = 1 is really necessary in statement (b) above.

1.6.6. The vector law for cohomology sequences. Let (X, A, E) and (X, A, B) be two triples of topological spaces, such that X)A)E, X) A) B, and A, E are closed subsets of X, and 1: jj are closed subsets of X. As in 1.6.4, A and E are thought of as subs paces of X, and similarly A and jj are thought of as subspaces of X. Now let I:X-'>-X be a continuous mapping such that

IA(.1, IECH. (1 )

We shall refer to these assumptions by writing

I: (X, A, E) -'>- (X, 1: B) .

Clearly, (1) implies the relations I: (X, A) -'>- (X, A), I: (X, E) -'>- (X, B), I IA : (A, E) -'>- (A, B). The corresponding co chain mappings will be denoted by

jP: CP (X, A) -'>- CP (X, A), If: CP (X, B) -'>- CP (X, E),

(fIA)P: 0(.1, B) -'>- CP(A, E).

In accordance with 1.6.4, we have the additional cochain mappings

iP: cP(X, A) -'>- CP(X, E), iP: CP(X, E) -'>- O(A, E).


The analogous co chain mappings relative to the triple (X, A, B) will be denoted by

[P; CP (X, A) ----'? CP(X, B), TP: cP (X, B) ----'? cP (A, B).

The corresponding induced homomorphisms for the cohomology groups will be denoted by f~, i~, ~:, .... Finally, J~-l will denote the homomorphism relative to the triple (X, A, B) that is associated with this triple in the same manner as the homomorphism <5~-1 was associated in 1.6.4 with the triple (X, A, B). The cohomology sequences of the triples (X, A, B) and (X, A, B) are then connected in the manner indicated by the following diagram.

-P-I 'P "':"P

"'- - HP-'cX,'P,1 E ..

.. HP(X}.) I" .. HP(X;a) J* , HP(A.B)~ ---

(f I A):-' fP " f;" (fIA):

-----HP-'(A,l) ~ HP(X.A) )' HP(X.B) ~ HP(A,B)-----SP-' • P .p

.,.. ' .. J* Fig. 17.

The vector law for cohomology sequences states that the vector law holds in each box of this diagram. To establish this statement, in view of 1.4.4, theorem 2 it is sufficient to verify that in the diagram

Tp ":"p CP(X,A) CP(X,i) J • CP(A,Bl

fP f'p I (fiAt

CP(X.A) ~ CP(X,!) • CP(A,B) i P jP

Fig. 18.

the vector law holds in each box. But this is a direct consequence of the vector law in 1.6.2, since i, j, 7, i are inclusion mappings and hence clearly ~t=fi, iUiA) =fj.

Important special cases of the general vector law for cohomology sequences are considered in the following remarks.


Remark 1. If B = 0 and 13 = 0, then we have the case of two pairs (X, A) and (X, A) related by a continuous mapping t: (X, A) -+ (X, A). The general diagram reduces now to the form

""p-' ... -0* o p o P

··-HP·'(A) . HP(i,A) 1..-• HP(X) J ..

~HP{A)-'"

I (fIA):" fP I,: (fIA): ,.

·~HP·'(A) p.' , HP(X,A) op

~ HPlX) o P • HP(A)~'

S* 1* J .. Fig. 19.

The vector law holds in each box.

Remark 2. If X is a subspace of X, then we can take A = A, B = B. Let us choose t as the inclusion mapping i: X -+ X (thus i x = x for xEX). Clearly i: (X, A, B) -+ (X, A, B). The general diagram contains now the partial diagram

'p 1*

g:.' ~HP(X.A) Fig. 20.

Now clearly ilA coincides with the identity mapping j:A-+A. Hence, by remark 1 in 1.6.2, one sees that (i I A )t-1 is merely the identity homomorphism in HP-l (A, B). It follows that the vector law holds in the following diagrams (where the one on the right concerns the case B= O).

S:;Y'" HP (X, A)

> HP-'(A,B)

> E:-'~ 'p 1*

HP(X.A) Fig. 21.

• P 1*


1. 6. 7. Extension and reduction I. In a fixed fully normal topological space 5, we shall consider pairs of closed subspaces (X, A). Let (Xl' AI), (X2' A 2) be two pairs such that (X2' A 2) ((Xl' AI) (see 1.1.1 for notations). If i:X2-+X2 is the identity map on X 2, then clearly i: (X2' A 2) -+(Xl' AI)' and thus we have the corresponding induced

homomorphisms 'P . HP (X A) HP (X A) z*. 1'1-+ 2' 2'

Definition 1. If an element h2 of HP(X2, A 2) is the image of some element hI of HP (Xl' AI) under i~, then h2 is termed extendable into HP(XI , AI)'

Definition 2. If an element hI of HP (Xl' AI) is carried into the zero element of HP (X2' A 2) by i~, then hI is said to reduce to zero in HP(X2, A 2)·

The following two remarks are direct consequences of the vector laws for induced homomorphisms.

Remark 1. If (X3,A3)((X2 , A 2)((XI , AI)' and an element hs of HP(Xs, A3) is extendable into HP (XI , AI)' then h3 is also extendable into HP(X2 ,A2).

Remark 2. If (X3' A3)((X2 , A 2)((Xl , AI)' and an element hI of HP (Xl' AI) reduces to zero in HP (X2' A 2), then it also reduces to zero III HP (X3' A3)'

Now let there be given in 5 a closed pair (Xo, Ao) and a family F of closed pairs (X, A) such that F==?(Xo' Ao), in the sense of 1.1.3, definition 25. Then the following statements hold.

Extension Theorem. Given an element ho of HP(Xo, Ao), there exists a pair (X, A) E F such that ho is extendable into HP (X, A).

Reduction Theorem. Given a pair (X, A) E F and an element h of HP (X, A) such that h reduces to zero in HP (Xo, Ao), then there exists a pair (Xl' AI) E F such that (Xl' AI) ( (X, A) and h reduces to zero in HP(XI , AI)'

For clarity, we break down the proof into several steps.

1 The extension theorem and the reduction theorem are due to A. D. WALLACE, The map excision theorem, Duke Math. Journal, vol. 19, 1952, pp. 177-182. We have stated here these theorems in a form convenient for our purposes. These two theorems summarize, in a sense, those aspects of the so-called continuity theorem (see the ElLENBERG-STEENROD treatise in the Bibliography) which are independent of the assumption of compactness. From the point of view of our needs in this volume, the chief virtue of the extension theorem and of the reduction theorem is that they represent an entirely adequate substitute for the conceptually more involved general continuity theorem. The detailed discussion in 1.6.7 to 1.6.16 is meant to provide opportunities for the reader to further familiarize himself with the use of diagrams. For this same reason, we adopted (in 1.6.15) a method of proof for the homotopy theorem (in terms of the reduction theorem) due to J. W. KEESEE, On the homotopy axiom [Ann. of Math. 54, 247-249 (1951)].


1.6.8. Lemma. Let Y be a closed subset of a fully normal topological space S, and let z be an element of ZP(S, Y). Then there exists an open set 0 in S and a co cycle Zl E ZP (S, Y) such that 0 ) Y, Zl vanishes on 0, and zl-zEBP(S, Y).

Prool. In view of remarks 1 and 2 in 1.6.1, the assertion is trivial for p:;;;; 0, and hence we assume that p;;;;:; 1. The given p-cocycle z vanishes on some open covering U of S. Since S is fully normal, by the M odilication Theorem (see 1.1.5) there exists an open set 0) Y, a star refinement m of U, and a (generally discontinuous) mapping I:S-+S such that

10(Y, (1 )

I V ( S4J V for V Em. (2)

Clearly, (1) implies that IY(Y, and thus

I: (S, Y) -+(S, Y). (3)

Accordingly, I induces a cochain mapping

IP: C$(S, Y) -+ ct(S, Y). (4)

Observe that the subscript F is necessary since I is not known to be continuous (see 1.5.1). We set

(5)

and we proceed to verify that 0 and Zl satisfy the requirements of the lemma. First, take points Xo ,"" xp EO. Then Zl (xo , ... , xp) = z(f xo, ... , I xp) = 0, because I xo, ... , I xpEY by (1) and z vanishes on Y. Observe next that in view of (3), zlEZ~(S, Y) (see 1.5.5). To see that

zlEZP(S,Y), (6)

there remains to show that Zl vanishes locally on S. In fact, it is immediate that Zl vanishes on each VE m. Indeed, if xo, ... , xpEV, then by (5) we have zdxo, ... , xp)=z(fxo, ... ,lxp)=O, since Ixo,'''' I xp E I V ( St;n V by (2), St;n V is a subset of some U E U by assumption, and Z vanishes on each U E U. Thus (6) is established. Now let i: S -+ 5 be the identity map on S. Then

i: (5, Y) -+(5, Y). (7)

Let DP:Ct(S, Y)-+q-l(S, Y) be the homotopy operator corresponding to the pair of mappings I and i (see 1.5.5). Let us set

c =DP Z. (8) Rado and Reichelderfer, Continuous Transformations. 6


Then, by 1.5.5, we know that CEq-liS, Y). We verify that actually

c E 0-1 (S, Y), (9)

by showing that c vanishes locally on S. In fact, it is immediate that c vanishes on each set V Em. Indeed, take xo , ... , xP_ l E V. Then [see 1.5.5 and 1.5.4 (4)J

P-l c(xo, ... , xp_1) = 1: (-1)i z(i xo, ... , i xi' lXi' ... , I xp- l). (10)

i~O

Now IXI, ... ,lxp-IEIV(St:nVby (2), and iXk=xkEV(St:nV. Thus ixo' ... ,ixi,lxi' ... ,lxp_1ESt:nV, and St:nV is a subset of some U E U by assumption. Since z vanishes on each U E U, it follows that each individual term of the summation in (10) vanishes, and hence c(xo, ... , xp- 1) =0, proving (9). Finally, by 1.5.5 (23) we have

(J~-l c = jP z - iP z = Zl - z.

In VIew of (9), this shows that zl-zEBP(S, Y), and the lemma is proved.

1.6.9. Lemma. Let Y be a closed subset of the fully normal topological space 5, and let an element h of HP (5, Y) be assigned. Then there exists an open set 0 containing Y such that h reduces to zero in HP(O, Y).

Prool. The given h is of the form h= [zJ, where zEZP(5, Y) and [z] denotes the cohomology class containing z. We apply the lemma in 1.6.8 to this p-cocycle z, obtaining 0 and Zl as described there. Consider the inclusion mapping j: (0, Y) -+(5, Y). Since zl-zEBP(5, Y), we have h= [zJ = [ZlJ, and hence j~h=j~ [ZlJ. Thus the lemma is proved if we can show that jPzlEBP(O, Y). Now for X o, ... , xpEO we have uP Zl) (xo, ... , xp) = Zl (xo' ... , xp) = 0, since Zl vanishes on O. Thus actually jP Zl = 0, and hence surely jP Zl E BP (0, Y).

1.6.10. Lemma. Let Y be a closed subset of the fully normal topological space 5, and let 0 be an open set containing Y. Let an element h of HP (0, Y) be assigned. Then there exists an open set U such that Y ( U (0 and h reduces to zero in HP (V, Y).

Prool. Setting S' = 0, we observe that 5' is again fully normal (see 1.1.3, exercise 10), and we apply the lemma in 1.6.9 to (5', Y). According to that lemma, we have a set 0' which is open relative to 5', satisfies Y (0' ( 5', and is such that h reduces to zero in HP (0', Y). Observe now that since 0' is open relative to 5' = 0, we have 0' = on 01 ,

where 0 1 is open in 5. On choosing U = 0 n 0 1 , clearly Y ( U (0', U is open in S, and Y (V (0'. Thus (V, Y) ((0', Y). Hence, by remark 2 in 1.6.7, h reduces to zero in HP (V, Y) also, and the lemma is proved.


Remark. Since 5' = ° is closed in 5, there was no need to distinguish between closure relative to 5 and closure relative to 5' in the preceeding argument.

1.6.11. The extension lemma. Let (X, A) be a closed pair in a fully normal space 5, and let an element h of HP(X, A) be assigned. Then there exist open sets 0 and U such that X ( 0, A (U, U ( 0, and h is extendable into ' HP(O, U). HP(O,O) J6

Proal. The diagram in Fig. 22 is meant to assist the reader in visualizing the situation.

Starting with the assigned element hE HP (X, A) , denote by 151 the homomorphism HP (X, A) -'>-

HP+O(S,X.)_. -i-, -H"(X,A)

HP+I (5, X) occurring in the cohomology sequence of the triple (5, X, A), and set hI = t5I h. By the lemma in 1.6.9,' applied

Fig. 22.

in dimension p + 1, we have an open set O)X such that j2 hI = 0, where i2 is induced by the inclusion mapping (0, X) -'>- (5, X). On denoting by 153 the homomorphism HP(X, A) -,>-HP+1(O, X) ocuning in the cohomology sequence of the triple (0, X, A), we have t5a h = j 2151 h = j 2 hI = 0, by the vector law for cohomology sequences. By the exactness of the cohomology sequence of the triple (0, X, A), the relation t53 h = ° implies the existence of an element h4 E HP (0, A) such that j4h4 = h, where i4 is induced by the inclusion mapping (X, A) -'>- (0, A). By the lemma in 1.6.10, there exists an open set U such that A ( U (0 and h4 reduces to zero in HP (U, A). This means that i5h4 = 0, where i5 is induced by the inclusion mapping(U, A) -)- (0, A). By the exactness of the cohomology sequence of the triple (0, U, A), there follows the existence of an element h6 E HP (0, U) such that i6h6=h4' where i6 is induced by the inclusion mapping (0, A)-,>-(O, U). On denoting by i7 the homomorphism induced by the inclusion mapping (X, A)-,>-(O, U), the vector law yields j7h6=j4j6h6=j4h4=h. Thus h is extendable into HP (0, U), and the extension lemma is proved.

Remark. For the special case when A = 0, the extension lemma reduces to the following statement: If X is a closed subset of the fully normal space 5, and if an element h of HP (X) is assigned, then there exists an open set O)X such that h is extendable into HP(O).

6*


Inspection of the preceding diagram reveals that in this special case the proof is complete at the moment when the homomorphism i, is brought into play.

1.6.12. The reduction lemma. In the fully normal space S, let (X, A) be a closed pair, and let 0 and U be open sets such that X(O, A ( U, U (0. Finally, let It, be an element of HP (0, U) such that k reduces to zero in HP(X, A). Then there exist open sets 01 , u;. such that X(Ol (0, A (U1 (U, U1 (01, and It, reduces to zero in HP(Ol' VI)'

The march of the proof is indicated by the following diagram.

Fig. 23.

The proof is made as follows. By assumption, i1k=O, where il is induced by the inclusion mapping (X, A) -+ (0, U). Let i2 and f3 be induced by the inclusion mappings (O,A)-+(O, U) and (X,A)-+(O, A) respectively, and set ~ =12k. By the vector law it follows that 13kl = 1312 k = 11 It, = O. By the exactness of the cohomology sequence of the triple (O,X,A), there follows the existence of an element k 2EHP(O, X) such that i4k2=kl' where 14 is induced by the inclusion mapping (0, A) _(0, X). By the lemma in 1.6.10 there exists an open set 01

such that X (01 (0, and 15k2=O, where 15 is induced by the inclusion mapping (01 , X)_(O, X). Let 10 and j7 be induced by the inclusion mappings (01 , A)_(OI' X) and (01 , A)_(O, A) respectively. By the vector law we have then 17k1=171·4k2=10j5k2=100=0. Now introduce the set U* = un 0 1 , Then U* is open and A ( U* (01 , Accordingly, we have homomorphisms 1s and 19 induced by the inclusion mappings (01 , V*)_(O, U) and (01, A)_(OI' V*) respectively. Set k3=18k. By the vector law it follows that 19ks=1918k=1712k=i7~=0. By the exactness of the cohomology sequence of the triple (01 , U*, A) there follows the existence of an element k4EHP-l (V*, A) such that ()lo k4 =k3' where ()IO is the homomorphism HP-I(V*, A)_HP(OI' V*) occurring in the cohomology sequence of the triple (01 , U*, A). By the lemma in 1.6.10, applied in the dimension p -1, there exists an open set UI

such that A ( UI (U* and 111 k4 = 0, where 111 is induced by the inclusion mapping (VI' A)_(U*, A). Let now 113 and 114 be induced by the


inclusion mappings (01' VI) -+(01, V *) and (01' VI) -+ (0, V) respectively. Finally, let 1512 denote the homomorphism HP-1(V1, A)-+HP(Ol' VI) occurring in the cohomology sequence of the triple (01' VI' A). We have then j1aha=j131510h4=1512jllh4=15120=0 by the vector law for cohomology sequences, and finally i14h =i13ish =i1aha = ° by the vector law for induced homomorphisms. The relation j14 h = ° means that h reduces to zero in HP(Ol' VI), proving the reduction lemma.

Remark. In the special case when U = 0 and A = 0, the reduction lemma yields the following statement: If X is closed and ° is open and X(O, and if h is an element of HP(O) such that h reduces to zero in HP (X), then there exists an open set 0 1 such that X (01 (0 and h reduces to zero in HP (01), For this special case, the homomorphisms i2 and ja are not needed in the preceding diagram, and the proof is completed at the moment when the homomorphism j7 has been reached.

1.6.13. Proof of the extension and reduction theorems. Returning to 1.6.7, consider first the extension theorem. Given hoEHP(Xo, Ao), by the extension lemma in 1.6.11 there exist open sets 0, U such that Xo (0, Ao ( U, U (0, and ho is extendable into HP(O, V). Since F =} (Xo, Ao), there exists a pair (X,A)EF such that X(O,A(U. Then we have the relations (Xo, Ao) ( (X, A) ( (0, V). By remark 1 in 1.6.7 it follows that ho is extendable into HP(X, A), and the extension theorem is proved.

Next, consider the situation assumed in the statement of the reduction theorem (see 1.6.7). The diagram in Fig. 24 illustrates the sequence of steps in the proof.

. /HPOt"Ao)

/~ j, H'(X,A) 'j. rO'UI

H~ /~.'U' HP(X"A,)

By assumption, we are given Fig. 24.

an element hE HP (X, A) such that j1h = 0, where j1 is induced by the inclusion mapping (Xo, Ao) -+ (X, A). By the extension lemma in 1.6.11, there exist open sets 0, U such that X (0, A ( U, U (0 and there exists an element hI E HP (0, V) such that j2h1=h, where i2 is induced by the inclusion mapping (X, A)-+(O, V). Let j3 be induced by the inclusion mapping (Xo, Ao)-+(O, V). Observe that this mapping is available since (Xo, Ao) ((X, A) ((0, V). The vector law yields now iah1 =iI1'2h1 =i1h= 0. By the reduction lemma in 1.6.12, there follows the existence of open sets °1 , U1 such that X O(Ol (0, Ao( U1 (U, U1 (°1 , and j4h1 =0, where i4 is induced by the


inclusion mapping (01' U1)-+(0, U). Now since F=}(Xo, Ao), there exists a pair (Xl' AI)EF such that (Xl' AI)((OI' UI ) and finally there exists a pair (X2' A 2) E F such that (X2' A 2) ((X, A), (X2' A 2) ((Xl' AI)' The inclusion mappings (Xl' AI)-+(Ol' U1), (X2' A2)-+(XI' AI) and (X2' A 2) -+ (X, A) induce homomorphisms that are designated by is, is, i7 respectively in the diagram. By the vector law we conclude now that i7h=i7i2hl=j6i5i4hl=j6jsO=O. Thus hreducestozeroin HP(X2, A 2), where (X2' A2)EF and (X2' A 2) ((X, A). This completes the proof of the reduction theorem.

1.6.14. Strong excision. Let A and U be two subsets of the fully normal space X, such that

U(A(X, A-U=I=0. (1)

The inclusion mapping (X - U, A - U) -+ (X, A) induces then (see 1.6.3 for terminology) the excision homomorphism which we now denote by

eP :HP(X, A) -+ HP (X - U, A - U). (2)

In 1.6.3 we proved that eP is an isomorphism onto, provided that U ( int A. We proceed to show that if

A closed, U open in X,

then the condition U ( int A can be dropped. Thus we have the

Strong excision theorem. If X is fully normal, and the conditions (1) and (3) are satisfied, then the excision homomorphism (2) is an isomorphism onto, for every p.

Proof. Let F be the family of all the pairs (X - V, A - V) where V denotes an arbitrary (perhaps empty) open set such that V (U. By 1.1.3, exercise 26 we have then the relations

F =} (X - U, A - U), (X, A) E F. (4)

Let (X-V, A - V)E F. The inclusion mappings (X - U, A - U)-+ (X-V,A-V), (X-V, A-V)-+(X, A) give rise to induced homo-

eP morphisms designated by i~, if re-

H~P(JX.~,'A)"" /.~,' HP(X-U,A-U) spectively in the diagram in Fig. 25 _ _ [where eP is the excision homomor-

phism (2)]. Since V(U and U(A, clearly

HP(X-V,A-V) V (intA, and A - V =1= 0 since A-Fig. 25. U =1= 0. Thus if is an isomorphism

onto byI.6.3,lemma3. The symbol ~ in the diagram refers to this fact. Now assign any element hE HP (X - U, A - U). In view of (4) and the extension theorem in 1.6.7, we can


choose V in such a way that h=iChl , ~EHP(X - V, A - V). Next, since if is onto, we have hl=ith2' h2EHP(X, A). By the vector law it follows that h=iCifh2=ePh2. Thus eP is onto. Next, let hE HP(X, A) and ePh=O. In view of (4), it follows from the reduction theorem that we can choose V in such a manner that if h = O. Since it is an isomorphism, it follows that h=O. Thus eP is also an isomorphism into, and the proof is complete.

Remark 1. The case when A - U = 0 has been considered in 1.6.3.

Remark 2. In a fully normal space X, let Xl' X 2, Y be closed subsets such that

(4*)

Then the inclusion mapping

(5)

is an eXCISIOn mapping. Indeed, set U = exl . Then U is open, and clearly UeX2 by (4*). Furthermore X-U=XI , and X 2-U= x2n e U =x2n Xl = Y. Thus (5) coincides with the excision mapping obtained by excising the open set exl . Observe that since X 2 - U = Y =F 0, the theorem derived in this section applies. Hence, on denoting by

(6)

the excision homomorphism induced by (5), we see that eC is an isomorphism onto for every p. Since the assumptions (4*) are symmetric with respect to Xl' X 2' it follows in the same manner that the excision homomorphism

(7)

induced by the inclusion mapping (X2' Y)-+(X, Xl)' is an isomorphism onto for every p.

1.6.15. The homotopy theorem. This basic theorem is actually valid in general topological spaces, but we restrict ourselves to the case of compact HAUSDORFF spaces which is adequate for our purposes. In this special case, several elegant proofs are available. The one to be presented here has been selected because it constitutes an instructive application of the reduction lemma.

The special homotopy theorem. Let A be a compact subspace of the compact HAUSDORFF space X. Denoting by I the closed unit interval O:;;;:u:~1, consider the compact HAUSDORFF pair (XxI, A xl). The


lirst homotopy diagram in 1.1.1 gives rise to the following diagram of induced homomorphisms.

'F' ;\Xl~~f: 'F' I .. It

= = HP{Xxs,Axs) hP

• HP(X,A) • h P

HP(Xxt,AlC.t) S t

Fig. 26.

In this diagram, the symbol gP, for example, denotes the homomorphism induced by the mapping g:(XXI,AXI)-+(X,A) explained in 1.1.1, with similar interpretations for the other homomorphisms appearing in the diagram. Since the mappings h .. ht are homeomorphisms, the homomorphisms hf, hf are isomorphisms onto by 1.6.2, remark 2. The symbols ~ in the diagram refer to this fact. The special homotopy theorem, to be proved presently, states that for any two numbers s, tE I we have

If = If. (1 )

The proof is based on the following series of observations. (i) By 1.1.1, we have the relation Is = ishs' By the vector law

it follows that If = hf if. Since hf is an isomorphism, we infer by 1.3.2, exercise 2 that If and if have the same nucleus, to be denoted by W.,. Similarly, If and, if have the same nucleus Wt.

(ii) Set ~* = gP HP (X, A). Then (see 1.3.1)

HP(XxI, A xl) = ~* + 1)4. (2)

Indeed, by 1.1.1 we have glt=i, the inclusion mapping (X, A)-+(X, A). Hence, by the vector law and remark 1 in 1.6.2 it follows that

(3 )

where 1 denotes the identity homomorphism in HP(X, A). Thus (2) is a direct consequence of 1.3,2, exercise 5.

(iii) For eEWt , there exists an 8=8(e,t»0, such that eEWs for I s - tl < 8. Indeed, the assumption means, by (i), that if e = 0. By the reduction lemma in 1.6.12 there follows the existence of open sets 0, U in XxI such that O)Xxt, U) A xt, U (0, and e reduces to zero in HP(O,U). By 1.1.3, exercise 40, we have then an 8>0 such that X X s ( 0, A x s (U for Is - tl < 8. If s is so restricted, then by remark 2 in 1.6.7 the element e reduces to zero in HP(Xxs, A xs) also. In other words, if e = 0 for Is - tl < 8, proving our assertion.


(iv) If eE~*, then Ife=/fe for any two numbers s, tEl. Indeed, the assumption means that e=gPa, aEHP(X, A). But then, by (3), it follows that Ife=/fgPa=a. Thus Ife is independent of t for eE~*.

(v) Given aEHP(XxI, A xl), bEHP(X, A), denote by E(a, b) the set of those numbers uEI for which I£a=b. Then the set E(a, b), which may be empty of course, is both open and closed (in I).

Indeed, in case E(a, b) is non-empty, take any tEE(a, b). By (ii) , we have

a = a' + a", a' E ~*, a" E 91/. (4)

By (iii) we have then

a"E91s for !S-t!<I3(a",t). (5)

Consider now any sEI such that! s-tl < 13 (a", t). Then, by (4), (5) and (iv), we obtain successively Ifa=jfa'=/fa'=/fa=b. Thus sEE(a, b) for !s-tl< 13 (a", t), proving that E(a, b) is open. Now clearly

I-E(a,b)=UE(a,c), cEHP(X,A), C=Fb.

By what we just proved, each set E(a, c) is open in I, hence I -E(a, b) is open as a union of open sets. Thus E (a, b) is also closed, proving our assertion.

(vi) Since I is connected, it follows (see 1.1.3, exercise 13) that either E(a,b)=I or else E(a, b) is empty. Now take any element aEHP(XxI, A xl), and set b=lg(a). Then E(a, b) is non-empty, and hence E (a, b) = I. In other words, If a = b = Ie a for every tE I, proving the special homotopy theorem.

The homotopy theorem. Let (X, A) and (y, B) be two pairs of compact HAUSDORFF spaces, and let the continuous mappings mo: (X, A) -+ (Y, B), 1nt: (X, A) -+ (Y, B) be homotopic. Denote by mg, mf the corresponding induced homomorphisms from HP(y, B) into HP(X, A). Then mg=mf.

Prool. The induced homomorphisms, corresponding to the transformations in the second homotopy diagram in 1.1.1, give rise to the diagram in Fig. 27.

Fig. 27.

Since mo=mlo' by the vector law we have mg= IgmP, and similarly mf = IfmP. Now I~ = If by the special homotopy theorem, and hence mg=mf.

Remark. In the preceding discussion, the only fact about I that we actually used was that I is compact and connected. This observation yields a generalized homotopy theorem which the reader will readily visualize.


1.6.16. Examples of trivial cohomology groups. In many applications, it is a matter of importance to know that certain cohomology groups involved are trivial. In this section, some theorems are presented that have a bearing upon this subject. For convenience, all the spaces involved are assumed to be compact HAUSDORFF spaces.

Lemma 1. If A is a retract of X, then the homomorphism jP: HP (X) ~ HP(A), induced by the inclusion mapping j:A~X, is onto.

Prool. By assumption, we have a continuous mapping I:X ~A such that I x = x for xE A. Then I induces a homomorphism IP: HP (A) ~ HP(X). Now fj=i, the identity mapping A~A. By the vector law and remark 1 in 1.6.2 it follows that jP jP = iP = 1, the identity homomorphism in HP (A). Now assign any element uE HP (A) and set v = jPu. Then jP v = jP fP U = iP U = u, proving the lemma.

Theorem 1. If HP (X) = 0 for a certain p, and A is a retract of X, then HP (A) = 0 for the same p.

This is a direct consequence of lemma 1.

t: f P Theorem 2. If A is a deformation H~P(X'A) HP(lI..A\ retract of X, then HP(X, A) = 0 for

/ everyp.

f:' Y Prool. Consider the diagram in

Fig. 28. H P(A. Al In this diagram, I denotes a reFig. 28.

traction from X onto A such that the mapping I:(X, A)~(X, A) is homotopic to the identity mapping i:(X, A)~(X, A). Such a mapping I exists by assumption (see 1.1.5, definition 8). The corresponding induced homomorphisms are denoted by IP, iP respectively. By the homotopy theorem we have then fP = iP. Since IX(A, we have also the relation I: (X, A)~(A, A), yielding the induced homomorphism If. Finally, jP is induced by the inclusion mapping (A, A)~(X, A). Clearly I=jl, since IX(A, and hence IP=/fjP by the vector law. Now take any element eEHP(X, A). Then, on the one hand, IP e = iP e = e by remark 1 in 1.6.2, and on the other hand IPe=lfjPe=/10=0, since HP(A, A) =0 by remark 3 in 1.6.1. Thus e = 0, proving the theorem.

Corollary to Theorem 2. If A is a deformation retract of X, then the homomorphism jP:HP(X)~HP(A), induced by the inclusion mapping j:A~X, is an isomorphism onto, for every p.

Proof. The groups HP(X, A), HP(X), HP(A), HP+l(X, A) are consecutive terms in the cohomology sequence of the pair (X, A). The first group and the fourth group are trivial by theorem 2, and thus the corollary follows directly from 1.3.3, lemma 4.


Theorem 3. Let Xl' X 2 , Y be subspaces of X such that Xl UX2 =X, Xl n X 2 = Y =j= 0. Suppose that, for some integer p,

HP(X1 ) = 0, HP(X2) = 0, HP-I(y) = 0. (1 )

Then HP (X) = 0, for the same p. Proal. By 1.6.14, remark 2, we have

(2)

Now the groups HP-I(y), HP (Xl ,Y), HP(XI) are consecutive terms in the cohomology sequence of the pair (Xl' Y). The first group and the third group are trivial by (1). Hence, by 1.3.3, lemma 3,

HP (Xl , Y) = 0. (3)

Next, the groups HP(X, X 2 ), HP(X), HP(X2) are consecutive terms in the cohomology sequence of the pair (X, X 2). The first group is trivial by (2) and (3), while the third group is trivial by (1). Hence HP(X) =0 by 1.3.3, lemma 3.

Theorem 4. Let Xl' X 2 , Y be subspaces of X such that Xl UX2 =X, Xl nx2 =Y=j=0. Suppose that, for some integer p,

(4)

Then HP (Xl) = 0, for the same integer p. Proal. By 1.6.14, remark 2, we have again (2). The groups HP-I (X2) ,

HP(X, X 2), HP(X) are consecutive terms in the cohomology sequence of the pair (X, X 2). The first and the third groups are trivial by (4). Hence, by 1.3.3, lemma 3 it follows that

(5)

Next, the groups HP (Xl , Y), HP(XI), HP(Y) are consecutive terms in the cohomology sequence of the pair (Xl' Y). The first group is trivial by (5) and (2), while the third group is trivial by (4). Hence, by 1.3.3, lemma 3, HP (Xl) = 0.

Theorem 5. Let X be a subspace of S. Suppose that, for some integer p, the following holds: for every open set O)X there exists a subspace Y such that X(Y(O and HP(Y)=O. Then HP(X) =0, for the same p (recall that all the spaces occurring in the present section are assumed to be compact HAUSDORFF spaces).

Proal. Select an element eEHP(X). By the extension lemma in 1.6.11 (see the remark there) we have then an open set O)X such that e


is extendable into HP (0). By assumption, we have next a subspace Y such that X (Y (0 and HP (Y) = O. The homomorphisms induced by the inclusion mappings X -+Y, Y -+0, X -+0 give rise to the following diagram.

jP

"\/H"'" HP(y)

Fig. 29.

Since e is extendable into HP(O), we have e=iPe', e'EHP(O). Then jP e' = 0 since HP(Y) = 0, and hence e = iP e' =kPjP e' = kPO =0. Since e was an arbitrary element of HP(X), it follows that HP(X) =0.

1.6.17. A structure theorem. It will be a matter of importance in the sequel to decompose additively certain cohomology groups, as well as homomorphisms connecting such groups. We proceed to derive the basic facts needed for this purpose. Let X be a compact HAUSDORFF

space and let Y be a closed (and hence compact) subset of X. We assume that

Y=l=0, Y=l=X. (1 )

Then 0 = CY is a non-empty open set. Let 0 be decomposed, in any manner, into a union of pair-wise disjoint open sets 0 1 , ... , On' ... , where the sequence {On} may be finite or infinite. Set

(2)

Then (see 1.1.3, exercise 41) Yn and Yn* are non-empty, closed (and hence compact) sets, and Yn> Y. The inclusion mapping (X, Y)-+ (X, Yn) induces a homomorphism

(3)

On setting (4)

we can formulate the basic structure theorem as follows.

1. i~ is an isomorphism into, and hence I(i~) ~HP(X, Y,.). II. HP (X, Y) is the weak direct sum or the direct sum of the sub

groups I (i~), according as the sequence {On} is infinite or finite (see 1.3.1).

§ I.6. General cohomology theory. 93

Prool. We fix the integer p, and will not display it in designating the various homomorphisms to be used. Thus i~ will be written simply as in. The proof is based now on the following diagram.

,* '* H"(X:r:) In

• HP(X.Y) In • HP(y: .y)

.. / 1m ~ en

HP(X,Y.,) rn~, rn HPeX,Yn)

/~ e~ HP (X,"Fn_,)

Un • HP(X,F,,) Vn • HP(F,,_, • Fnl Fig. 30.

All the homomorphisms involved are induced by inclusion mappings'. in is the i~ in (3), and im is the analogous homomorphism corresponding to an integer m=f=n. By 1.1.3, exercise 41 and (2) we have

(5)

One has thus the inclusion mappings (Yn*' Y)-+(X, Y), (X, Y)-+(X, Y,,*) , (Yn* , Y)-+(X,Y,,), and (X,Yn*)-+(X,YI1l ) for m=f=n, yielding the induced homomorphisms i:, i:, en, knm respectively. Next, on setting

F" = n 1';" k = 1 , ... , n, (6) it is clear that

(7)

By 1.1.3, exercise 41 it follows that

(X, F,,) ~ (X, Y) . (8)

By (7) one has the inclusion mappings (X, Fn) -+ (X, Y,,), (X, Y) -+ (X, F,,), yielding the induced homomorphisms s,., Yn' Finally, for the case when n;;;;;; 2, one has also the inclusion mappings (X, Y) -+ (X, Fn~l)' (F,'~l' F,,) -+ (X, Fn), (X, Fn) -+ (X, F,'-l)' (Fn~l' Fn) -+ (X, Yn), yielding the induced homomorphisms Y"~l , Vn , U," e~ respectively. Clearly

Yn U Y: = X, Yn n Y,,* = Y =f= 0,

YnUFn~l=X, YnnFn~l=Fn=f= 0.


Accordingly (see 1.6.14, remark 2) en and e~ are excision homomorphisms and hence isomorphisms onto, as indicated by the symbols R::I.

The proof of the structure theorem is now made by the following series of observations.

(i) By the vector law, i! in = en, an isomorphism. Hence clearly in is an isomorphism into.

(ii) If bm EHP (X, Ym), then

.*. b {O if m =1= n, 1n ~m m = l·f en bn m = n.

The second assertion is obvious since i! i" = en. On the other hand, if m =1= n, then i! im bm = i! i! k"m bm = 0, because of the exactness of the cohomology sequence of the triple (X, Y,,*, Y).

(iii) The subgroups I (in) = i"HP (X, Yn ) of HP (X, Y) form an independent system in the following sense: if am E I (im), m = 1, ... , N, and N

Lam = 0, (9) m=1

then a1 = ... = aN = 0. Indeed, by assumption

a",=imbm , bmEHP(X,Ym ), m=1, ... ,N, (10)

and hence (9) yields

m=1

Take any integer n such that 1 -;;;. n -;;;'N. Then (11) yields N

'\' .* . b ° L.." 1" ~m '" = . m=1

(11)

By (ii) this formula reduces to ell bn = 0. Since en is an isomorphism, it follows that bn=O and hence an=O, 1-;;;'n-;;;.N, proving the assertion (iii).

(iv) If a E HP (X, Y) is such that

aEI(r,,} = r"HP(X, Fn),

then a can be written in the form

" a=Likbk , bkEHP(X,Yk)· k=1

(12)

(13 )

Denoting this assertion by An, we make its proof by induction. For n = 1, we have 1\ = 1;, r1 = iI' and thus Al is obvious. So let n:;;;; 2, and assume that A n - 1 is known to be true. Observe that since


e~ is an isomorphism onto, the relation e~ = v,.s,. (which holds by the vector law) can be re-written in the form

((e~)-lvn) s,. = 1,

the identity homomorphism in HP(X, Y,.). By 1.3.2, exercise 5, it follows that any element Cn of HP (X, Fn) can be written in the form

c~ = snbn + c~, bnE HP(X, Y,.), c~E N((e~)-lvn).

Since e~ is an isomorphism, it follows that c~EN(v,,), the nucleus of v,.. But N (v,,) = u,.HP (X, P,.-I) by the exactness of the cohomology sequence of the triple (X, F..-l, Fn). Hence c~ can be written in the form c~=U"C"_I' C .. - 1 EHP(X, P,.-I). Thus we obtain the representation

(14)

Now if a satisfies (12), then a=r,.c", cnEHP(X, Fn). By (14), we obtain

By the induction assumption An-I' the term rn - 1 C .. - 1 can be written in the form ,.-1

Z.ik bk, bk E HP(X, Yk), k=1

and thus An follows, completing the induction.

(v) The structure theorem is now proved as follows. In view of (i) and (iii), there remains to show that any assigned element aEHP(X, Y) can be written as a (finite) sum of the form

n

a=Likbk , bkEHP(X,Yk )· k=1

( 15)

To verify this, observe that by (8) and the extension theorem in 1.6.7 there exists an integer n (depending upon a) such that aEI(rn), and hence for this n there exists a representation of the form (15) by (iv).

Remark 1. If the sequence {On} is finite, and contains say N terms, then for n = N one has FN = Y and rN = 1 [the identity homomorphism in HP (X, Y) J. Accordingly, I (rN) = HP (X, Y), and thus by (iv) every element a of HP (X, Y) can be written in the form (1~) with n=N.

./H~:. ;" ~

HP(X,Yn ) - H·(Y~, Y)

Remark 2. Consider Fig.31 (which Fig. 31.

is a portion of Fig. 30). Since en is an isomorphism onto, e;;-1 is also a (single-valued) iso

morphism. We assert that for every element aEHP(X, Y) we have


the relation ",. -1'* a = L... ~,.e,. 1,. a, (16)

where the prime attached to the summation symbol means that the summation is finite (that is, all but a finite number of the terms are equal to zero). To prove (16), observe that by the structure theorem we have a representation of the form

N

a = L ik bk , bk E HP (X, Yk ) • k~1

( 17)

Take any positive integer n. From (17) we get N

. -1'* ". -1'*' b ~nen In a = L...~nen In ~k k' (18)

k~1

By (ii) above, it follows that

.* . b {O In Zk k = en b"

Thus (18) reduces to

if k=t=n, if k = n.

. -1'* _ {in b" if 1;;;;;' n ;;;;;. N, } ~"en In a - l'f N On> .

(19)

Thus all the terms in (16) vanish for n> N, and hence that summation is"'finite. From (17) and (19) we infer now that

.&

N ",. -1'* -". b -L... ~nen In a-£...J~" ,,-a.

n~l

Remark 3. By 1.1.3, exercise 41, we have the relations O,,(Yn*, frO,,(Y(Yn , yielding the inclusion mapping (0", frOn) --+(Yn*' Y).

t., r·~.O") 0~~

Accordingly, we can complete the last diagram as indicated in Fig. 32.

All the homomorphisms are induced by inclusion mappings. Let us now add the assumption

frO" =t= 0 for all n. (20)

HP(X,Yn ) ------''''---~. HP(y~ ,Yl By 1.1.3, exercise 41 we have X = eo Fig. 32. Y"UO", frO"=Y,,nOn' Thus (see 1.6.14,

remark 2) it follows that t" is an excision homomorphism, and hence in view of (20) it is an isomorphism onto, as indicated by the symbol ~. By the vector law we have t" = z" en, and hence t;:l z" en = 1, the identity homomorphism in HP (X, Y,,). Thus e;:l = t;:1 z". We conclude that

(21)


From (16) and (21) we obtain, for aEHP(X,Y), the representation

(22)

which will be of importance later on. Recall that the summation IS

finite (all but a finite number of the terms are equal to zero), a fact referred to by the prime attached to the summation symbol.

Remark 4. Since tn is an isomorphism onto, in view of part I of the structure theorem it follows that I (i,,) R>O HP (On' IrO,,). Part I! of the structure theorems yields then the following fundamental result.

Theorem. HP (X, Y) is the weak direct sum (or direct sum if the sequence {On} is finite) of a set of its subgroups which are isomorphic to the groups HP(On' IrO,,).

This theorem describes the structure of HP (X, Y) in terms of the holes 0".

Remark 5. Take a second pair (X*, Y*) of compact HAUSDORFF

spaces and consider a continuous mapping

I: (X, Y) -+(X*, Y*). (23)

Also, take a closed subset Z of X such that Z (Y. Then we have also

I: (X, Z) -+ (X*, Y*). (24)

Furthermore, since Ir 0" (Y, we have the relation

Consider now the diagram in Fig. 3 3. The homomorphisms [P, IP, If.

are induced by the mappings (23), (24). (25) respectively. The homo-morphisms Wn, tn, i" are taken from

Fig. 33.

the diagram used in remark 3. Finally, the homomorphisms k and kn

are induced by the inclusion mappings (X, Z) -+ (X, Y), (X, Z) -+ (X, 1',,) respectively. Now take an element a*EHP(X*, Y*). Using (22), we calculate

P a* = kl-P a* = k 'VI i t-1w I-P a* = k 'VI it-liP a* ~nn n ~nntJ.

- 'V'k' t-1 / P * - ""k t-1 / P * - ~ 't n n n a - L n n n a .

Symbolically, this result may be written in the form

IP = L' kn t;;l/f.· (26) Rado and Reichelderfer, Continuous Transformations. 7


This formula will play an important part later on. It yields a decomposition of the homomorphism fP in terms of the holes 0" corresponding to the subspace Y) Z.

Remark 6. Due to the fact that the case p = 1 required special treatment in connection with excision homomorphisms (see 1.6.3), we excluded in the preceding discussion the cases when Y = 0 or fr On = 0. However, if p =i= 1, then the excision homomorphisms involved are isomorphisms onto even if Y = 0 or fr 0" = 0, by the lemmas 1 and 2 in 1.6.3. It follows that for P =i= 1 the structure theorem and the various other results derived from it remain valid even if Y = 0 or fr 0" = 0, as inspection of the proofs reveals.

§ 1.7. Cohomology groups in Euclidean spaces l •

I. 7.1. Cells and spheres. The cohomology groups of the n-cell E" and of the n-sphere 5" (see 1.2.2) are given by the following theorems for all integers n::?; O.

Theorem 1. HP (P) = 0 for all p. Theorem 2. HP(5") =0 for p=i=n+ 1, and H"+l(5") f':::! I. Proof. Since EO is a single point, we have HP (EO) = 0 by 1.6.1,

remark 5. For n:2; 1, the origin P = (0, ... , 0) is a deformation retract of E" (see 1.2.2, exercise 13). Thus HP(E", P) =0 by theorem 2 in 1.6.16, and hence HP(P) =HP(P, P) =0 by 1.6.1, remark 4, proving theorem 1.

Theorem 2 will be proved by induction on n. For clarity, denote by T" that theorem stated for a specific n. Since 50 consists of precisely two points, To is a direct consequence of remark 2 in 1.6.3. So take n:;;;; 1 and assume that T,,- 1 has been already established. From the basic diagram in R,,+l (see 1.2.2) select first the pair (E"_, .~"-I), and choose an integer q. The groups

Hq(E"-) , HQ(S"-l), Hq+I(E,,-,sn-I), Hq+l(E'~)

are consecutive terms in the cohomology sequence of the pair (E''-, .S·n-I). Since E'~ is homeomorphic to En, the first and the last groups are trivial by theorem 1. By 1.3.3, lemma 4, it follows that

(1 )

1 For historical comments and further details, see the various treatises in Algebraic Topology listed in the Bibliography. If this volume is used in conjunction with other treatises, then the shift in dimension referred to on p. 64 in footnote 1 should be kept in mind. Many of the results derived in § r. 7 are treated in the literature as applications of the ALExA'mER Duality Theorem. The.inductive proofs given here provide instructive exercises for the reader in making effective use of elementary properties of Euclidean spaces.

§ I. 7. Cohomology groups in Euclidean spaces. 99

In 1.2.2, exercise 1 we noted the relation 51! = E+. U E":..., 5n- 1 =

E+. n E":... =l= 0. The excision theorem (see 1.6.14, remark 2) yields therefore

Hq+1(5", E+.) ~ Hq+1 (E":..., 5n- 1). (2)

Observe now that the groups

Hq (E+.) , Hq+1 (5", E+.) , Hq+1 (5n) , Hq+1 (E'+)

are consecutive terms in the cohomology sequence of the pair (5n, E+.). Since E+. is homeomorphic to En, the first and the last groups are trivial by theorem 1. Hence, by 1.3.3, lemma 4,

(3)

As 5,,-1 is homeomorphic to 5,,-1, we have

(4)

From (1) to (4) we infer finally that

Hq (5n- 1) ~ Hq+1 (5n). (5)

By the induction assumption Tn-I' the group on the left in (5) is trivial or isomorphic to I according as q =l= n or q = n, and thus (5) yields directly T", completing the induction.

Remark. For the case q=n of (5) it is a matter of interest to have an explicit isomorphism from H" (5n- 1) onto Hn+1 (5"). Consider the following diagram.

Hn (s n-l ) -------'-:::::=-----___ ... , Hn + I (5 n)

ine~~nf;' Fig. 34.

The homomorphisms On' en, ill are those that yielded the relations (1), (2), (3). As regards In' it is induced by the mapping f: 5,,-1-+5"-\ given by I (Xl, ... , x") = (xl, ... , x", 0). Since f is clearly a homeomorphism, I" is also an isomorphism onto, as indicated by the symbol R::!, in view of 1.6.2, remark 2. The diagram yields the desired explicit isomorphism in the form ill e;l 0" 1;1 .

7*


1. 7.2. A general property of compact sets in R n and S". For any topological space X, we have HP(X) =0 for p;;;;:,O by 1.6.1, remarks 1 and 2. For compact sets in Rn and S", the following two theorems yield an upper limitation for the dimension of non-trivial cohomology groups.

Theorem 1. If F is a non-empty, compact subset of R", then HP(F)=O for P> n.

Theorem 2. If F is a non-empty, compact subset of S" such that S"-F=l=0, then HP(F) =0 for p>n.

Since the proof will be made by induction on n, it is convenient to denote these theorems by T (R") and T (S") respectively. The following auxiliary theorem, to be denoted by T(Rn, m, s), will also be proved and utilized in the course of the proof. Let n:;;;;: 1, m:;;;;: 1, s:;;;;: 1 be integers, and let F (n, m, s) be a generic notation for a set in R" that is the union of precisely s different cubes of the subdivision L/(n, m) of R" (see 1.2.3, definition 1). Then T(R", m, s) is the theorem that HP(F(n, m, s)) =0 for P> n.

The proof of T (R") , T (sn), T (R", m, s) is made by simultaneous induction, and is based on the following series of observations.

(i) T (RO) and T (SO) are both true. Indeed, RO consists of a single point, and thus T(RO) is a direct consequence of remark 5 in 1.6.1. As regards T(SO), observe that a non-empty, proper subset of SO consists of a single point, and thus T (SO) is also a direct consequence of remark 5 in 1.6.1.

(ii) T(R",m, 1) is true. Indeed, a set F(n,m, 1) is merely a cube of the subdivision J(n, m), and hence it is homeomorphic to the ncell E". By theorem 1 in 1.7.1 and remark 2 in 1.6.2 it follows that HP (F (n, m, 1)) = 0, and indeed not only for P> n but for every p.

(iii) For n:;;;; 1, T (R") and T (5") mutually imply each other. Indeed, in view of the availability of the stereo graphic projection (see 1.2.2) it is evident that a non-empty compact set F (Rn is homeomorphic to a non-empty, compact set F*(S" such that S"-F*=l= 0, and vice versa. Since then HP (F) ~ HP (F*) for every p by remark 2 in 1.6.2, the equivalence of T(R") and T(sn) for n;;;:1 is obvious.

(iv) If n;;;;, 1, and T(Rn-l) is already established, then T(R", m, s) follows for all integers m:;;;;: 1, s:;;;;: 1. To see this, observe first that T(sn-l) is also available [by (i) if n=1 and by (iii) if n> 1]. Fix now the integer m:;;;;: 1. Noting that T (R", m, 1) holds by (ii), we proceed by induction on s. So assume that s:;;;;2, and that T(R", m, s -1) has already been established. Take a set

§ 1.7. Cohomology groups in Euclidean spaces. 101

where q1 • ...• qs are different cubes of the subdivision L1(n, m) of RH. Set s-l

X 1= U qi' X 2 =qs' i~l

Then Xl is a set F (n, m, s -1), and hence

HP(X1 ) = 0 for p > n, (1)

by the induction assumption T (Rn , m, s -1). Also, smce X 2 = qs is homeomorphic to the n-cell En, we have

(2)

and indeed for every p, by theorem 1 in 1.7.1. By 1.2.3, exercise 3, the notations may be so chosen that Y is a proper subset (perhaps empty) of Irqs. Since Irqs is homeomorphic to 5,,-1 (see 1.2.2, exercise 12), and since T (5"-1) is available as noted above, we have the relation

HP-l (Y) = 0 for Y =t= 0, P - 1 > n - 1.

Assume first that Y =t= 0. Noting that

X=F(n,m,s)=X1 UX2 , Y=X1 nX2 ,

the relations (1) to (4) yield, by theorem 3 in 1.6.16,

HP(F(n, m, s)) = 0 for p > n.

(3)

(4)

(5)

On the other hand, if Y = 0, then (5) follows already from (1), (2) and (4) by 1.6.5 (b), since p> n?:; 1 and hence p =t= 1. Thus the inductive proof is complete.

(v) If T(R", m, s) is known to hold for some n;:;:, 1 and for all integers m;:;:, 1, s;:;:' 1, then T(Rn) follows. To see this, take a non-empty compact set F (Rn, and let 0 be any open set such that 0 )F. Consider the subdivision L1(n, m) of Rn, and denote by F* the union of all those cubes qEL1(n, m) that intersect F. Then F* is a set F(n, nt, s), and hence by T (Rn, m, s) we have

HP(F*) = 0 for p > n. (6)

For m sufficiently large, we have however (see 1.2.3, exercise 5) the inclusions F (F* (0. Since the open set O)F was arbitrary, (6) implies by theorem 5 in 1.6.16 that HP (F) = 0 for p> n. Thus the inductive proof of T (R") is complete. Hence, in view of (iii), T (5") is also established.

Remark 1. The condition S"-F=t= 0 in theorem 2 is necessary. Indeed, if F=S", then Hn+l{F)=Hn+l(sn)=t=O by theorem 2 in 1.7.1.


Remark 2. As an immediate corollary to theorem 1, we obtain the statement that if X is a topological space which is homeomorphic to R" and Y is a non-empty compact subset of X, then HP (Y) = ° for p > n. Indeed, the assumption concerning X implies that Y is homeomorphic to a non-empty compact subset F of R". Hence by remark 2 in 1.6.2 we conclude that, for p > n, HP (Y) R:::! HP (F) = 0. A similar argument, starting with theorem 2, yields the following corollary: if X is a topological space which is homeomorphic to sn and Y is a non-empty, compact, proper subset of X, then HP (Y) = ° for p > n.

I. 7.3. A lemma. Let n;;;;;' 2, m"2 1 be arbitrary integers which will be kept fixed in this section. Consider the subdivision L1*(n, m) of the unit cube Qn in R" (see 1.2.3, definition 2). The letter q is used as a generic notation for a cube of L1*(n, m), and q* refers to the unique cube q which contains the point (0, 0, ... ,0). Then clearly

(O,O, ... ,O)E/rq*, (0,0, ... ,0)4q if q='Fq*. (1)

Let N be the number of cubes q and s be an integer such that

oSs<N. (2)

The symbol Ks is used to refer to a collection of s cubes q with the following properties (see 1.2.3, definition 7).

(i) K, is strongly connected. (ii) q* E Ks if s:;::;; 1 .

The symbol K; is used to refer to the collection complementary to Ks (that is, K; contains those cubes q that are not contained in Ksl. The union of the cubes of Ks ' K; will be denoted by 1 K,J , 1 K; I respectively. Observe that 1 K; 1 ='F 0 in view of (2).

Lemma L (s). H" (I Kil) = 0.

Proof. As the notation L (s) indicates, we proceed by induction on s. For s = 0, the lemma is true since 1 Ki 1 = Qn, and HI> (Qn) = ° by 1.7.1 and remark 2 in 1.6.2, in view of the fact that Qn is homeomorphic to the n-cell E" (see 1.2.2, exercise 12). Now that L (0) is established, take s:2: 1 and assume that L (s - 1) has been proved. We arrange the s cubes of K5 in the following manner into a sequence

ql' ... ,q s' (3 )

(a) If s=1, then Kl consists [in view of (ii)] of the single cube q*, and (3) is of course the one-term sequence where ql = q*.

(b) If s"2 2, then by 1.2.3, exercise 8 we can choose (3) in such a manner that the cubes ql' ... , qs-l form a strongly connected collection containing q* and the cubes qs-J and qs are strongly adjacent. Observe that these properties of (3) imply that qs ='F q*.

§ 1. 7. Cohomology groups in Euclidean spaces. 103

Introduce the notations

Since X 2 is then homeomorphic to the n-cell En, we have

(5) Next we verify that

(6)

If s = 1, then X = Q", and (6) follows since Q" is homeomorphic to E". If s:2; 2, then in view of (b) it follows that X is a set of the form IK;-ll, where Ks*-l is the complementary collection to the collection K s_l consisting of ql' ... , qs 1· Thus (6) holds by the induction assumption L(s-1).

Assume now first that Y = O. Then by 1.6.5 (a) it follows from (5) and (6) that H" (Xl) = 0, proving L (s) in this case. Assume next that

Y=t=0. (7)

Since q/'lK; and Y(Xl=IK;I, clearly

(8)

We proceed to verify that

(9)

If s = 1, observe that in this case qs = ql = q*, and hence by (1)

(0,0, ... , 0) E Irq* -IKtl (trq* - Y,

showing that (9) holds. If s:2; 2, then by (b) the cubes q,l and q, are strongly adjacent, and hence (see 1.2.3, exercise 6) there exists a point x such that

( 10)

Since all the cubes q E K; satisfy the condition q =F qs' qs- l' we infer from (4) and (10) that

x E Ir q, - Xl ( Ir qs - Y,

and thus (9) holds again. In view of (7), (8), (9) it follows that Y is a non-empty, compact, proper subset of Irq,. Since Irqs is homeomorphic to the (n-1)-sphere 5,,-1 (see 1.2.2, exercise 12) we conclude by remark 2 in 1.7.2 that

H"(Y) = O. (11)

Finally, by theorem 4 in 1.6.16 the relations (7), (5), (6), (11) imply that H" (Xl) = H" (I K; I) = 0, and the inductive proof of the lemma is complete.


I. 7.4. Applications. Theorem 1. Let F be a non-empty compact subset of the interior of the unit cube Q", n::2; 2, such that Q" -F is connected. Then H n (F) = 0.

Proof. Let there be assigned any set U* such that F ( U* ( Qn and U* is open relative to Q". Setting U = U* n int Qn, we have F ( U (U*. Clearly, the point (0, 0, ... ,0) does not lie in U. Since Qn_F is connected, we have (see 1.1.3, exercise 35) an open set 0 such that

(i) F (0 ( U, (ii) Qn - 0 is connected.

Since (0, 0, ... , 0) Ef u, it follows that

(iii) (0,0, ... , O)EQ"-O.

In view of (i), F and Q" -0 are non-empty, compact, disjoint subsets of Q". Consider now the subdivision L1*(n, m) of Qn (see 1.2.3). Denote again by q a generic cube of this subdivision, and by q* the unique cube q which contains the point (0, 0, ... ,0). Let Ks be the collection of those cubes q which intersect Qn -0, where s is the number of the cubes of Ks> and let K: be the complementary collection (thus K: consists of those cubes q which do not intersect Qn -0). For m large enough, by 1.2.3, exercise 4 we have then

(1 )

Let m be so chosen. In view of (iii), clearly q*EKs' In view of (ii), it follows by 1.2.3, exercise 7 that Ks is strongly connected. Since F =1= 0, (1) implies that K: is non-empty. Thus Ks and K: satisfy the assumptions of the lemma in 1.7.3, and hence

In view of (1), (i), and the definition of K:, it follows that

F( IK:I (O( U( U*.

(2)

(3 )

Since the (relative to Q") open set U*) F was arbitrary, the relations (2) and (3) imply, by theorem 5 in 1.6.16, that H" (F) = 0.

Theorem 2. If F is a non-empty compact subset of R", n::2; 2, such that R" - F is connected, then H" (F) = 0.

Proof. By 1.2.2, exercise 11, F is homeomorphic to a non-empty, compact subset F* of the interior of the unit cube Q" such that Qn - F* is connected. In view of theorem 1 and 1.6.2, remark 2, it follows that Hn (F) ~ Hn (F*) = 0.

Theorem 3. If F is a non-empty, compact subset of sn, n::2; 1, such that S"-F is connected, then H"(F) =0.


Proof. If S"-F= 0, then H"(F)=Hn(sn)=O by 1.7.1. If sn-F=F0 and n;;;;;; 2, then F is homeomorphic (see 1.2.2, exercise 10) to a non-empty compact subset F* of R" such that Rn - F* is connected. By theorem 2 and 1.6.2, remark 2 it follows that Hn (F) R:::! H n (F*) = O. Finally, if n = 1 and Sl-F =F 0, then either F is a single point and HI (F) =0 by 1.6.1, remark 5, or else F is homeomorphic (see 1.2.2, exercise 14) to the i-cell £1, and HI (F) R:::!HI(£1) =0 by 1.7.1 and 1.6.2, remark 2.

Theorem 4. Let X and Y (X be non-empty, compact subsets of the n-sphere sn, n;;;;;; 1, such that X - Y is connected and non-empty. Then Hn+l (X, Y) R:::! I or Hn+l (X, Y) = 0, according as X - Y is or is not open, or equivalently, according as Y does or does not contain fr X.

Proof. Since X - Y is a non-empty connected subset of CY, it is contained in a certain component 0 1 of CY. Let" O2 be the union of the components of CY other than 0 1 (thus O2 is empty if C Y is connected). Since S" is locally connected, 0 1 and O2 are disjoint open sets, and 0 1

is connected. Thus we are dealing with the situation studied in 1.1.3, exercise 19. Accordingly, on setting X* = X U O2 , y* = Y U O2 , the sets X*, y* are closed (and hence compact since sn is compact). We first verify tha t

(4)

Of course, (4) is obvious if O2 = 0, since then X* =X, y* = Y. In any case, by 1.1.3, exercise 19 we have the relations

x u y* = X*, X n y* = Y =F 0,

and (4) follows from the excision theorem as stated in 1.6.14, remark 2. Next (see 1.1.3, exercise 19), S"-Y*=01=F0. Thus y* is a nonempty, compact subset of S" such that sn - y* is connected. By theorem 3 it follows that

Furthermore, by 1.7.2, theorem 2 we have

Hn+l (Y*) = o.

(5)

(6)

Observe now that the groups Hn(y*), HIl+1(X*, Y*), Hn+1(X*), Hn+1(Y*) are consecutive terms in the cohomology sequence of the pair (X*, Y*). The first and the last group are trivial by (5) and (6). Hence, by 1.3.3, lemma 4,

Hn+1 (X*, Y*) R:::! Hn+l (X*) . (7)

By 1.7.1, theorem 2 and 1.7.2, theorem 2 we have Hn+1(X*)R:::!1 or Hn+1 (X*) = 0 according as S" - X* = 0 or sn - X* =F 0, and hence (see 1.1.3, exercise 19) according as X - Y is or is not open, or equiv-


alently, according as Y does or does not contain fr X. In view of the isomorphisms (7) and (4), the proof is complete.

Theorem 5. Let X and Y (X be non-empty, compact subsets of R", n:2; 1, such that X - Y is connected and non-empty. Then H"+l (X, Y) Ri I or H,,+l (X, Y) = 0, according as X - Y is or is not open, or equivalently, according as Y does or does not contain Ir X.

Prool. Let f: R"---7 S" - P be the inverse of the stereographic projection (see 1.2.2). Thus I is a homeomorphism from R" onto S" - P (the n-sphere with the north pole P deleted). The pair (X, Y) is carried by I into a pair (X*, Y*) satisfying the assumptions of theorem 4. Observe now that X - Y is open in R" if and only if f (X - Y) = X* - y* is open in S" - P (since I is a homeomorphism). On the other hand, since S" - P is op~n in S", clearly X* - y* is open in S" - P if and only if it is open in S". Thus X - Y is open in R" if and only if X* - y* is open in S". Since Hn+l(X*, Y*) RiH"+l(X, Y) by 1.6.2, remark 2, and since by 1.1.3, exercise 19 the set X - Y is open if and only if jr X (Y, theorem 5 appears as a corollary of theorem 4.

Theorem 6. Let Y be a non-empty, compact, proper subset of S", n?;. 1. Denote by q(CY) the number of components of CY. Then (see 1.3.1)

(8)

Proof. Since 5" is locally connected and separable, the components of CY constitute a finite or countably infinite sequence of non-empty, pair-wise disjoint domains 1-;., ... , r;., ... (see I.1.3, exercise 28). The set Y" = C 1; is a non-empty, compact, proper subset of S", and C Y" = I~ is connected. By I. 7.4, theorem 3 and I. 7 .2, theorem 2 it follows that

(9)

The groups H"(Y,J, H"-,-l(S", Yk ), Hn-l-l(S"), H,,+l(y,) are consecutive terms of the cohomology sequence of the pair (S", YJ- The first and the last group are trivial by (9), and hence by I.3.3, lemma 4 and I.7.1,

(10)

By the structure theorem in 1.6.17, the group H"+l(S", Y) is the weak direct sum or the direct sum of a sequence of its subgroups which are isomorphic, term for term, to the groups of the sequence Hn+l (S", Yk ),

k=1, 2, ... , and thus (8) follows in view of (10).

Theorem 7. Let Y be a non-empty, compact, proper subset of S", n"2 1, and denote again by q (C Y) the n urn ber of com ponen ts of C Y. Then

H" (Y) ~ 1'i(CY)-l. ( 11)

§ 1. 7 Cohomology groups in Euclidean spaces. 107

Proof. Let r be a component of CY, and let 0 be the union of the other components of CY (thus 0 is empty if CY is connected). As noted in the proof of theorem 6, the components of CY are open sets. It follows that the sets X = Y U 0, y* = Y U T are closed (and hence compact). Indeed, clearly

. CX=T, Cy* =0 ( 12)

are open. (12) shows that X is a non-empty, compact, proper subset of 5" such CX is connected. Hence, by 1.7.2, theorem 2 and 1.7.4, theorem 3,

H"(X) = 0, H"+1 (X) = 0. ( 13)

The groups Hn(x), H"(Y), H,,+l(X, Y), H"+1(X) are consecutive terms in the cohomology sequence of the pair (X, Y). The first and the last group are trivial by (13). Hence, by 1.3.3, lemma 4,

H" (Y) R:j Hn+l (X, Y). (14)

We verify next that

H,,+l (X, Y) R:j Hn+l (5", Y*). (15 )

Clearly S"-T=X, Y*-T=Y, and thus (15) follows from the eXClSlOn theorem in 1.6.14. Obviously, the number of components of CY* is one less than the number of components of CY. By theorem 6 it follows that

( 16)

and (11) is obtained finally by means of the isomorphisms (14) to (16).

Theorem 8. Let Y be a non-empty, compact subset of R", n~ 1, and denote by k (CY) the number of boztnded components of CY. Then

H" (Y) R:j Ik (CY).

Proof. By 1.2.2, exercise 10, the set Y is homeomorphic to a nonempty, compact, proper subset y* of 5" such that k (CY) = q (CY*) -1, where q(CY*) denotes the number of the components of CY*. By 1.6.2, remark 2 and the preceding theorem 7 we conclude that

H" (Y) R:j H" (Y*) R::; IQ(Cl'*)-l = Ik(Cy).

Theorem 9. If the non-empty, compact subsets Y1 , 1'2 of R", n~ 1, are homeomorphic, then the number of the components of CYr is equal to the number of the components of CY2 •

Proof. Let again k (CY,), i = 1,2, denote the number of the bounded components of CYi . Since (see 1.2.1, exercise 6) the total number of the components of CYi is k(CYi ) +2 if n=1 and k(CYi)+1 if


n~2, it is sufficient to show that

k (Cii) = k (CY2).

Now we have, by 1.6.2, remark 2,

Hn (ii) ~ Hn (Y2).

In view of theorem 8 it follows that

Ik(CY,) ~ Ik(CY,).

By 1.3.1, exercises 5 and 6, the isomorphism (19) implies (17).

(17)

(18)

(19)

Theorem 10 (the JORDAN-BROUWER theorem). If the subset Y of R", n:;:;; 2, is homeomorphic to the (n -i)-sphere S"-1, then the following holds.

(i) CY has precisely two components D1 , D 2 •

(ii) One of these components is bounded, while the other one is unbounded. Both of these components are domains.

(iii) Ir Dl = Ir D2 = Y. Proof. Since Y is homeomorphic to 5"-1, n:2 2, Y is a non-empty,

compact set which does not reduce to a single point, and precisely one component of CY is unbounded (see 1.2.1, exercise 6). Furthermore, by 1.2.1, exercise 4, each component of CY is open and connected, and hence it is a domain. By 1.1.3, exercise 33, for each component D of CY we have the inclusion Ir D (Y. Thus the theorem will be verified if we show yet that (i) is true and that

Y(/rD;, i=1,2. (20)

Now since Y is homeomorphic to sn-l, by 1.7.1 and 1.6.2, remark 2 it follows that H" (Y) R3Hn (sn-l) ~ I. By theorem 8 it follows that the number of bounded components of CY is equal to one. Since n~2, we conclude by 1.2.1, exercise 6 that the total number of components of CY is equal to two, and (i) is proved. Now let Xo be any point of Y, and consider the spherical neighborhood

where r> 0 is a real number. We show first that

(21)

Suppose that (21) fails to hold for some ro>O. Then it also fails to hold for any r such that 0 < r < ro' since LI,(xo) decreases as r decreases. Choose now r < ro in such a manner that Y -Ll,(xo) =l= 0 (this is possible since Y does not reduce to a single point). Consider the set

F = C (Dl U D2 U LI,(xo»).


This defining formula can be re-written in the form

It follows that F is a non-empty, compact subset of Y. Furthermore, F is a proper subset of Y, since it does not contain the point xoEY. Since Y is homeomorphic to sn-l, we conclude that F is homeomorphic to a non-empty, compact, proper subset of sn-l, and hence by 1.7.2, theorem 2, we have H n (F) = O. By theorem 8 it follows that CF is connected. Now

Since (21) fails to hold by assumption, ,1,(xo) does not intersect both Dl and D 2 • As D 1 nD2 = 0, it follows from 1.1.3, exercise 15, that CF is disconnected, a contradiction. Thus (21) is established. Since (21) holds for every l' > 0, by 1.2.1, exercise 3 we conclude that

xoED;, i=1,2. (22)

On the other hand, Xo E Y = C (Dl U D 2) = C Dl n C D 2 , and thus

xoECD;, i=1,2. (23)

Since Di is open, (22) and (23) imply that

Xo ED; n CD, = 11' D" i = 1, 2.

As Xo was an arbitrary point of Y, (20) follows, and the proof is complete.

Theorem 11. Let Y be a non-empty set in Rn, n?;;, 1, and let Xo

be a point of Y. Then xoE int Y if and only if there exists a pair of subsets A, B of Y with the following properties.

(i) 0 =f= B e A e Y. (ii) A and B are compact. (iii) A - B is non-empty and connected. (iv) Xo E A - B. (v) Hn+l(A, B) R::!I.

Prool. Suppose first that xoE int Y. Then for r >0 sufficiently small the spherical neighborhood ,1 =,1 r (xo) satisfies the inclusions xoE,1, 3 e int Y. Define the sets A, B as follows: A = 3, B = 11' 3. Then (i), (iij, (iii), (iv) hold (see 1.2.1, exercise 1). As A - B =,1 is open, we conclude from 1.7.4, theorem 5 that (v) also holds.

Suppose now that there exists, for the point xoEY, a pair of sets A, B satisfying the conditions (i) to (v). From 1.7.4, theorem 5 we conclude that A - B is open, and hence A - Be int Y (see 1.1.3, definition 4). Thus xoEA -Be int Y, and the theorem is proved.

Theorem 12 (invariance 01 the interior and 01 the Irontier). If the subsets Y1 , Y2 of Rn, n:;;;:, 1, are homeomorphic and if T: Y1 --J> Y2 is

110 Part II. Topological study of continuous transformations in Rn.

a topological mapping from 1'1 onto Y2 , then Tint Yl = int Y2 •

Furthermore, if Yl (and hence also Y2) is compact, then T Ir Yl = Ir Y2 .

Prool. Take a point xlf int Yl . By theorem 11, we have then a pair of compact, non-empty subsets AI' BI of YI such that Bl (AI, Al - BI is non-empty and connected, Xl E Al - BI , and HnH (AI' Bl ) ~ I . Put x2 = T Xl' A2 = TAl' B2 = T BI . Since T is a topological mapping, by 1.1.5, exercise 8 it follows that A 2 , B2 are compact, non-empty subsets of Y2 , B 2(A 2, A 2 - B2 is non-empty and connected, x2EA 2 - B 2 ,

and (by 1.6.2, remark 2)

Hn+I(A2' B 2) ~ Hn+l(AI' BI ) ~ I.

By theorem 11 we conclude that X2t int Y2 . Since Xl was an arbitrary point of int Yl , it follows that Tint Yl ( int Y2 . The same argument applied to T-I yields the complt;mentary inclusion Tint Y; )int Y2 •

If Yl , Y2 are compact, then Ir Yi = Yi -int Y;, i = 1, 2, and the relation T Ir Yl = Ir Y2 appears as a direct consequence of the relation Tint Y1 = int Y2 •

Part II. Topological study of continuous transformations in Rn.

§ 11.1. Orientation in R n 1.

11.1.1. Bounded domains in Rn. The integer n:2;1 will be arbitrary but fixed until further notice. A bounded domain D in R" is a nonempty, bounded, connected, open set. A pair of non-empty compact sets X, Y in R" will be said to form a Irame for D if X) Y and X - Y = D. The symbol [X, Y, DJ will be used to refer to this situation. For brevity, we shall speak simply of the frame [X, Y, DJ. Thus the use of this term is merely an abbreviation for the following set of statements.

(i) X and Yare non-empty compact sets in Rn. (ii) X )Y. (iii) X - Y = D is non-empty, open, and connected.

By 1.7.4, theorem 5 we have the relation

Hn+l (X, Y) ~ I

for every frame [X, Y, DJ in R".

(1 )

1 The relationship between the abstract concept of orientation and the intuitive idea of orientation should become clearer to the reader after he had completed the study of Parts V and VI which deal with the case of the line and of the plane.

§ II.1. Orientation in R". 111

For any frame [X, Y, D], the following holds by 1.1.3, exercise 31.

(iv) Ir D and D are non-empty and compact. (v) X=DUY,lrD=Dny. (vi) D)/rD, D-jrD=D.

In view of (iv) and (vi), we have the special frame [D, Ir D, DJ. From (v) it follows that

(vii) (D, jrD) ((X, Y) for any frame [X, Y, D].

Accordingly, for a given D, the frame [D, Ir D, D] is the smallest available frame [in the sense of the inclusion stated in (vii)]. This smallest frame [D, jrD, D] will be termed the reduced Irame for D. From (iv) and (v) we obtain, by the strong excision theorem as stated in 1.6.14, remark 2, the excision isomorphism

e: H"+1 (X, Y) ~~ Hn+1 (D, Ir D) , (2)

where the symbol i":::! indicates that e is an isomorphism onto. Applying (1) to the reduced frame, there follows the relation

Consider now a pair of bounded domains D1, D2 such that

On setting F12 = D2 - D1,

we have (see 1.1.3, exercise 7) the following relations.

D2 ) F12 ) 11' D2 =l= 0, D2 - F12 = D1,

D2 =[51 UF12 , 11' D1 = [51 n F12 =l= 0.

(3)

(4)

(5)

(6)

(7)

From (5) and (7) we see that F12 is a non-empty compact set. In view of (6), we obtain a homomorphism

(8)

induced by the inclusion mapping (D2' Ir D2) -+ (D2' F12). Furthermore, the relations (7) yield, by the strong excision theorem in 1.6.14, remark 2, the excision isomorphism

(9)

The relations (6) also yield the frame [D2' F12 , D1]. Hence [see (1)J

Hn+1(D2' F12) i":::! I. (10)

Lemma 1. The homomorphism i12 occurring in (8) is an isomorphism onto.


Proof. Assume first that 11, = 1. Then D1 , D2 are merely bounded open intervals in Rl (the real number-line), and clearly fr D2 is a deformation retract ofF12 . Accordingly, H1(F12' frD 2) = 0, H2(F12' frD 2) = 0 by 1.6.16, theorem 2. The cohomology sequence of the triple (152, F 12 , fr D2) contains the section

As we just observed, the first and the last group in this section are trivial. Hence (see 1.3.3, lemma 4) i12 is an isomorphism onto, and the lemma is proved for the case 11, = 1. Assume now that n ~ 2. Choose a point xOED1 , and consider the open spherical neighborhood

Do:l!x-xoll<r.

Since Dl is open, we have the inclusion 15o(Dl and hence also 15o(D2' if r > 0 is chosen sufficiently small. Furthermore (see 1.2.1, exercise 1), Do is a bounded domain, and fr Do is connected. As a consequence (see 1.1.3, exercise 34) the set D2 - Do is non-empty, connected, and not open. We set now F02 = 152 - Do, and assert that

( 11)

To see this, observe that Fo 2, fr D2 are non-empty compact sets such that F o2 ) fr D2 [by (6) applied to the domains Do, D2J. Also,

F02 - f1' D2 = (152 n C Do) n C jr D2 = (D2 n C Dol n (C D2 U D2)

= D2 n D2 n C Do = D2 n C Do = D2 - Do·

Thus F02 - jr D2 is non-empty, connected, and not open. Accordingly, (11) follows by 1.7.4, theorem 5. Next, by (10) applied to the domains Do, D2,

(12)

Consider now the following diagram, where all the homomorphisms are induced by inclusion mappings (which are available by virtue of inclusions already established).

IH~ Hn+l(Dl ,Fat) io t ' H"·' (D t ,fI"D,)

Fig. 35.

§ 11.1. Orientation in Rft. 113

The portion of this diagram containing i02 and k is a three-term section of the cohomology sequence of the triple (D2' F02 ' IrD2)' The third group of this section is trivial by (11). Hence, by 1.3.3, lemma 1, io 2

is onto. Since i02 =i12l by the vectorlaw, it follows (see 1.3.2, exercise 3) that i12 is also onto. But the groups joined by i 12 are both isomorphic to I [see (10) and (3)]. By 1.3.2, exercise 4, the fact that i l2 is onto implies therefore that i l2 is an isomorphism onto, and the lemma is proved.

Our next lemma is concerned with two frames [Xl' li, DIJ, [X2' Y2' D2J satisfying the condition

(13)

The inclusion mapping (X 2' Y2) ~ (Xl' li) induces then a homomorphism

(14)

In view of 1.1.3, exercise 42, we have either DI (D2 or else DI n D2 = 0 under the present circumstances.

Lemma 2. The homomorphism il2 occurring in (14) is an isomorphism onto or a zero homomorphism (see 1.3.2) according as DI (D2 or D1 nD2 = 0.

Prool. For the case DI (D 2 we use the following diagram.

Fig. 36.

Since DI (D2 by assumption, the isomorphisms el2 and i l2 are available, in the sense of (9), (8) and lemma 1. The excision isomorphisms el

and e2 are used on the basis of (2). Finally, t is obtained as follows. We have obviously 152 (X1, since 152 (X2 (X1. Also,

Fl2 = 152 - DI(Xln CD! (X1n (CXI Uli) = 1;,

since Xl) li. Thus the inclusion mapping (152 , Fl2) ~ (Xl' li) is available, and t is the homomorphism induced by this mapping. In summary, all the homomorphisms in the diagram are induced by inclusion mappings. The vector law yields the relations e2 il2 = i 12 t

Rado and Reichelderfer I Continuous Transformations. 8


and el2 t = el , and hence

Since each factor on the right is an

k

(15 )

isomorphism onto, it follows that j12 is also an isomorphism onto.

For the case when DI n D2 = 0, we use the diagram in Fig .37.

The excision isomorphism e2 is used again in the sense

H"·' (Y, .",) HnT'(-D f D ) .1. ------. z, r 2. of (2), while k is induced m Fig. 37. by the inclusion mapping

(1";., 1";.) --+ (Xl' YI ). Finally, m is induced by the inclusion mapping (D2' fr D 2) --+ (1";., 1";.), which is available in view of the inclusion

(16)

To verify (16), note first thatD2(X2 (XI , and alsoD2 (CDI = CXI U YI ,

since now DI n D2 = 0. Thus clearly D2 ( 1";., and hence a fortiori fr D2 (152 (Yr, establishing (16). Take now any element u E Hn+I (Xl' 1";.). The vector law yields

(17)

since kuE Hn+l (YI , 1";.) = 0 (see 1.6.1, remark 3). As e2 is an isomorphism onto, (17) implies that jl2u=O for every uEHn+l(Xl' 1";.) and the proof of lemma 2 is complete.

n.1.2. Standard isomorphisms. Given in R" two bounded domains D2 and DI (D 2 , we introduce the auxiliary isomorphism

Hn"(D, .f,.D, }---='---~ .. Hn"(Dz .f,.D.) s,.

Fig. 38.

SI2:Hn+I(~I,frDI) "'''} (1) Hn+l (D2' fr D 2 )

on the basis of the diagram in Fig. 38.

The isomorphisms eI2 , i l "

are used in the sense of (9), (8), and lemma 1 in n.1.1, and Sl2 is defined by

sl2 = i l2 ell. (2)

Since both of the factors on the right are isomorphisms onto, s12 is also an isomorphism onto.

Consider now three bounded domains D I , D2 , Da such that DI (D 2 (D 3 • There arise, in the sense of the preceding definition,

§ 11.1. Orientation in R". 115

three auxiliary isomorphisms S12' S23' S13' We proceed to verify the vector law

(3)

To save space, Hn+1 is written merely as H in this diagram. The sets Fl3 , F23 and the isomorphisms el3 , il3 , e23 , in are associated with the pairs of domains Dl (D3 and D2 (D3 in the same manner as F 12 , e12 , i l2

were associated with the pair of domains Dl (D 2 • Accordingly

Finally, the homomorphisms k, l, m are induced by the inclusion mappings

([52' F12) ---'? ([53' Fd, ([53' F 23) ---'? ([53' F1S) , ([52' Ir D 2 ) ---'? ([53' F13 ) ,

which are available in view of the obvious inclusions

Thus all the homomorphisms in the diagram are induced by inclusion mappings. By the vector law we conclude now that

S23 S12 = (i23 e;l) (i12 eli) = i23 e;li12 k ell = i 23 e;J m ell = i23 l ell = i 13 ell = S13'

and (3) IS proved. 8*

116 Part II. Topological study of continuous transformations in R".

Consider now the special case of (1) when D1 =D2. Then clearly F12 = 152 - D1 = 151 - D1 = /r D1. Accordingly, by remark 1 in 1.6.2, it follows that e12 and i12 reduce in this case to the identity homomorphism in the group Hn+1(151' /r D1), and hence the same is true for S12' We express this fact by the formula

Su = 1. (4)

Let now D1 , D2 be two bounded domains in arbitrary relative position (thus neither D1(D2 nor D2(D1 is required). We proceed to define a standard isomorphism

in the following manner. Take any bounded domain D3) D1, D3) D2. Then 0"12 is give~, in terms of the morphisms S13 and S2 3' by means of the agreement

(5)

D3 such that auxiliary iso-

(6)

To justify this definition, we have to show that on taking another bounded domain D4 such that D4 )D1, D4)D2, we have the relation

(7)

To prove (7), take a bounded domain D5 such that D5)D3, D5)D4 •

By the vector law (3), applied with the appropriate subscripts, we obtain

and similarly

and (7) follows. Thus 0"12' as defined by (6), is independent of the choice of D 3 .

Lemma 1. If D1 =D2 , then 0"12 reduces to the identity homomorphism in the group H,,+1 (151 , /rD1l. Symbolically O"ll = 1. This is a direct consequence of the defining formula (6).

Lemma 2.

Proof· By the definition (6), we have 0"12 = S2"i S13 , 0"21 = sli s23 , and the assertion follows.

Lemma 3 (the vector law). If D1, D 2 , D3 are any three bounded domains, then 0"13=0"230"12'

Proof. Take a bounded domain D4 )D1 UD2UD3. Then 0"12 = s2"is14 , 0"2:J~s2"ls21' 0"13= S;;i S14 , and the assertion follows.

§ H.1. Orientation in R". 117

Consider now two frames [Xl' Y1' D I], [X2' Y2, D 2] in arbitrary relative position. We proceed to define a standard isomorphism

1'12: Hn+1(XI' Yr) ~~ Hn+1(X2' Y2)

by means of the formula

In this definition, el and e2 are the excision isomorphisms

e1 : Hn+1 (Xl' Yr) ~ ~ Hn+1 (D1' Ir D1) ,

e2: Hn+1(X2' Y2) !':!..~Hn+1(D2' Ir D2),

(8)

(9)

explained in connection with (2) in 11.1.1, while al2 IS the standard isomorphism

a12 : Hn+1 (D1' Ir D1) "" > Hn+ 1 (D2' Ir D2) .

Lemma 4. If both of the frames [Xl' Yr, DI ] and [X2' Y2, D2] are reduced, then 1'12 = a1 2.

Prool. The assumption means that X;=D;,Y;=lrD;,i=1,2. By 1.6.2, remark 1, e; in (9) reduces then to the identity homomorphism in Hn+1 (D" Ir D i)' i = 1, 2, and the assertion follows from the definition (9).

Lemma 5.

This is a direct consequence of the definition, in view of lemma 2.

Lemma 6 (the vector law). If [X;, Y" D;], i = 1,2,3, are any three frames, then 1'13 = 1'231'12.

Proof. In terms of the excision isomorphisms

we have by definition the relations 1'13 = eJ1a13e1' 1'23 = e31a23e2' 1'12 = e21 a12 e1, and the assertion follows in view of lemma 3.

Lemma 7. Let [Xl' Yr, D1], [X2' Y2, D2J be two frames such that (Xl' Yr)) (X2' Y2)· Then the homomorphism

112: H"+1(X1' Yr) -+Hn+1(X2' Y2),

induced by the inclusion mapping (X2' Y2) -+ (Xl , Yr)' is related to the standard isomorphism 1'12 [given by (9)] as follows: i12 = 1'12 if D1(D2, and 112 is a zero-homomorphism if Dl nD2= 0.

Proof. Let us recall that the assumption (Xl' Yr)) (X2' Y2) implies that either D1 (D2 or else D1 n D2 = 0. In the second case, i12 is a zero-homomorphism by 11.1.1, lemma 2. On the other hand, if D1 (D 2 ,

then by (15) in 11.1.1 we have j12=e21i12ellel. Since D1,~D2' we


can choose Da=D2 in the formula (6) for (]1Z, obtaining in view of (4) the relation (]12 = S;:lS12 = S12' Finally, S12 = i12 e1l by the definition (2). Hence by (9)

Lemma 8. If [Xl' 1';"D1]=[X2 , Y2 ,D2], then the standard isomorphism 1'12 reduces to the identity homomorphism in the group H"+!(X1' 1';,). Symbolically: 1'11 = 1.

This follows directly from lemma 7, in view of remark 1 in 1.6.2 (or else more directly from the definition of Td.

11.1.3. Simultaneous orientation of frames in R U • For any frame [X, Y, D] in Rn, the group H"+!(X, Y) is infinite cyclic by H.1.1 (1). Accordingly, this group contains precisely two elements g', gil each of which generates the group, and these two elements satisfy the relation gil = - g'. Orienting the frame [X, Y, D] means, for our purposes, the selection of one of g', gil as the preferred generator for the group. We use the following process to orient simultaneously all the frames in R". Consider the (n -1)-sphere 5 n - 1 and the n-cell En in R" (see 1.2.2). The cohomology sequence of the pair (En,5,,-1) contains the section

H" (En) -!>- H" (5"-1) 0';-1 ~ Hn+! (En, 5 n - 1) -!>- Hn+1 (En) ,

where the un designated homomorphisms are induced by inclusion mappings (see 1.6.4). The first and the last group are trivial by 1.7.1. Accordingly, by 1.3.3, lemma 4, O~-1 is an isomorphism onto. In symbols:

(1)

By 1.7.1, the group H"(5n - 1) is infinite cyclic. Letg"_.1 denote one of the two elements of this group which can be used to generate it. Once chosen, this generator gn-l will be kept fixed, and will be termed the pre/erred generator for 5 .. - 1. Observe now that fr En = sn- 1, and int En is non-empty, open, and connected (see 1.2.2, exercise 5). Thus we have the frame [En, 5"-1, int E"]. Accordingly, the group H,,+1(E",5,,-1) is infinite cyclic [a fact which is clear from (1) also], and for this group the isomorphism (1) yields a preferred generator in the form O~-1 (1"-1'

Consider now any frame [X, Y, D] in R". The standard isomorphism l' (see H.1.2) from H,,-1(E", 5n- 1) onto H"+1(X, Y) yields the preferred generator TO~-1 g,,-1 for the (infinite cyclic) group H"+l (X, Y). This preferred generator will be denoted, if such explicit notation is needed in some situation, by g [X, Y, DJ. In view of lemma 8 in II.1.2, this definition leads to no discrepancy for the special frame [E", 5,,-1, int PJ.

§ 11.1. Orientation in R". 119

Lemma 1. If [Xl' Y;., D1], [X2' Y2' D2] are two frames in R", then

i"12 9 [Xl' Y;., D I ] = 9 [X2' Y2 , D2],

where i"12 is the standard isomorphism defined by (8), (9) in 11.1.2.

Prool. Denote by i";, i = 1, 2, the standard isomorphism from H"+1(E", 5"-1) onto H"+1(X;, Y;), in the sense of 11.1.2 (8), (9). By definition

On the other hand, the vector law yields (see 11.1.2, lemma 6) i"12 i"l = i 2 •

We conclude that

i12 9 [Xl' Y;., D1] = i12 i l 15:-1 g,,-l = i2 d~-l g,,-1 = 9 [X2' Y2 , D2]·

Lemma 2. Let [Xl' Y;., DI ], [X2' Y 2 , D2] be two frames in Rn such that (Xl' Y;.) (X2' Y2). On denoting by

j12: Hn+1 (Xl' Y;.) -+ Hn+1 (X 2' Y2)

the homomorphism induced by the inclusion mapping (X2' Y2) -+ (Xl' YI ),

one has

jI2 9 [Xl' Yl , D1] = 9 [X2' Y2 , D 2] or j12 9 [Xl' YI , Dl] = 0,

according as Dl (D 2 or Dl nD2 = 0. This is a direct consequence of the preceding lemma 1 and of 11.1.2,

lemma 7.

11.1.4. Positive orientation of frames in Rn. In the preceding discussion, the choice of the preferred generator 9n-l for the group Hn (5"-1) was quite arbitrary. For certain purposes, it is convenient to make this choice a definite one (corresponding to the choice of the positive or counter-clockwise orientation in the Euclidean plane R2, for example). Starting with the case n = 1, we have to consider first the matter of making a definite selection for a generator for the infinite cyclic group HI (50). Now 5° consists of the two points -1 and +1 on the real number-line Rl. By 1.6.1, remark 8, the group Hl(5°) is isomorphic to (and indeed may be considered as identical with) the group P of the 1-cocycles Zl of 5°. Furthermore, as it has been proved there, we have at our disposal the explicit isomorphism h:Zl~ .. I, given by the formula hZl=Zl(-1, 1). t Thus the (unique) 1-cocycle Zl for which Zl (-1, 1) = 1 corresponds to he generator + 1 of I, and hence this particular 1-cocycle is a generator

of P=Hl(5°). We select this generator as the preferred generator 90 for HI (50). Thus (see 1.6.1, remark 8) 90 is the 1-cocycle of 5° which is fully determined by the following table of its values:

go (- 1, 1) = 1 , go (1, - 1) = - 1, go (- 1, - 1) = 0, go (1, 1) = 0.


There follows a natural selection of a preferred generator 9,,-1 for Hn(sn-1), for all values of n;;;;; 2, based on the explicit isomorphism

H" (sn-1) ~~ Hn+1 (S") ,

discussed in the remark to 1.7.1. Denoting that explicit isomorphism by hn - 1 for brevity, we define successively

B1=ho go, ... , 9,,= h"_l gn-l' ... ,

as preferred generators for H2 (SI) , ... ,Hn+1(S"), .... Once 90' gl'''' have been selected in this manner, the process described in II.1.3 yields, for each n;;;;;1, preferred generators 9 [X, Y, DJ for the frames in R". For convenience, the orientation so constructed will be referred to as the positive orientation of the frames in R".

§ 11.2. The topological index 1.

11.2.1. Preliminaries. We shall operate in Euclidean n-space R", where the integer n?:.1 is arbitrary but fixed. Let

T: U -+Rn (1)

be a continuous mapping from a non-empty subset U of R" into R". To increase clarity of notations, x will be used as a generic notation for a point of R", while u will be used to refer to a generic point of U (thus a point denoted by x mayor may not lie in U). Let us recall (see 1.1.1) that the concept of a mapping T has been so formulated that it includes the set from which T operates. Accordingly, if we have occasion to consider the mapping T in (1) as operating from a subset U* (U, then we shall write TI U* to refer to T cut down to U*. At times, when results from general cohomology theory will be applied, U will play the role of space, and to avoid ambiguity we agree that terms like open set, closed set, closure will always be used relative to RI!. If we wish to consider, for example, frontier relative to U, then some special notation like fr* will be employed to secure clarity.

In the present § II.2 we shall be concerned mainly with the following situation. Given the continuous mapping T as in (1), a bounded domain D will be considered such that

I5 (U. (2)

1 While it is now customary to treat the topological index in terms of cohomology groups, a reader desiring to achieve a better understanding of the geometrical motivation should study the excellent presentation of the degree of a mapping (Abbildungsgrad) in terms of singular homology groups in the ALEXA1\"

DROFF-HoPF treatise listed in the Bibliography. In Part VI we shall discuss the description of the topological index in terms of the variation of the argument for the case of the plane.

§ II.2. The topological index. 121

The following definitions will be used.

Definition 1. Given T and D as in (1), (2), a point x is termed (T, D)-admissible if x(f TfrD.

Definition 2. Given T and D as in (1), (2), a frame [A, B, Ll] in Rn (see 11.1.1) is termed (T, D)-admissible if the relation

TID: CD, fr D) -+ (A, B) (3 )

holds. Thus the frame is (T, D)-admissible if and only if T15CA, TfrDCB.

Definition 3. Given T and D as in (1), (2) and a point x, a frame [A, B, Ll] in Rn is termed (x, T, D)-admissible if it is (T, D)-admissible and xELl.

Definition 4. Given a point x and a real number r such that 0< r< 1, the spherical frame [Ar(x), B,(x), Ll,(x)] with center x is defined as follows:

A,(x) = {x' Illx' - xii;;:;; 1fr},

B,(x) = {x'lr;;:;;llx' - xii ;;:;;1/r},

Ll,(x) = {x' Illx' -- xii < r}.

Clearly (see 1.2.1, exercise 2), the spherical frame is a frame in the sense of 11.1.1. The domain Ll,(x) is the open spherical neighborhood of radius r for x (see 1.1.4, definition 2).

Lemma 1. If x is (T, D)-admissible, then there exist (x, T, D)admissible frames. In particular, if r>O is sufficiently small, then the spherical frame [Ar(x), Br(x), L1r(x)] is (x, T, D)-admissible.

Proof. Since D in (2) is bounded, by 1.2.1, exercise 4, the sets 15 and frD are compact and non-empty (see 1.1.3, exercise 18). Hence (see 1.1.5, exercise 8) the sets T 15 and T fr D are also compact. By assumption, x(fTfrD. Since CTfrD is open, for r small enough we have the inclusion TfrDCCLl,(x). Also, since T15 is compact and hence bounded, for r > 0 sufficiently small we have T 15 C Ar (x). On choosing r sufficiently small for both of these purposes, we shall have T15(A r(x), TfrD(Br(x), and the lemma is proved.

Lemma 2. A point x is (T, D)-admissible if and only if .on T- 1 xCD.

Proof. Since .o=DUfrD and DnfrD= 0, one has

.0 nT-l x = (D nT- 1 x) U (/r Dn T-1x) , (D n T- 1 x) n (/1' D nT-l x) = 0.

Now x is (T, D)-admissible if and only if fr D n T-l x = 0, so the lemma is obvious.

Definition 5. Given T and D as in (1), (2), let {D j } be a (finite or infinite) sequence of pair-wise disjoint domains in D. Then the sequence {DJ is termed (x, T, D)-complete if .on T-l x(UD j •


Remark J. If {Dj} is (x, T, D)-complete, then Dn T-1x(UDj(D, and hence x is (T, D)-admissible by lemma 2.

Remark 2. If xEf TD, then Dn T- 1x=0, and hence any sequence {DJ of pair-wise disjoint domains in D is (x, T, D)-complete.

Remark 3. If x is (T, D)-admissible, then in view of lemma 2 the sequence consisting of D alone is (x, T, D)-complete.

Definition 6. Given T and D as in (1), (2), a domain D*(D is (x, T)-admissible if xEf TfrD*. Equivalently, the domain D*(D is (x, T)-admissible if x is (T, D*)-admissible.

Lemma 3. Let the sequence {Dj} be (x, T, D)-complete, in the sense of definition 5. Set D - U D j = Y. Then the following holds.

(i) 0=t=frDj(Y' (ii) Each domain D j is (x, T)-admissible. (iii) fr D j coincides with the frontier of D j relative to D. (iv) The number of those domains D j for which Djn T-1x=t= 0

is finite.

Proof. Set 0 = UD j . By 1.1.3, exercise 4, it follows that IrDj( CO. Also, fr Dj(Dj(D. Thus

fr D j (15 nco = D - 0 = Y,

and (i) is proved (note that Ir D j =t= 0 by 1.1.3, exercise 18). Observe now that DnT-1x(UDj by assumption, and hence YnT-- 1x=0. Thus xEf TY, and in view of (i) (which we already proved) it follows that xEf TfrDj, proving (ii). To verify (iii), note that since 15 is closed in Rn, the terms closed set and closure 01 a set have the same meaning relative to D as relative to R". Accordingly, if fr* Di is the frontier of Dj relative to D, then

fr* Di = Din (15 - Dj) = Dj n D n C Dj = D j n C Dj = fr Dj ,

where we used the fact that Dj n 15 = Dj as a consequence of the inclusion Di ( D.

Finally, to prove (iv), observe that the set

Dn T-1x=(TID)-lX

is a closed subset of D (see 1.1.5, exercise 4), and hence it is compact sinceD is compact (see 1.1.3, exercise 22, 1.2.1, exercise 4). Byassumption, the domains D j cover Dn T-l x. Since each D j is open and D n T-l x is compact, it follows that D n T-l x is covered by a finite subsequence of the sequence {D j }, and (iv) is proved.

Lemma 4. Let x be (T, D)-admissible and let {D1}, j = 1, ... , m, be a finite sequence of pair-wise disjoint (x, T)-admissible domains

§ 11.2. The topological index. 123

in D such that {D j } is not (x, T, D)-complete. Then {D j } can be extended to a sequence D1 , ... , Dm , Dm +1' . .. of pair-wise disjoint domains in D such that the extended sequence is (x, T, D)-complete.

Proof. Introduce the auxiliary set

F = fr D U 151 U ... U 15m .

Then F, as a finite union of compact sets, is compact and contains frD. Hence, by 1.1.3, exercise 5, the set O=15-F IS open. Since 15 =FUO, we obtain

15 n T- 1 x = (F n T-l x) U (0 n T-l x) .

By assumption, fr D n T- l x = 0, fr Din T- l x = 0, and hence

Fn T-lx=(UDj)n T-1x.

From (4) and (5) we infer that

Dn T-lx=((UD j ) UO) n T-lx, and hence

(4)

(5)

(6)

Since the sequence {D j } is not (x, T, D)-complete, (6) shows that 0=1= 0. Furthermore, since UDj(F and OnF= 0, we have

D j n 0 = 0, i = 1 , ... , m. (7)

Now let Dm+1 , ... , DmH , '" be the sequence of the components of o =1= 0. Note that each D m H is a domain and that the set of the components of 0 is countable (see 1.1.3, exercise 28). By (6) and (7) it follows that

Dl , ... , D"" Dm +1 , ...

is a sequence of pair-wise disjoint domains in D whose union contains Dn T- 1 x, and the lemma is proved.

Remark 4. In view of part (iv) of lemma 3, only a finite number of the new domains DmH intersect T-l x. Hence only a finite number of these new domains are needed to obtain an (x, T, D)-complete sequence. Accordingly, lemma 4 can be strengthened to the statement that the initial finite sequence {D j } can be extended to a finite (x, T, D)complete sequence of pair-wise disjoint domains in D.

II.2.2. Definition of the topological index. Given T and D as in 11.2.1 (1), (2), take a point x which is (T, D)-admissible, and select a frame [A, B, LlJ which is (x, T, D)-admissible (see 11.2.1, lemma 1). Then the mapping

TID: (15, fr D) --)o-(A, B) (1 )


induces a homomorphism

(2)

The image of the preferred generator 9 [A, E, J] of the group H,,+l (A, B) is then an element of the group Hn+1(15, fr D), and hence it is a certain integral multiple of the preferred generator 9 [D, fr D, D] (see 11.1.3). Thus we have a relation of the form

(TID)* 9 [A, B, J] = f-l 9 [.0, fr D, D],

where f-l is an integer which is uniquely determined by T, D, and the frame [A, B, J]. Let us recall that this frame must be (x, T, D)admissible.

Lemma. The integer f-l is independent of the choice of the (x, T, D)admissible frame [A, E, J].

Proof. An (x, T, D)-admissible frame [A, B, J] having been selected, consider the spherical frame [Ar (x), By (x), J r (x)]. By lemma 1 in 11.2.1, this frame is (x, T, D)-admissible if r>O is sufficiently small. If r is so chosen, then the mapping

TID: (.0, fr D) --+ (Ar (x), Br (x)) (4)

induces a homomorphism

(5)

which in tum gives rise to the relation [analogous to (3)]

(TID)~ g[Ay(x), By(x), Jy(x)] =f-l,. 9 [.0, fr D, D]. (6)

Observe now that since [A, B, il] is (x, T, D)-admissible, we have the inclusion xE J. Hence, if r> 0 is sufficiently small, we have the inclusion ilr (x) (J. Furthermore, since A is compact and hence bounded, for r sufficiently small we have the inclusion Ay(x) )A, and consequently also

Summing up: for r > 0 sufficiently small, we have the inclusions

Ar(x))A, By(x))B, Jy(x)(J. (7)

The identity mapping (A,E)--+(Ay(x), Br(x)) induces then a homomorphism

j:Hn+1(Ay(x), Br(x)) --+Hn+l(A, B),

for which lemma -2 in 11.1.3 yields the relation

j 9 [A, (x), B,(x), Jr(x)] = 9 [A, B, J]. (8)


The vector law yields (TID)~ = (TID)* j. In view of (6), (8) and (3) we conclude that

fir 9 [D, fr D, D] = fl 9 [D, fr D, D]. (9)

Since 9 [D, fr D, D] is a generator of the infinite cyclic group Hn+1 (D, fr D), (9) implies that

fl = U,. ( 10)

Now consider any two (x, T, D)-admissible frames [Ak, Bk, L1k], k = 1, 2, and denote by flt, fi2 the corresponding integers fl. If r> 0 is sufficiently small in relation to both of these frames, then (10) holds for either frame, and hence fil =fi, =fl2, proving the lemma.

In view of this lemma, the integer fi determined by (1), (2), (3) depends only upon T, D, and the point x [which must be (T, D)admissible J. Accordingly, we are justified in denoting this integer by fl(x, T,D).

Definition. fl (x, T, D) is termed the topological index of the point x, with respect to the mapping T and the domain D.

Remark. According to this definition, fl (x, T, D) may be thought of as an integral-valued function of three variables x, T, D which are subject to the following' conditions: (i) T is a continuous mapping from a set U (R" into Rn. (ii) D is a bounded domain such that D ( U. (iii) xEf Tfr D.

II.2.3. Properties of the topological index. Given T and D as in 11.2.1 (1), (2), consider a (T, D)-admissible point Xo and an (xo, T, D)admissible frame [A, B, .1]. Then T fr D ( CL1, and hence .1 n T fr D = 0. It follows that every point xEL1 is (T, D)-admissible and that [A, B, .1] is (x, T, D)-admissible for xE .1. Thus we can use this same frame [A, B, .1] in evaluating fl(x, T, D) for all the points xEL1. In vIew of the defining formulas (1), (2), (3) in 11.2.2 it follows that

fl(x, T, D) =fl(xo, T, D) for xEL1. (1 )

Consider now a component r of the (open) set C TfrD, and take any point xoEr. Then Xo is (T, D)-admissible, and by (1) there exists an open set .1 such that xoEL1 andfl(x, T, D) is constant on .1. Since r is also open, it follows that for every point xoEr there exists an open set o=rnL1 such that xoEO and fl (x, T, D) is constant on O. Since r is connected and fi(x, T, D) is integral-valued, by 1.1.3, exercise 37 we obtain the following statement.

Theorem 1. fl(x, T, D) is defined and has a constant value on each com ponent of the set C T fr D.


Remark 1. Let E be any connected subset of C TlrD. Then E is contained in a certain component r of C T Ir D, and by theorem 1 we conclude that p, (x, T, D) is defined and has a constant value on each connected subset of C T Ir D.

Remark 2. Suppose xEf TV. Then xEf TlrD, and so p,(x, T, D) is defined. Since T D is compact, for r> 0 sufficiently small the spherical frame [A,(x), B,(x), Ll,(x)] satisfies now not merely the inclusions (see 11.2.1, lemma 1) TD(A,(x), TlrD(B,(x), but also the inclusion TD(Br(x). By 1.6.2, remark 3 it follows that the induced homomorphism (T!D)* in 11.2.2 (2) is a zero-homomorphism, and thus 11.2.2 (3) yields p,(x, T, D) =0. Summarizing: if xEf TD, then x is (T, D)-admissible and p, (x, T, D) = O.

Theorem 2. If x is (T, D)-admissible and p,(x, T, D) =F 0, then xETD.

Prool. Observe first that by remark 1 it follows that xE TD. But since xEf T Ir D by assumption, one must have xE T D.

Remark 3. Consider again a component r of CTlrD, and suppose that r is not completely covered by T D. Then there exists a point xoEr - T D. Clearly Xo is (T, D)-admissible, and p, (xo, T, D) = 0 by theorem 2. Since p, (x, T, D) is constant on r by theorem 1, it follows that p, (x, T, D) == 0 on r. This discussion is applicable, in particular, to the unbounded component (see 1.2.1, exercise 6) of C T Ir D. Indeed, T D is a subset of the compact set TD, and hence T D is a bounded set. Accordingly, the unbounded component of C T Ir D is certainly not completely covered by T D, and hence p (x, T, D) == 0 on this particular component. If n = 1, then C T Ir D has two unbounded components (see 1.2.1, exercise 6), and the preceding comments apply to both of them.

Theorem 3. If the sequence {D j } is (x, T, D)-complete (see 11.2.1, definition 5), then

(1 *)

where the prime attached to the summation symbol indicates that the summation is actually finite (that is, only a finite number of terms can be different from zero).

Proof. In view of remark 1 of 11.2.1 the point x is (T, D)-admissible and p(x, T, D) is defined. By lemma 3 in 11.2.1, each domain Dj is (x, T)-admissible and hence p, (x, T, Dj ) is defined. To simplify notations, put p=p(x, T, D), Pj=p(x, T, D j ), and introduce the auxiliary sets

Y=V-UD j , Yj=D-D j •


By assumption .on T-1 x( UDj , so clearly xEf TY. Since TY is compact, it follows that for r> 0 sufficiently small the spherical neighborhood L1,{x) satisfies the inclusion

In view of the compactness of TD we conclude that for r> 0 sufficiently small the spherical frame [A,{x), B,{x), L1,(x)] satisfies the following inclusions: TD(A,(x), TY(B,(x). Since frDj(Ybylemma3 inII.2.1, there follows the inclusion

Furthermore, since Dj(D, we also have the inclusion

TDi(A,(x), j=1,2, ....

These inclusions mean that for r> 0 sufficiently small the spherical frame [A, (x), B,(x), L1,(x)] is (x, T, Di)-admissible for every j. From lemma 1 in II.2.1 we also know that this frame is (x, T, D)-admissible if r> 0 is sufficiently small. Choose 1'> 0 sufficiently small for all these purposes, and write [A, B, L1] for [A,(x), B,{x), L1,{x)] to simplify notations. For the frame [A, B, .1] we have then the relations

and the corresponding induced homomorphisms

(T!D)* :Hn+1 (A, B) ~Hn+l (.0, fr D),

(T!Dj)* :HHI (A, B) ~Hn+l (.oj, fr Dj ).

By the definition of the topological index we obtain

(T!D)* g [A, B, L1] = /k g [.0, jr D, D],

(T!Dj )* g [A, B, L1] =/kj g [.oj, fr Dj , Dj].

(2)

(3)

To this situation we apply the result in 1.6.17, remark 5. The sets now denoted by .0, jr D, A, B, Yj, Dj correspond to the sets denoted there by X, Z, X*, Y*, Yn , 0" respectively. The inclusions already noted in this section yield readily the verification of the fact that the assumptions made in 1.6.17, remark 5 are all satisfied. While the term jrontier is used at present relative to Rfl, we noted in II.2.1, lemma 3 that the frontier of D j relative to .0 is the same as its frontier relative


to R", and thus no discrepancy arises on this account. The needed portion of the diagram of 1.6.17, remark 5 appears in the present context in the following form.

Hn"(A.~)

~~. H~ /,H."'D"f'D)

Hn+I(D.~)

Fig. 40.

By formula (26) in 1.6.17, we have the relation

(TID)* 9 [A, B, Ll] = 2:' kifl(TIDj)* 9 [A, B, Ll], (4)

where the notation 2:' indicates that the summation is actually finite. In view of (3) we have

kjfl(TIDj )* g[A, B,Ll] =fJjkjtj- 1 g[Dj' /rDj' DjJ. (5)

Observe that tj is induced by the inclusion mapping (Dj' /r Dj) --+ (D, YJ Since Dj-/rDj=Dj=D- Yj, lemma 2 in II.1.3 yields

tj 9 [D, Yj, D j ] = 9 [Dj' /r D j , DjJ.

Since tj is an isomorphism onto, it follows that

tj- 1 9 [Di' /r Di , DiJ = 9 [D, Yj, DjJ. (6)

Observe next that kj is induced by the inclusion mapping (D, /r D)--+ (15, Yj). Since D-Yj=DjCD=D-/rD, from lemma 2 in II.1.3 it follows that

(7)

From (2), (4), (5), (6), (7) we obtain

fl 9 [V, fr D, DJ = (2:' fJj) 9 [V, /r D, DJ.

Since 9 [D, fr D, DJ is a generator of the infinite cyclic group H"+1(D, jr D), it follows that fJ = L'fJj, and (1 *) is proved.

Remark 4. Consider a (T, D)-admissible point x. Then Dn T- 1 xCD. Let now D* be a domain such that DnT-1xCD*(D. Then the one-term sequence consisting of D* alone is (x, T, D)-complete, and theorem 3 yields therefore fJ (x, T, D) = fJ (x, T, D*).


Definition. Given T and D as in 11.2.1 (1), (2) and a (T, D)admissible point x, let D1 , .•• , Dm be a finite sequence of pair-wise disjoint (x, T)-admissible domains in D. The difference

m

d = fl(x, T, D) - Lfl(X, T, D j ) j~l

is termed the defect of the sequence {D j }, relative to x, T, D.

(8)

Theorem 4. If the defect d (in the sense of the preceding definition) is different from zero, then there exists in D a domain D* with the following properties. (i) D* is (x, T)-admissible. (ii) D* n D j = 0, j = 1 , ... , m. (iii) sgn fl (x, T, D*) = sgn d.

Proof. Since d =f= 0, we infer from theorem 3 that the sequence D1 , •.. , Dm is not (x, T, D)-complete. By lemma 4 in 11.2.1 there follows the existence of an extension D1 , ... , D m' D m + l' ... to an (x, T, D)-complete sequence. Applying theorem 3 to this extended sequence, we obtain the formula

m

fl (x, T, D) = L fl (x, T, D j ) + L' fl (x, T, DmH) . i~l k


d = L' ,u (x, T, DmH) . k

Since d =f= 0, we must have at least one integer k such that

sgnfl(x, T, DmH ) = sgnd.

(9)

(10)

If k satisfies this condition, then clearly the domain D* = Dm+k possesses the required properties (i), (ii) , (iii).

Remark 5. If D#- is an (x, T)-admissible domain in D such that

fl(X, T, D#-) =f=fl(x, T, D),

then D# by itself constitutes a one-term sequence satisfying the assumptions of the preceding theorem. Accordingly, there follows the existence of an (x, T)-admissible domain D* (D such that D* n D #- = 0 and

sgnfl(x, T, D*) = sgn[fl(x, T, D) -fl(x, T, D#-)J.

Remark 6. Consider two continuous mappings

T: U --+Rn, T:fJ --+Rn, (11)

where U, fJ (R n , and a bounded domain D such that

D(UnfJ. ( 12) Rado and Reichelderfer, Continuous Transformations. 9


Suppose that a frame [A, B, LI] is both (T, D)-admissible and Ct, D)admissible. Then the mappings

TID,1'ID:(D,lrD)-+(A,B) (13)

induce homomorphisms

(TID)*, (1'ID)* :Hn+1(A, B) -+ Hn+1(D, Ir D). (14)

If x, x are any two points in L1, then by the definition of the topological index

(TID)d [A, B, LlJ =p,(x, T, D) 9 [D, Ir D, D], (15 )

(1'ID)* 9 [A, B, LI] = p,(x, T, D) 9 [D, Ir D, D]. (16)

Now suppose that the mappings (13) are homotopic (see 1.6.15). Then (TID)* = (TID)*, and from (15) and (16) it follows that p,(x, T, D) = p,(x,T,D) for any two points x,xELI. In particular, p(x, T,D)= p,(x, 1', D) for xELI.

Theorem 5. Given T, T and D as in (11) and (12), let LI be a bounded domain in R" such that the following holds: for every point uElrD there exists a convex set K(u)(Rn such that

Tu, TUEK(u), K(u) nLl = 0.

Then p, (x, T, D) =p, (x, T, D) for any two points x, xE LI .

Proal. For r > 0 large enough, the set A = {x II! xii:;;;; r} satisfies the conditions

A)TDUTDULI,

since the sets T D, 1'D, LI are bounded. Let r be so chosen, and put B=A -Ll, obtaining a frame [A, B, LI] which is both (T, D)-admissible and (J, D)-admissible. For this choice of [A, B, LI], we assert that the mappings (13) are homotopic (note that if this fact is established, then the present theorem follows by remark 6). For O:;;;;t:;;;;1, consider the continuous mapping

defined for uE D by the formula (see 1.2.1)

1; u = t T u + (1 - t) T u.

Since Tu, TuEA and A is convex, clearly 1;uEA, and hence

1; D ( A for 0:;;;; t :;;;; 1 . ( 17)

For uElrD, we have by assumption Tu, TUEK(u)(CLI, and hence

TIt, TltEA nK(u) (A n CLI = B. ( 18)


Noting that A nK(u), as the intersection of convex sets, is itself convex, we conclude from (18) that I;uEB if uE/rD. Hence

1; Ir D (B for 0 ~ t ;:;;; 1-

By (17) and (19) we have the relation

I;:(15,lrD)-+(A,B) for o~t:;;;;:1.

(19)

(20)

Clearly I;u is continuous on15xI, where I is the unit interval O:;;:;:t ~1, and To = ]'115, :z;. = TI15. Thus (20) implies that the mappings (13) are homotopic, and the proof is complete.

Theorem 6. Given T, T, D as in (11) and (12), let xo, Xo be two points such that Xo is (T, D)-admissible and (see 1.1.5, definition 5 and 1.1.4, definition 7),

Ilx - xoll + e (T, T, Ir D) < e(xo, T Ir D). (21)

Then,u(xo, T,D) =,u(xo,T,D).

Prool. In view of (21) we can choose r to satisfy the inequalities

Ilxo - xoll < r < e(xo, T Ir D) - e(T, T, Ir D). (22)

To simplify notations, put

e = e (T, T, Ir D).

For uE/rD, introduce the set

K(u) = {xlllTu - xII S::e}. (23)

In view of the definition of e we have then Tu, TUEK(u). For the spherical neighborhood LI,(xo), where r satisfies (22), we have the relation

K (u) n LIT (xo) = 0. (24)

Indeed, if one assumes that a point x lies in both K(u) and LI,(xo), then in view of (22) it would follow that

a contradiction. Thus (24) holds. Since K (u) is convex by 1.2.2, exercise 7, the assumptions of theorem 5 are satisfied, with LI =LI,(xo). In view of (22), clearly xo, xoE LI,(xo), and hence,u (xo, T, D) =,u (xo, T, D) by theorem 5.

Remark 7. If TllrD=TllrD, then e(T,T,lrD)=O, and on choosing Xo = xoEt T Ir D the preceding theorem yields the relation ,u(xo, T,D)=,u(xo/I,D). This result shows that the topological

9*

132 Part II. Topological study of continuous transformations in RH.

index ft (x, T, D) depends only upon the behavior of Ton jr D. Explicitly: if the continuous mappings T, T are both defined on 15, and if TI/r D =

TI/r D, then ft (x, T, D) and ft (x, T, D) are defined on the same set CT/rD=CTjrD andft(x, T,D) =ft(x,f,D) on this set.

Remark 8. On choosing Xo = Xo Ef T /r D in theorem 6, we obtain the following statement. If the continuous mappings T, T are both defined on 15, then

ft(xo, T, D) = ft(xo, T, D) if e(xo, T /r D) > e(T, T, fr D).

II.2.4. Applications to homeomorphisms. Given T and D as in 11.2.1 (1), (2), assume now that TID is one-to-one (that is, U 1 , u2 E:D and T U 1 = T U 2 imply that u1 = u2). Since D is bounded by assumption, the set D is compact (see 1.2.1, exercise 4). By 1.1.5, exercise 8 it follows that the image set

is also compact, and by 1.1.5, exercise 7 it follows further that

TID:D-+X

is a homeomorphism from 15 onto X. Put

TD=tJ,

T jrD=Y.

(1 )

(2)

(3)

(4)

Now since the mapping (2) is a homeomorphism and DCD, the mapping

TID:D-+tJ

is a homeomorphism from D onto tJ. As D is open, we have the relation D =int D. By 1.7.4, theorem 12 it follows that

tJ = TD= TintD=inttJ. (5)

Since int tJ is open, we conclude that tJ itself is open (relative to Rn). As D is connected, tJ = T D is also connected (see 1.1.5, exercise 8). Finally, tJ = T DC T D implies (since T 15 is compact and hence bounded) that tJ is bounded. Thus tJ = T D is a bounded domain. Since 15 is compact, by 1.1.5, exercise 9 we infer that

.3 =TD = T15 =X. (6)

Since D and tJ are bounded domains, and TID is one-to-one, we conclude now [see (1), (5), (6)J that

T frD = TeJ5 - D) = T15 - T D =.3 - tJ = fr tJ. (7)

In view of (1), (6), (7), we conclude that

TID: (1), fr D) -+ (.3, fr tJ) (8)


is a homeomorphism. from the pair (D, frD) onto the pair (3, fr ..1). Accordingly, the induced homomorphism

(9)

is an isomorphism onto (see 1.6.2, remark 2), a fact indicated by the symbol ~ in (9). It follows that

(TID). 9 [3, fr ..1, ..1] = t(T, D) 9 [D, fr D, D], (10)

where t(T, D) is an integer equal to 1 or to -1. In view of the defining formula (10), this integer is determined by T and D and thus the notation t(T, D) is justified.

Definition. t(T, D) is termed the index of T relative to D, and T is termed even or odd in D according as t(T, D) = 1 or t(T, D) =-1.

Remark 1. For the symbol t(T, D) to be meaningful in the sense of this definition, T must be a continuous mapping from a subset U of Rn into R", D must be a bounded domain such that the inclusion D( U holds, and TID must be one-to-one. In view of (3), (6), (7) the defining equation (10) may be re-written in the form

(TID). 9 [TD, T fr D, T DJ = t(T, D) 9 [D, fr D, DJ. (11)

Observe also that [(T, D) is determined as soon as T is known on D, and thus

t(T, D) = t{TID, D).

Remark 2. Since ..1 = TD is a bounded domain, [3, frLl, ..1] is actually a frame, and (8) implies that this frame is (T, D)-admissible. By 11.2.3 it follows that

(TID)* 9 [3, fr ..1, ..1] = J.l (x, T, D) 9 [D, fr D, DJ for x ELI. (12)

Comparison of (10) and (12) yields the following statement.

Theorem 1. t(T, D) = J.l (x, T, D) for xELI = T D.

Remark 3. Since J.l (x, T, D) is undefined for xE T tr D and vanishes if xEf TD (see 11.2.3, theorem 2), we obtain the following picture under the present circumstances. If J.l (x, T, D) is defined for a point x, then either xE T D or xE C T D. On T D, J.l (x, T, D) is defined and has the constant value t(T, D), and thus on TD it has the constant value 1 or the constant value -1. On C T D, J.l (x, T, D) is defined and has the constant value zero.

Theorem 2. If TID is the identity mapping on D (that is, if Tu=u for uE D), then t (T, D) = 1.

Proof. In this case, TID is certainly one-to-one on D, and hence t(T, D) is defined. Also, TD=D, TfrD=frD,TD=D. Accordingly,


in the defining fonnula (11) the homomorphism (TIV)* reduces now to the identity homomorphism by 1.6.2, remark 1, and thus (11) reduces to

g[V, IrD, DJ = t(T, D) g[V, IrD, DJ.

Since g[V,/r D,DJ is a generator of the infinite cyclic group Hn+1(V,/r D), it follows that t(T, D) = 1-

Theorem 3. If t(T, D) is defined (see remark 1) and D* is any domain in D, then t(T, D*) is also defined and t(T, D*) = t(T, D).

Proof. The assumptions imply that T is defined, continuous, and one-to-one on V, and that D is bounded. The inclusion D* (D implies that D* is also bounded and V* (V, and thus T is defined, continuous, and one-to-one on V*. Hence l(T, D*) is defined. Take a point u*ED* and put x*= Tu*. Then vn T-I X*=u*(D*(D, since TIV is one-to-one. By remark 4 in 11.2.3 and theorem 1 in the present section we conclude that

l(T, D*) = fl(x*, T, D*) = f-l (x*, T, D) = l(T, D).

Theorem 4. Let U1 , ~ be subsets of Rn, and let

~ : U1 -+ Rn, 7; : U2 -+ R"

be continuous mappings. Let D1 , D2, Da be bounded domains in R", such that (i) VI ( U1 , V 2 ( U2, (ii) the rela.tions

T1 IV l: (VI' Ir D1) -+(V2' Ir D2). (13)

T2Iv2:(V2• IrD2)-+(Va, IrD3) (14)

hold. and (iii) the mappings (13). (14) are homeomorphisms from the pairs (VI' IrD1), (V2' IrD2) onto the pairs (V2' IrD2). (V3• IrD3) respectively. Then l(T1 ,D1).l(T2,D2), t(T2~.Dl) are defined. and

(15 )

Proof. In view of remark 1. it is immediate that the three indices t occurring in (15) are defined. Observe now that the mappings (13). (14) yield

From the defining fonnula (10), in combination with the vector law for induced homomorphisms, we conclude that

t(T2 T1 , D1) 9 [VI' Ir D1 , D1J = (T2 ~IVl)* 9 [V3' Ir D3• D3J

= (~IVl)* (T2 IV 2)* 9 [V3' IrD3,D3J = t(7;,D2) (T1 IV1)* 9 [1)2' IrD2,D2J

= l(T2' D2) t(~, D1 ) 9 [VI' Ir D1 , D1J,

and (15) follows by 1.3.1, exercise 4.


Theorem 5. Let D be a (not necessarily bounded) domain in Rn, and let

(16)

be a one-to-one continuous mapping. Then the following holds.

(i) TD is a domain. (ii) T is a homeomorphism from D onto TD. (iii) If D1 , D2 are any two bounded domains 1D D such that

D1 ,D2(D, then t(T,D1). t(T,D2) are both defined, and

( 17)

Proof. In view of remark 1 (applied with U = D), I (T, D1) and t(T, D2 ) are both defined. Denote now by Q+ (Q-) the class of all those bounded domains D in D for which D (D and t (T, D) = 1 (- 1). Put

Then G+, G- are open (perhaps empty) sets in D. We first verify that

( 18)

Indeed, consider any point uoED. Then there exists a bounded domain D (uo) such that uoE D (uo) and D (uo) (D (for example, the spherical neighborhood LI, (uo) may be used if r is sufficiently small). Then l (T, D (uo)) is defined, and clearly uoE G+ or uoE G- according as l(T, D(uo)) equals 1 or -1. Thus (18) holds. Next we verify that

G+nG-=0. (19)

If this assertion is denied, then we should have two bounded domains D', D" in D such that D', D" (D and

l(T,D')=1, l(T,D")=-1, D'nD"=j=0. (20)

Then D=D'UD" would be a bounded domain in n such that D(D, and theorem 3 would yield

I (T, D') = t (T, D) = t (T, D"),

in contradiction with (20), and (19) is proved. Since D is connected, (18) and (19) imply that one of the sets G+, G- must be empty. Accordingly, one of the classes Q+, Q- must be empty, and part (iii) of the theorem follows.

Consider now any open set OeD, and take any point xoE TO. Then Xo = Tuo, uoEO. Since 0 is open, there exists a bounded domain Do such that uoEDo, [50(0 [for example, the spherical neighborhood Llr(uo) will serve if r is sufficiently small]. Then, since T is one-to-one,


T Do is a domain (by the discussion at the beginning of the present section), and Xo = TUoE T Do( TO. Thus every point Xo of TO is contained in an open set (namely, in T Do) which is contained in TO. Hence TO is open. Thus the mapping (16) is continuous, one-to-one, and carries open sets into open sets, and hence part (ii) of the theorem follows by 1.1.5, exercise 6. Observe that the term open set is used here relative to Rn. However, if a set G is open relative to Rn and is contained in a subset S of R", then G is also open relative to S, and hence there arises no discrepancy. Finally, part (i) of the theorem follows from part (ii) [see (5)].

We turn presently to a study of one-parameter families of hom eo-morphisms

(21 )

defined as follows. D is a bounded domain in R n, and for each tEl, where I is the unit interval 0 ~t ~ 1, 7; is continuous and one-to-one on 15. Finally, we require that 7; U, interpreted as a mapping from 15xI into R", be continuous on 15xI.

Theorem 6. Given a one-parameter family of homeomorphisms as in (21), the index l(7;, D) is defined for each tEl and is independent of t. In particular, l(To, D) = l(~, D).

Proof. In view of remark 1, applied with U = 15, l (7;, D) is defined for tEl. Observe now that l(7;, D) may be considered as an integralvalued function of t on I. Since I is connected, in view of 1.1.3, exercise 37, it is sufficient to show that d7;, D) is locally constant on I. To simplify notations, put L1(t)= 7;D. Choose a point u*ED and set 7; U* = x (t). Now take any to E I. Then (see the discussion at the beginning of the present section), x (to) E L1 (to), and thus x (to) Ef frLl (to). Since

fr L1 (to) = 7;, fr D , it follows that

e ( x (to), 7;,tr D) > o.

As 7; u is continuous on 15 X I, there follows the existence of an 8> 0 such that

Ilx(t) - x(to)11 + e(7;, 7;" fr D) < e(x(to), 7;, fr D), (22)

for It - tol < 8, tE I. Let t be so restricted. By theorem 6 in II.2.3, (22) implies that

p(x(t), 7;,D) =p,(x(to), 7;"D). (23)

By theorem 1 we have

p, (x (to), 7;" D) = I (7;" D) = ± 1. (24)



J-l(x(t), 7;, D) =1= o. (25)

By remark 3, we infer from (25) that l(7;, D) is defined and

l(7;,D)=,u(x(t),7;,D). (26)

From (26), (23), (24) we see that

l(7;,D) = l(7;., D) for It-tol<£,tEI.

Thus l(7;, D) is locally constant on I, and the proof is complete.

11.2.5. Linear transformations in Rn. Returning to the mapping T: U -+Rn in II.2.1 (1), let us note that for certain purposes it is convenient to describe T in the form

T: Xi = fi (ul , ... , Un), i = 1 , ... , n, (1 )

by giving the coordinates Xl, ... , x" of the image point x as functions of the coordinates ul , ... , un of the point uE U. We shall study presently the important special case when U = Rn and the functions fi in (1) are linear functions of 1~1, ... , un. The mapping T is then termed a linear transformation from R" into R" and will be denoted, generically, by B. Thus a linear transformation is determined by formulas of the type n

B:xi=bi+La;kuk, i=1, ... ,n, (2) k=l

where bi, aik are real numbers. Clearly B is a continuous mapping from R" into Rn.

In the following discussion of linear transformations, the integer n~ 1 is arbitrary, but fixed until further notice.

The matrix (aik) will be termed the matrix we (B) of B. The determinant det (a; k) of this matrix will be termed the determinant of B and will be denoted by det B.

Definition 1. A linear transformation B is termed non-singular or singular according as det B =1= 0 or det B = o.

Remark 1. The non-singular linear transformations B fall naturally into two categories, according as det B> 0 or det B < O. The first category contains the identity mapping in R", given by Xi = ui, i = 1 , ... , n. An important linear transformation in the second category is that given by Xl = - ul, Xi = ui for i =1= 1. For convenient reference, these two particular transformations will be denoted by B+ and Brespectively. Thus the defining formulas are as follows.

B + : Xi = ui , i = 1, ... , n, (3 )

(4)


Clearly det 2+ = 1, det 2- = - 1. (5)

The following two lemmas are direct consequences of standard theorems on systems of linear equations.

Lemma 1. If 2 is non-singular, then 2 is a homeomorphism from R n onto R", and 2-1 is linear and non-singular.

Lemma 2. If 21 , 22 are linear transformations, then their product (see 1.1.1) 2221 is a linear transformation and det 2221 = det 22 det 21 .

In particular, if 21, 22 are both non-singular, then 22 £!1 is also nonsingular.

Definition 2. Let I denote the unit interval 0:;;;;; t:;;; 1, and let bi(t), aik(t), i, k=1, ... , n, be real-valued continuous functions on I such that det aik(t) =Fa for all tEl. For each tEl we have then the non-singular linear transformation

n

2t:xi=bi (t) + 1: aik(t) Uk, i=1, ... ,n, tEl. (6) k~1

These linear transformations will be said to constitute a one-parameter family of non-singular linear transformations.

Remark 2. Since det 2t is a continuous function of t which is different from zero on I, it follows that either det2t>0 for all tEl or det2t <0 for all tEl.

Definition 3. Two non-singular linear transformations 2', 5!," will be termed equivalent, in symbols 2'", 2", if and only if there exists a one-parameter family 2t of non-singular linear transformations such that 20= 2',21 = 2". For brevity, such a family £!t will be termed an admissible family connecting 2' to 2".

Remark 3. In view of remark 2, the relation 2'", 2" implies that sgn det 2' = sgn det 2". Thus this condition is necessary for the relation 2' ",2" to hold. It follows that the transformations 2+,2-[see (3), (4)J are not equivalent in the sense of definition 3.

Lemma 3. The binary relation 5!,' '" 2" is reflexive, symmetric, and transitive.

This is an obvious consequence of the definitions involved.

Lemma 4. Given a non-singular linear transformation s:! as in (2), let 5!,* be given by

" \I . Xi - '\' a Uk .. - 1 11 ....... * 4 - ~ ik , "- , ... , '. (7) k=I

That is, s:!* is obtained from 2 by dropping the constant terms bi •

Then 2* is non-singular and 2* '" 2.

§ I1.2. The topological index. 139

Prool. Since det £*=det £=1=0, £* is non-singular. Now define, for O;;::;;t;;::;; 1,

n

£/ : Xi = (1 - t) bi + L ai k Uk, i = 1, ... , n. k~1

Clearly, £/ is an admissible family connecting £ to £*' and the lemma is proved.

Remark 4. The next few lemmas all assert that certain pairs of linear transformations £, £* are equivalent. The pattern of the proof will be the same as in lemma 4: it will be clear that £* is non-singular, and there will remain to exhibit an admissible family £/ connecting £ to £*. Accordingly, we shall merely indicate in each case the connecting admissible family £/, leaving trivial verifications to the reader.

Lemma 5. Given a non-singular linear transformation of the form

(8)

let 52* be the linear transformation obtained by multiplying the elements of the r-th column (row) of the matrix (aik) by the same real number ,1>0. Then £,....,£*.

Proof. Use (for the column case) the connecting family

where

n

£/:xi=Laik(t)u}, i=1, ... ,n, O;;::;;t;;::;;1, k~1

aik(t)==a ik for k=l=r, i=1, ... ,n,

ai, (t) = (1 - t + ), t) ai,' i = 1, ... , n.

Lemma 6. Given a non-singular linear transformation £ as in (8), denote by £* the linear transformation obtained by multiplying the elements of the r-th row (column) of the matrix (aik) by an arbitrary real number A and adding to the corresponding elements of the s-th row (column), where r=l=s. Then £,....., £*.

Proof. Use (in the row case) the connecting family

" 52/: Xi = L aik (t) Uk, i = 1, ... , n, 0;;::;; t ;;::;; 1 , k~1

where k=1, ... ,n, k=1, ... ,n.

Definition 4. Let E1 , .•. , En be integers such that I Eil = 1, i = 1, ... , n (thus each Ei is equal to either 1 or -1). Then the linear transformation

(9)


is non-singular, since det B = el ... e" = ± 1. The transformation Q

given by (9) will be denoted by [el' ... , en]. Lemma 7. Suppose that n;;;;;2, and let j be any integer such that

1 :;;, j :;;, n - 1. Then

Proof. Use the connecting family Bt defined as follows:

Xi = ei ui for i =l= j, j + 1 ,

xi = ej (cos n t) u i - ej (sin n t) ui+I,

xi+! = ej+! (sin n t) u i + ej+! (cos n t) u i +!.

Theorem 1. Given a non-singular linear transformation Q as in (2), one has B '" Q+ or B '" B- [see (3), (4) J according as det B> 0 or det B < O.

Proof. The idea of the proof is to exhibit a finite sequence of nonsingular linear transformations Bl , ... , Bm , such that B '" ~l' ~l '" B2, .... Bm - l '"" Bm, and Bm coincides either with B+ or B-. By lemma 3 it fpllows then that ~ '"" Bm , and hence ~ '" B+ or B '" B-. By remark 3 we must have in the first case det B> 0 and in the second case det ~ < 0 [see (5)J, and the theorem will be proved. We proceed now to exhibit the sequence Bl , B2 , . ..• Starting with B given as in (2), we first drop the constant terms bi, obtaining by lemma 4 a non-singular linear transformation Br '" B, where Bl is of the form

i=1, ... ,n.

Observe now that the matrix (a ik) can be brought into the diagonal form by repeated application of the process of multiplying all the elements of a column (row) by a real number and adding to the corresponding elements of another column (row). In view of lemma 6, it follows that we obtain in this manner a sequence ~1' ... , By of nonsingular linear transformations such that BI ,-....,,··· ",By, and By is of the form

i = 1, ... , n.

Repeated application of lemma 5 (using successively A = 1/lcl l, ... , A = 1/Ic"l) , yields a non-singular linear transformation BY +1 '"" By of the form [e1' ... , enJ (see definition 4). If n=1, then the proof is complete, since [e1J = B+ or [elJ = B- according as e1 = 1 or e1 =-1. So assume that n;;;;;2. We treat then

as follows. Suppose that for some index j we have Sj = 1, ej+! = - 1-

By lemma 7 we have then (since now - Sj = ej+I' - Sj+ 1 = Sj)

[ ... , ef' ej + l' ... J r-..' [ ... , ej + l' ei' ... J .


In other words, if in the sequence e1' ... , en a tenn 1 is followed by a tenn - 1, then the process of exchanging these two consecutive tenns yields an equivalent transformation. Repeated application of this remark yields a linear transformation

£'+2 = [e~, ... , e~],

such that £d 2 '" £, +1 and ei ther e~ = ... = e~ = 1 (in which case £,+2= £+ and the proof is complete), or else we have an integer j such that e~= ... =e;=-1, and (in case j<n) e;+1="·=e~=1. If j = 1, then clearly £,+2 = £-, and the proof is complete. If j~2, then e;_l = e; = -1, and by lemma 7 we have

[ ... , CJ~ -1, sf ' ... ] I"'-' [ ••• , - Sf -1' - e; , ... ] . Since now - ef -1 = - ef = 1, the number of tenns equal to -1 has been reduced by two. Repeated application of this remark yields a linear transfonnation

h th t oo d 'th" "1 s~c a ~;+3"'~'+2' an :1 er e1 = "'=en =. ' "2 = ... = en = 1. In the hrst case £'+3 = £ +, III S!'+3 = S!-, and the proof is complete.

or else e~ = - 1 , the second case

Theorem 2. Two non-singular linear transfonnations £', £" are equivalent if and only if

sgn det £' = sgn det £". ( 10)

Proof. The necessity of the condition (10) follows by remark 3 To prove the sufficiency, assume that (10) holds. If det S!', det S!" are both positive (negative), then by theorem 1 we have £', S!" ""S!+ (£-), and hence £' "-' £'., in view of the transitivity of the relation "-' (see lemma 3).

Consider now a non-singular linear transformation £. Then £ is a homeomorphism from Rn onto Rn (see lemma 1) and hence theorem 5 in 11.2.4 applies (with D = R", T = S!). We conclude that if D, D' are any two bounded domains in Rn, then t (L, D) = t{L, D'). This shows that the index £1£, D) is independent of the choice of the bounded domain D (Rn, and hence this index depends only upon S!. Thus the following definition is justified.

Definition 5. The index t (£) of a non-singular linear transformation £ is defined by the formula

t(£) = t(S!, D), (11 )

where D is any bounded domain in Rn.


Lemma 8. If S}', 53/1 are non-singular linear transformations such that 53' ",53/1, then t(53') = t(53/1).

Proof. Choose a bounded domain D (Rn• By definition

t(53') = l(53', D), t(53/1) = t(53/1, D). ( 12)

By assumption (see definition 3) there exists an admissible family 53( connecting 53' to 53/1. On setting 7;= 5311.0, clearly 7; is a one-parameter family of homeomorphisms [see 11.2.4 (21)] such that

To = 53'1.0, Tl = S}/II.o· Hence, by theorem 6 and remark 1 in 11.2.4,

l(53', D) = t(53/1, D). (13)

From (12) and (13) it follows that t(53') = t(53/1).

Theorem 3. If 53', 53/1 are non-singular linear transformations such that sgn det 53' = sgn det 53/1, then t(53') = t(53/1).

Proof. By theorem 2 the assumption implies that 53' '" 53/1, and hence t (S}') = t (53/1) by lemma 8.

In view of theorem 1 it is clear that t(53) can be explicitly calculated for any non-singular linear transformation 53 provided that one knows the index t for the particular transformations 53 +, 53 - [see (3), (4) ]. We proceed to derive the lemmas to be used in determining t(53+), t(53-).

Consider the (n -1)-sphere sn-l in Rn and the linear transformation 53- [see (4)]. Put

(14)

Displaying the integer n in the notation Tn is now necessary since we shall employ induction on n later. Clearly Til is a homeomorphism from sn-l onto sn-l. Accordingly, the induced homomorphism

(15 )

is an isomorphism onto (see 1.6.2, remark 2), as indicated by the symbol ~'.

Lemma 9. The isomorphism T; is odd (see 1.3.2).

Proof. Consider first the case n = '1. Recall that So consists of the two real numbers 1 and -1, and that (by 1.6.1, remark 7) HI (S0) can be identified with the group Zl(SO) of the 1-cocycles of S°. Let z be any element of Zl (SO). We first note that

z(1, -1) = -z(-1, 1). ( 16)

Indeed, since z is a 1-cocycle, one has

0= (bIZ) (-1, 1, -1) = z(1, -1) - z(-1, -1) + z(- 'I, 1).


The term z(-1, -1) vanishes by 1.6.1, remark 6, and (16) follows. We noted in 1.6.1 that we have at our disposal the isomorphism

h:ZI(SO)~~I,

defined by the formula hz=z(-1, 1). By (14), (4), (16) and 1.6.2 it follows that

h Ti z = (Ti z) (- 1, 1) = z (1, - 1) = - z (- 1, 1) = - h z.

Since h is an isomorphism, we conclude that Ti z = - z. Thus Ti is odd.

Proceeding by induction, we assume that for a certain integer n;;;;: 1 we already know that T*n is odd, and we show that T;+l is also odd by using the following diagram.

( ) f n ,. 'n ,.. en In "5"-' .. H"(S"-') =' H"+'(E~.5"-')-==Hn+'(S".E,!) -:::: • H"+'(5")

= == k.

= r-Hn·'(E~.Sn-')~Hn+'(sn.E'!) 'n en Fig. 41.

The isomorphisms In, ~,., en, in. as well as the cohomology groups involved are taken from the diagram used at the end of section 1.7.1. The isomorphism T;+l corresponds to T; in the dimension n + 1, while T; itself is defined in (15). The isomorphisms k1' k2' k3 are induced by the homeomorphisms

T n +llsn-1:sn-1-l>sn-1,

yn+1IE~: (E~, 5,.-1) -l> (E~ , S»-l),

yn+l:(sn, E~) -l> (sn, E~)

respectively. By 1.6.2, remark 2 it follows that k1, '~2' k3 are isomorphisms onto. Finally, the vector law holds in all four boxes of the diagram (by 1.6.6 in the second box from the left, and by 1.6.2 in the other three boxes). Now. since T; is odd by the induction assumption, it follows by proceeding from the left to the right in the preceding diagram that k1, k2 , k3, T;+l are odd, and the inductive proof of the lemma is complete.

In preparation for our next lemma, we consider the mapping

(17)


where En is the n-cell in Rn. Clearly, T" is a topological mapping from the pair (En, sn-1) onto itself. Hence, by remark 2 in 1.6.2, the induced homomorphism

( 18)

IS an isomorphism onto, as indicated by the symbol ~.

K Lemma 10. H"(S n-'}---==,---_.-H"·' (E".S "-')

The iso-

H"(Sn-')-----="'''--~. Hn+,(E. n ,5"-') b

Fig. 42.

(see 1.6.4) of the pair (En, S"-1). In

morphism t;: is odd (see 1.3.2).

Proof. Consider the diagram in Fig. 42.

The isomorphisms T;, t; are those defined in (1 5) and (18) respectively. The homomorphism <5 is taken from the cohomology sequence

the four-term section

H"(E"), Hn(sn-1), H"+1(E", sn-1), Hn+1(En)

of that cohomology sequence the first and the fourth group are trivial by 1.7.1, and hence we conclude by 1.3.3, lemma 4 that the homomorphism <5 from the second to the third group is an isomorphism onto. Observe finally that the vector law holds in the diagram by I.6.6. Now since T; is odd by lemma 9, we conclude that t; is also odd, and the lemma is proved.

Lemma 11. {-t(x, ~-, int £'I) = -1 for xE intE" [see (4) and 11.2.2J.

Proof. Note first that [E", S"-1, int E"] is a frame in the sense of II.1.1. We observed above that ~-IEn=tn is a topological mapping from the pair (En, sn-1) onto itself. Clearly, ~- int En= int E". Thus if xE int En, then x is (~-, int E")-admissible and the frame [£'I, sn-1, int £'I] is (x, ~-, int £'I)-admissible (see II.2.1). By the definition of the topological index (see II.2.2) we conclude that

{-t (x, ~-, int E") n [£'I, S,,-1, int E"] = 1: g [£'I, S,,-1, int £'I]. (19)

Since T;: is odd by lemma 10, we have

t;: g [En, S"-1, int En] = - g [En, S"-1, int En]. (20)

As the g occurring in (19) and (20) is a generator of the infinite cyclic group Hn+1 (En, sn-1), it follows from (19) and (20) that {-t (x, ~-, int £'I) = - 1, and the lemma is proved.

Lemma 12. t(~-) = -1.

§ 11.3. Multiplicity functions and index functions.

Proof. In view of definition 5 we have

l(S:n = l(£-, int E").

Theorem 1 in 11.2.4 yields

145

(21)

l(£-,intE")=,u(x,£-,intE") for xEintE". (22)

Finally, by lemma 11,

It (x, £-, int P) = - 1, x E int P. (23)

From (21) to (23) one sees that l(£-) =-1. Lemma 13. l(£+)=1.

Proof. If D is any bounded domain in RI>, then £+ID is the identity mapping on D. By definition 5 and 11.2.4, theorem 2 we conclude that

Theorem 4. If £ is a non-singular linear transformation III R", then the following holds.

(i) l (£) = sgn det £. (ii) If D is any bounded domain, then ,u (x, £, D) = sgn det £

for xE £D and = ° for xE C £D. Proof. If det £> 0, then l (£) = l (£ +) = 1 by theorem 3 and lemma 13.

Similarly, if det £ < 0, then t (£) = l (£-) = -1 by theorem 3 and lemma 12. Thus (i) is proved. Finally, (ii) follows from (i) by definition 5 and 11.2.4, remark 3.

Remark 5. In view of the definition in 11.2.4 and definition 5 in 11.2.5, part (i) of the preceding theorem may be re-stated as follows: a non-singular linear transformation £ is even or odd according as det £ > ° or det £ < 0.

§ II.3. Multiplicity functions and index functions!.

II.3.t. Preliminaries. We shall consider a continuous mapping

(1 )

using the notations agreed upon in 11.2.1. Let D be a bounded domain such that

D(U. (2)

1 The concepts and results discussed in § 11.3 were first developed for the case n = 2 (for historical comments and a comprehensive discussion, see the treatise on Length and Area listed in the Bibliography). For the extension to a general n"2 2 see H. FEDERER, Essential multiplicity and Lebesgue area [Proc. Nat. Acad. Sci. U.S.A. 34, 611-616 (1948)J, and T. RADO and P.V. REICHELDERFER, On n-dimensional concepts of bounded variation, absolute continuity and generalized Jacobian [Proc. Nat. Acad. Sci. U.S.A. 35, 678-681 (1949)J.



We also assume that the mapping

TID:D-+Rn (3 )

is bounded; that is, there exists a (finite) positive constant A such that

II T u II < A for u ED. (4)

Observe that we do not require now the inclusion jj (U. As a consequence, T may not be defined on fr D (the special case when U = D illustrates the generality of the situation to be studied).

A point xE R n mayor may not lie in T D. Our main objective at present is to assign a multiplicity number to each point x ERn which should indicate, in a useful and convenient manner, the number of times that x is covered under the mapping (3). If xEf T D, then clearly the multiplicity number should be equal to zero. On the other hand, if xE T D, then various plausible approaches may be considered in defining a multiplicity number. The number of distinct points in the set D n 1'-1 x would be the most obvious choice, but the literature on this subject reveals that this choice is inadequate beyond the most elementary cases. Experience has shown that a detailed study of the components of the set D n T-1 x is necessary to gain useful information concerning multiplicity numbers. In the present section we discuss background material needed in the subsequent study of multiplicity numbers. It will be assumed throughout that T and D are given as in (1) to (4).

Lemma 1. Let G be an open set in R", 0 an open set in D, and suppose that on T-1G=l= 0. Let D be a component of on T-1G such that D (0. Then the following holds.

(i) D is a bounded domain. (ii) TD(G. (iii) T D ( G. (iv) TfrDCfrG(CG. (v) D is a component of Dn T-1G.

Proof. Since D (0 (D and D is bounded, D is also bounded. Since G is open, the set 0 n T-1 G is open relative to 0 and hence also relative to Rn, because 0 is open in Rn. As D is a component of an open set in R", it follows that D is connected and open (see 1.2.1, exercise 4). Thus (i) is proved. The inclusion D ( T-1 G yields T D (G, and (ii) is proved. By 1.1.5, exercise 9 we conclude (since D(D) that T D = T D (G, and (iii) is proved. Since the components of the open set 0 n T-1G constitute a family of disjoint open sets, from 1.1.3, exerClse 33 we infer that

fr D (fr(O n T-1 G) ( C (0 n T-1 G) = CO U C T-1 G. (5)

§ 11.3. Multiplicity fUllctions and index functions.

By assumption, we have D(O, and hence

frD(D(O.

From (5) and (6) we conclude that fr D (C T-IG, and hence

TfrD( CG.

By (6) and (iii) (which we already proved) we have

T fr D ( T D ( G.

From (7) and (8) we obtain (since G is open)

T fr D ( C G n G = fr G ( C G,

and (iv) is proved. To verify (v), note that

D (0 n T-l G (D n T-l G.

147

(6)

(7)

(8)

Thus D is a connected subset of D n T-IG. Accordingly, there exists a component D* of Dn T-IG such that D(D*, and (v) will be proved if we can show that D = D*. If this relation is denied, then D should be a proper subset of the connected set D*, and hence (see 1.1.3, exercise 17) one would have

D*n frD =f=. 0.

As D* (T-IG, in view of (iv) (already proved) one obtains

T (D* n fr D) ( T D* n T fr D ( G n C G = 0,

(9)

and hence D*nfrD= 0, in contradiction with (9). Thus the assumption that D=f=.D* leads to a contradiction, and the proof is complete.

Lemma 2. Let G be an open set in Rn such that Dn T-IG=f=. 0, and let D be a component of Dn T-IG such that D(D. Then every point xE G is (T, D)-admissible, or equivalently, D is (x, T)-admissible for every point xEG (see II.2.1).

Proof. By lemma 1 (applied with O=D), we have TfrD(CG. Hence, if x E G, then x Ef T fr D, and the lemma is proved.

Lemma 3. Let D be a domain such that D (D and let x be a point such that D is (x, T)-admissible (that is, xEf T fr D). Finally, let G be an open set such that xEG(CTfrD. Then those components of the set Dn T-IG that lie in D constitute a sequence of domains which is (x, T, D)-complete (see 11.2.1, definition 5).

Proof. The lemma is trivial if D n T-l x = 0 (see II.2.1, remark 21. So assume that Dn T-1x=f=. 0. Then Dn T-IG=f=. 0, since D)D, and hence

10*


The components of the open set D n T-IG constitute a sequence {DI}

of disjoint domains (see 1.1.3, exercise 28). We assert that

D j n Ir D = 0 for every j. (10)

Indeed, since D j ( T-IG and G( C TlrD, we have

T (D j n Ir D) ( T Dj n T Ir D ( G n T Ir D = 0,

and (10) follows. We next assert that

D j n 15 =j= 0 implies D j e D. (11 )

Indeed, if D j n15=j= 0, then (10) shows that DinD=j= 0. If the relation Di(D is denied, then Di should intersect both D and CD, and hence by 1.1.3, exercise 17, one would have Din IrD=j= 0, in contradiction with (10). Thus (11) holds. Turning to the proof of the lemma, we start with the relations

where the used the relations D(D, Dn T-IG= UD j • Now in view of (11) a term 15nD j in (12) is either empty or else DjeD and then DnDi=Di . Thus (12) reduces to

Dn T-1x( UD j , Di(D.

In other words, the union of those components D j that lie in D contains the set 15 n T-l x, which means precisely that these components constitute an (x, T, D)-complete sequence.

Lemma 4. Let x be a point and let G, G* be open sets in R" such that xEG,G(G*. Let D* be a component of DnT-IG* such that D* (D. Finally, let {Dk} be the sequence (perhaps empty) of those components of Dn T-IG that lie in D*. Then the following holds.

(i) D* is (x, T)-admissible (see II.2.l). (ii) 15k (D* for every k. (iii) The sequence {Dk} is (x, T, D*)-complete (see II.2.l, defini

tion 5).

Proal. (i) is a direct consequence of lemma 2 (applied to G*). As regards (ii), note that Dk (D* implies that 15k (D*, and hence (ii) is proved if we can show that

15k n Ir D* = 0. (13 )

To verify this, we start with the relation

T(15k n Ir D*) ( T15k n T Ir D*. (14)

§ 11.3. Multiplicity functions and index functions. 149

By lemma 1 we have the inclusions TDk ( G, Tlr D* (CG*. Hence (14) yields

T(DknlrD*)(GnCG*= 0, ( 15)

since G (G* by assumption. Clearly (15) implies (13). Observe now that (by lemma 1)

T Ir D* ( Ir G* ( C G*, and hence

x E G ( G* ( C T Ir D*.

Thus x, G, D* satisfy the assumptions of lemma ), and hence (iii) follows directly from that lemma.

Remark 1. Take a point xETD. Then DnT-lx is a non-empty bounded set which is generally not closed (even though it is closed relative to D by 1.1.5, exercise 4). Accordingly, if C is a component of D n T-l x, then C is a connected bounded set which mayor may not be closed. If C happens to be closed, then it is also compact (see 1.2.1, exercise 4) and hence it is a continuum (see 1.1.3, definition 15) which may reduce to a single point.

Delinition 1. A maximal model continuum (abbreviated to m.m.c.) for (x, T, D) is a component of Dn T-1x which is a continuum.

Remark 2. If C is a component of D n T-l x, then C is an m.m.c. for (x, T, D) if and only if C (D. Indeed, if C is a continuum then C = C (D. Conversely, suppose that C (D. Then (see 1.1.5, exercise 9)

TC = TC (TT-1x = x, and thus

C (C (Dn T-l x. ( 16)

Now since C is connected (see 1.1.3, exercise 16) and C is a component of D n T-l x, (16) implies (see 1.1.3, exercise 11) that C = C. Thus C is a continuum, and hence an m.m.c. for (x, T, D).

Lemma 5. Given a point x, let D be a domain such that D (D and D is (x, T)-admissible, and finally let rbe a component of Dn T-l x. Then either reD or r( CD.

Proal. Since T r= xEf TlrD, clearly

rnlrD = 0. (17)

Si nee r is connected, by 1.1.3, exercise 17 we infer from (17) that r ca nnot intersect both D and CD, and hence either re D or else r( CD. Th ere remains to show that

r( CD implies r( CD. ( 18)


Now since D=DUfrD, we have

CD = CD n C fr D.

By (17) we have rCCfrD. Hence, if rCCD, then [see (19)J

r C CD n C fr D = CD, proving (18).

(19)

Definition 2. A determining sequence {Dj } for (x, T, D) is a sequence of (non-empty) domains Dj with the following properties.

(i) D)D1 , Dj )Dj +1 for every j. (ii) Each Dj is (x, T)-admissible (see IT.2.1). (iii) 0 TDj--'7-0 for j--'7-00 (see 1.1.4, definition 4). (iv) xE T Dj for every j. Lemma 6. If {D j } is a determining sequence for (x, T, D), then

the set (20)

IS an m.m.c. for (x, T, D), and C CDj for every j.

Proof. Since D j is a bounded domain, by 1.1.3, exercise 16 and 1.2.1, exercise 4 it follows that Dj is a continuum. By (i) in definition 2, the sequence {Dj } is a nested sequence of continua, and hence the set C in (20) is a continuum by 1.1.3, exercise 30. If r > 0 is arbitrarily assigned, then (iii) and (iv) in definition 2 imply that T Dj lies in the spherical neighborhood LIT (x) (see IT.2.1) for j sufficiently large. In view of (20) it follows that TC(Llr(x) for every r>O, and this clearly implies that TC = x. Hence

C C D n T-l x.

By (20) we have C C Dj +1' Hence, by (i) III definition 2,

C C D i for every j.

(21 )

(22)

There remains to show that C is a component of D n T-l x. Now since C is a connected subset of DnT-1x [see (21)J, by 1.1.3, exercise 11 there exists a component r of D n T-l x such that C Cr, and the proof will be complete if we can show that C =r. If this relation is denied, then there should exist a point uEr - C. In view of (20) there follows the existence of an integer j such that uE CD j . Thus the component r of D n T-l x would intersect both D j [in view of (22)] and CDj • Since DiCD and Di is (x, T)-admissible [see (ii) in definition 2), this conclusion contradicts lemma 5, and the proof is complete.

Remark 3. In view of lemma 6 there arises the question whether every m.m.c. C for (x, T, D) can be represented in the form (20) in terms of a properly chosen determining sequence {D j } for (x, T, D).

§ II.3. Multiplicity functions and index functions. 151

This is indeed the case, as we shall see presently. For the purpose of subsequent applications, we shall exhibit an especially convenient special determining sequence for each m.m.c. The following preliminary comments will facilitate the discussion. Let C be an m.m.c. for (x, T, D). Consider the spherical neighborhood Llr(x) (see I1.2.1, definition 4). Then D n T-1L1,(X) is an open set containing C, and hence this set has a unique component Dr containing C (since C is connected). Then Dr is a domain and

(23)

If 0 < s < r, then clearly

D n T-l Lis (x) ( D n T-l LI, (x) .

Thus Ds is a connected subset, containing C, of D n T-IL1,(X), and hence Ds is contained in Dr> the unique component of D n T-l LI, (x) containing C. So we have .

(24)

Observe that the inclusion 15,(D may not hold.

Lemma 7. Given an m.m.c. C for (x, T, D) and an open set 0 such that C ( 0, 0 ( D, there exists a number 'YJ > Q such that

(25)

where Dr has the meaning explained in remark 3. Proof. Put

Then 0; is an open set containing C, and since C is connected there exists a unique component D: of 0: containing C. We have then

C ( D;, D; (0 ( D. (26) Clearly

C ( D: ( D;, C ( D: (l5: for 0 < s < r. (27)

Furthermore, smce D: (T-1L1 r (X), it follows (see 1.1.5, exercise 9) that

(28)

Consider now any sequence r 1 > ... > rj > ... > 0, Yf -+ O. In view of (27), the sets D: constitute a nested sequence of continua containing C.

1

Hence (see 1.1.3, exercise 30) the set

C* = n D* 'j

is a continuum containing C. From (28) we infer that

TC* ('fD;/ 1\ (x) .

(29)


Since ri~O, it follows that T C* = x, and hence C* (D n T-l x. Now C*, which is connected, contains the component C of D n T-l x. By 1.1.3, exercise 11, we conclude that C = C*, and hence (29) yields

As C (0 and 0 is open, (30) implies (see 1.1.3, exercise 24) that

lJ: ( 0 for j large. , Choose an integer j such that (31) holds, and set

1] = rio

For 0< r< IJ we have then, in view of (27) and (31),

D: (D~ (0.

(30)

(32)

(33)

By lemma 1 [applied with G=Llr(x)] it follows from (33) that D; is also a component of D n T-l Ll, (x), and hence D; must coincide with Dr' the unique component of D n T-ILl r (x) containing C [observe that D~ contains C by (26)J. Thus (33) yields Dr(O for O<r<IJ, and the proof is complete.

Lemma 8. Let C be an m.m.c. for (x, T, D), and let 0 be an open set such that C(O, OeD. Then there exists a sequence

r 1 > ... > rj > ... > 0, ri ~ 0, (34)

such that the following holds.

(i) On denoting by Dr; the (unique) component of D n T-l Ll,; (x) that contains C (see remark 3), the sequence {Dr;} is a determining sequence for (x, T, D).

(ii) C= nDr;. (iii) D'j (0 for every j.

Proof. By lemma 7, we have a number 1]>0 such that (25) holds. Choose any sequence {rJ satisfying (34) and also the additional condition r1 < 1]. Then

ri < 1] for every j. Since clearly

Llr;+l (x) (Ll,; (x),

we infer from lemma 4, part (ii) that

(35)

(36)

From part (i) of the same lemma it follows that Dr; is (x, T)-admissible. By lemma 1 we have


and hence flTD'j""*O by (34). Since C(Drj , clearly

x= TC( TDrj ( TD'j'

This completes the proof of part (i) of the lemma. By lemma 6 it follows that the set C* = nD'j is an m.m.c. for (x, T, D). Now since C (Drf

for every j, we have the inclusion C (C*. As C and C* are components of the same set D n T-l x, it follows that C = C* = n 1\, and part (ii) of the lemma is proved. Finally, part (iii) follows directly from (35) and (25).

Lemma 9. Let U o be a point in D and {D",} a sequence of domains with the following properties.

(i) uoEDm,D",(D for every m. (ii) Each Dm is (Tuo, T)-admissible (see 11.2.1). (iii) fl T 15", < 1/m for every m.

Then the sequence {Dm} contains a subsequence {D"'j}' j = 1, 2, ... , which is a determining sequence for (Tuo, T, D).

Proof. If {Dm.} is any subsequence, then TUoE TD",. by (i) and _ J J

fl T Dm. ""* 0 by (iii). Accordingly (see definition 2) we have to show J

only that the subsequence can be made to satisfy the condition

To see this, take any positive integer m. We assert that

D",) 15M for M large.

(37)

(38)

Indeed, take any M>m. Then uoED",nDM and hence D",n15M =l= 0. Since 15M is connected, it follows [see 1.1.3, exercise (17)J that either (38) holds or else

(39)

In view of (ii) we have TUot± TfrD"" and hence (see 1.1.4, exercise 3)

e (T uo, T fr D m) > o.

If (39) holds, then we have a point um E15M nfrD",. Then Tu",E TfrDm

and hence

Also, Tu",ET15M , and hence liTuo-Tu",Ii<1/M, since TUoETDM and fl T 15M < 1JM. It follows that (39) implies the inequality

1 e (T 1to, T jr D m) < M .

Accordingly, (38) must hold if M is large enough to satisfy the inequality

1 M ;;;;, e (T Ito, T jr D m) .


The existence of a subsequence {DmJ satisfying (37) is now immediate. Indeed, on choosing Dm , = D1 , (38) yields the existence of an integer m2 >m1 such that Dm , ) i5m, , and it is obvious that successive application of (38) yields a sequence of sUbscripts m1 < m2 < ms < ... such that (37) holds.

11.3.2. The multiplicity functions K, K+, K-. Given T and D as in 11.3.1 and a point xER", an indicator domain D for (x, T, D) is defined by the following conditions. (i) l5 (D. (ii) D is (x, T)-admissible (that is, xtfTfrD). (iii),u(x, T,D) =l=O (see 11.2.2). If,u(x, T,D»O«O), then D is termed a positive (negative) indicator domain for (x, T, D). A finite (perhaps empty) collection of pair-wise disjoint indicator domains for (x, T, D) will be called an indicator system for (x, T, D), and will be denoted generically by lS (x, T, D). If all the domains comprised in an indicator system lS (x, T, D) are positive (negative) indicator domains, then lS will be termed a positive (negative) indicator system, and the notation lS+ (x, T, D) (\S- (x, T, D)) will be used to indicate this fact. The symbol M[lS(x, T, D)] is defined by the formula

M[lS(x, T,D)]=LI,u(x, T,D)I, DElS(x, T,D)' (1 )

with the understanding that

M [lS (x, T, D)] = 0 if ~ (x, T, D) is empty. (2)

The symbol N [lS (x, T, D)] is defined by

N [lS (x, T, D)J = number of domains DE lS (x, T, D), (3)

with the understanding that

N[lS(x, T, D)] = 0 if lS(x, T, D) is empty. (4)

Since every system lS+ (x, T, D) is also a system ® (x, T, D), the preceding definitions yield

M[lS+(x, T, D)] = L,u(X, T, D), DE ®+(x, T, D), (5)

and similarly

M[~- (x, T, D)] = - L,u(X, T, D), DE ®- (x, T, D), (6)

with the understanding that M = 0 if the system IS+ (lS-) involved is empty.

If D is an indicator domain for (x, T, D), then ,u (x, T, D) IS an integer different from zero, and thus clearly

M [lS (x, T, D)] ~ N [IS (x, T, D)],

M [lS+ (x, T, D)] ~ N [e+ (x, T, D)],

M[lS-(x, T, D)J~N[lS-(x, T, D)].

(7) (8)

(9)


Of course, for empty indicator systems these inequalities are obvious, since then M = N = o.

Remark 1. Let E be a finite, non-empty set of real numbers. Then l.u.b. E is an element of E. Indeed, E contains a largest element R, and (see 1.1.2) we have l.u.b. E = R.

Consider now the set of all the numbers M[6(x, T, D)], as defined by (1) and (2), corresponding to all possible systems 6(x, T, D). This set is non-empty .. Indeed, we have at least one system ®(x, T, D), namely the empty system, the corresponding M being equal to zero by (2). We define

and similarly

K(x, T, D) = l.u.b. M [6(x, T, D)], (10) 16

K+ (x, T, D) = l.u.b. M [6+ (x, T, D)], 1\2;+

K- (x, T, D) = l.u.b. M [6- (x, T, D)]. 16-

(11)

(12)

The letter 6 under the symbol l.u.b. in (10) is meant to indicate that the least upper bound is taken for the set of all the numbers M corresponding to all possible systems 6 (x, T, D). The formulas (11), (12) are to be interpreted in a similar manner in terms of 6+, 6- respectively.

Remark 2. Observing that M> 0 for a non-empty system 6(6+, 6-), one sees that K (x, T, D) = ° if and only if there exists no indicator domain for (x, T, D), with similar statements holding for K+ (x, T, D), K-(x, T, D).

Consider now a point x such that K (x, T, D) > o. Then there exists some indicator domain D for (x, T,D). Sincef1,(x, T,D)=t=O for such a domain D, by 11.2.3, theorem 2 we conclude that xETD(TD. Hence: if x!f.TD, then K(x, T, D)=O, and thus obviously also K+ (x, T, D) = 0, K- (x, T, D) = o.

Remark 3. If K (x, T, D) < 00 for a point x, then the numbers M corresponding to all possible systems 6 (x, T, D) satisfy the inequalities O:;;;;M :;;;;K(x, T, D) <00. Since every M is an integer, it follows that the set of these numbers M is now finite, and hence from remark 1 we conclude that there exists a system 6 (x, T, D) such that

K(x, T, D) =M[6(x, T, D)].

In a similar manner one sees that if K+ (x, T, D) < 00, then there exists a system 6+ (x, T, D) such that

K+(x, T, D) =M [6+ (x, T, D)].

An analogous statement holds if K- (x, T, D) < 00.


Remark 4. If K (x, T, D) < 00, then from (7) and (10) we conclude that

N[6(x, T, D)] ;;;;'K(x, T, D) < 00

for every system 6 (x, T, D). Thus in this case the number of domains in any indicator system 6 (x, T, D) remains under a fixed finite bound. Similar statements hold if K+ (x, T, D) < 00 or K- (x, T, D) < 00, for systems 6+, 6- respectively.

Remark 5. Fix a point xERn, and consider a system 6+(x, T, D). Then this system is also a system 6(x, T, D), and hence in view of (10)

M [6+ (x, T, D)] ;;;;, K (x, T, D) .

By (11) it follows that

K+ (x, T, D) ;;;;, K (x, T, D) . Similarly,

K-(x, T, D) ;;;;,K(x, T, D).

(13)

(14)

Consider now any system 6(x, T, D). The positive and the negative indicator domains contained in 6 (x, T, D) constitute then systems 6+ (x, T, D), 6- (x, T, D) respectively, one or both of which may be empty. Clearly, for these three systems we have

M[6 (x, T, D)] =M [6+ (x, T, D)] +M [6- (x, T, D)],

and hence [in view of (11) and (12)]

M[6(x, T, D)] ;;;;,K+(x, T, D) +K-(x, T, D).

By (10) we conclude that

K(x, T, D) ;;;;'K+(x, T, D) + K- (x, T, D).

Remark 6. Take a point xE Rn such that

K (x, T, D) < 00 .

(15 )

( 16)

Consider the open spherical neighborhood Llr(x) (see 1.1.4, definition 2). Since Llr (x) is open, the set

Or = D n T-1 LIT (X) ( 17)

is also open. Let Q r be the class (perhaps empty) of those components of OT which are indicator domains for (x, T, D). If D1, ... , Dm is any finite set of elements of QT' then these elements constitute an indicator system 6 (x, T, D), and hence m;;;;,K (x, T, D) < 00 by remark 4. It follows that the class Q r is finite and contains at most K (x, T, D) elements. Accordingly, the elements of Q r form an indicator system for (x, T, D) which will be denoted by 6 r (x, T, D). The positive and


the negative indicator domains contained in @i,(x, T, D) form systems to be denoted by @i; (x, T, D), @i; (x, T, D) respectively. Clearly, for any r>O,

M[@ir (x,T,D)]=M[@i;(x,T,D)J+M[@5;(x,T,D)]. (18)

Since @i,(x, T, D) is merely a special indicator system for (x, T, D), we have

and similarly M[@ir(x, T, D)J :;;;'K(x, T, D),

M [@i;(x, T, D)J;;;;;: K+ (x, T, D),

M [@i;(x, T, D)] ;;;;;: K- (x, T, D).

( 19)

(20)

(21)

Some or all of the systems @i,(x, T, D), @i: (x, T, D), @5;(x, T, D) may be empty.

Lemma 1. Let x be a point such that K (x, T, D) < 00. Then there exists a number 1]=1](x, T, D) such that (see remark 6)

for O<r<1].

K(x, T, D) =M[@i,(x, T, D)],

K+ (x, T, D) =M[@5:(x, T, D)J,

K- (x, T, D) =M [@i;(x, T, D)],

(22)

(23)

(24)

Proof. Since the argument is entirely analogous for each one of (22), (23), (24), we indicate merely the proof of (23) in the way of illustration. The assumption K(x, T, D)<oo implies, by (13), that

K+ (x, T, D) < 00. (25)

If K+ (x, T, D) = 0, then by (20) it is clear that M [@i; (x, T, D)] = ° for every r > 0, and thus (23) is obvious. So assume that

K+(x, T, D) > 0. (26)

In view of remark 3, the relations (25), (26) imply the existence of a system @i+ (x, T, D) such that

M [@i+(x, T, D)] = K+ (x, T, D). (27)

Let {Dk}, k = 1, ... , m be the (pair-wise disjoint) positive indicator domains for (x, T, D) which constitute @i+ (x, T, D). In view of (5), the relation (27) may then be re-written in the form

K+ (x, T, D) = L f1 (x, T, D k ). (28) k

Introduce now the auxiliary set

F = T fr Dl U ... U T fr Dm. (29)


Then F is compact and xEfF, since the domains D" are (x, T)-admissible. Accordingly (see 1.1.4, exercise 3)

e(x, F) > o. (30)

Choose now any real number r such that

o < r < e (x, F) . (31 )

We proceed to verify that (23) holds if r is so chosen. Observe first that if (31) is satisfied, then

xEJr(x)(CTjrDk , k=1, ... ,m.

Accordingly, we can apply II.3.t, lemma 3 with G = J, (x), D = Dk •

On denoting by Q~ the class of those components of D n T-1 J r (x) that lie in Dk , by that lemma the domains of Q~ constitute an (x, T, Dk)complete system. By II.2.3, theorem 3 we conclude that

/u(x, T,Dk)='ifl(X, T,dk ) , dkEQ~.

Summation with respect to k yields, in view of (28),

K+ (x, T, D) = 'i ('ifl (x, T, dk )) , dk Em. (32) k

Let us use the symbol 'i+ to indicate summation restricted to only those of the terms involved that are positive. Then (32) yields

K+ (x, T, D) ::;;, 'i ('i+ fl(x, T, dk )) , dk E Q~. (r~) k

Observe now that those domains dk E Q:, k = 1, ... , m, for which fl (x, T, dk ) > 0 are included amongst the domains that constitute the system is; (x, T, D) (see remark 6). Hence the double summation in (33) cannot exceed M [®; (x, T, Dll, and we conclude that

K+ (x, T, D) ::;;, M [®; (x, T, D)].

Jointly with (20), this inequality yields (23), under the assumption (31). Thus (23) is shown to hold for 1} = e(x, F). The proof of (22) and (24) (with a similar choice for 1}) is made in an entirely analogous manner.

Remark 7. Actually, in establishing (23), we used only the condition (25). This remark yields the following statement: if x is a point such that K+ (x, T, D) < 00, then (23) holds if r is sufficiently small. Similarly it follows that if K- (x, T, D) < 00, then (24) holds for r sufficiently small.

Theorem 1. K(x, T,D)=K+(x, T,D)+K-(x, T,D).

Prooj. Assume first that K(x, T, D) =00. By (15) we conclude that one at least of K+ (x, T, D), K- (x, T, D) must be equal to 00,


and thus the theorem is obvious in this case. Assume next that K(x, T, D) < 00. Using lemma 1, choose r>O so that (22), (23), (24) hold. The theorem follows then directly from (18).

In defining the multiplicity function K(x, T, D), we assumed that T was defined and continuous on some set U)D. Accordingly, if D is any domain in D, then the definition of K applies and we obtain the multiplicity function K (x, T, D). Thus, for fixed x and T, K (x, T, D) may be thought of as a function of the variable domain D (D. Similar comments apply to K+ (x, T, D), K- (x, T, D).

Theorem 2. Given T and D as in II.3.1, let D be a domain in D, and let {Dk} be a (finite or infinite) sequence of pair-wise disjoint domains in D. Then

L K (x, T, D k ) ;;;;; K (x, T, D) , (34) k

L K+ (x, T, D k ) ;;;;; K+ (x, T, D), k

L K- (x, T, D k ) ;;;;; K- (x, T, D) . k

(35)

(36)

Proof. The argument being entirely analogous for K, K+, K-, we indicate the proof merely for (35) to illustrate the method. Noting that (35) is trivial if K+ (x, T, D) = 00, we can assume that

K+(x, T, D) < 00 (37)

for the point x under consideration. Observe next that if every partial sum of a series of non-negative terms is less than or equal to a fixed finite number, then the series is convergent and its sum is less than or equal to the same number. Accordingly it is sufficient to prove (35) for the case of a finite sequence D1 , ... , Dm of pair-wise disjoint domains in D. Let then {Dk;}, j = 1, ... , mk be a finite sequence of pair-wise disjoint domains in Dk such that each Dk; is (x, T)-admissihle and

Dk;:::: Dk , fl(x, T, Dk;l > 0, j = 1, ... , mh •

Then {D k;} is a system ®+ (x, T, D k ) and also a system (5+ (x, T, D). Accordingly [see (11)]

M[@3+(x, T, D k )] ;;;;;K+(x, T, D) < 00.

Since this holds for every system (5+ (x, T, D k ), by (11) (applied to D k ) it follows that

K+ (x, T, D k );;;;; K+ (x, T, D) < 00.

Thus K+ (x, T, D k ) is finite. By remark 3 (applied to D.) we have therefore a system (5+ (x, T, D k ) such that

(38)


Choose such a system 6+ (x, T, D k ) for each k = 1, ... , m. Since D1 , ••• , Dm are pair-wise disjoint domains in D, clearly the totality of the domains constituting the systems 6+ (x, T, D k ), k = 1, ... , m, is a system 6+ (x, T, D) such that

M [6+ (x, T, D)] = L M [6+ (x, T, Dk)]' (39) k

By (11), applied to D, we have

M [6+ (x, T, D)]:;;;; K+ (x, T, D). (40)

From (38), (39), (40) we conclude that

L K+ (x, T, D k ) ':;;;;K+ (x, T, D), k

and the proof of (35) is complete. (34) and (36) are proved in a similar manner.

Remark 8. For fixed T and D, given as in 11.3.1, the multiplicity functions K (x, T, D), K+ (x, T, D), K-(x, T, D) are functions of the point xER". In our next theorem, we shall show that these multiplicity functions are lower semi-continuous with respect to x (see I.1.3, definition 28). In the sequel we shall use the following fact. Suppose that I (x) is a function in Rn which assumes only values that are integers, except that at certain points I(x) may have the value + 00. If I(x) is lower semi-continuous at a point Xo where I (xo) < 00, then there exists an open set 0 containing X o such that I (x);;;./(xo) for xEO. This is an immediate consequence of the definition of lower semicontinuity at a point.

Theorem 3. The multiplicity functions K (x, T, D), K+ (x, T, D), K- (x, T, D) are lower semi-continuous functions of x (for fixed T, D).

Proal. Again, the proof is entirely similar in all three cases, and this time we indicate the method of proof for the case of K (x, T, D). Take a point xo' If K(xo, T, D) =0, then K is clearly lower semi-cont.inuous at xo, since always K;;;'O. So we can assume that K(xo, T, D»O. Let then any positive real number A <K (xo, T, D) be assigned. In view of (10) there exists then a system 6(xo, T, D) such that

M[6(xo, T,D)] > A. (41)

Let D1 , ..• , Dm be the indicator domains for (xo, T, D) that constitute €l(xo, T, D). Then (41) means that

m

L l.u(xo, T, Dk)1 > A. (42) k~l

By the definition of an indicator domain, xoE C T Ir Dk . Since C T Ir Dk

is open, the set m

0= n C TlrDk k~l


is an open set containing xo. Hence 0 has a (unique) component Ll containing xo, and by 1.2.1, exercise 4, Ll is a domain. The lower semicontinuity of K (x, T, D) at Xo will be established if we can show that

K(x, T, D) > A for xELI. (43)

Now Ll is a connected subset of CTfrDk , k=1, ... , m, and hence (by 11.2.3, remark 1)

(44)

smce Dk is an indicator domain for (xo, T, D). From (44) we see that Dk is also an indicator domain for (x, T, D) if xELI, and hence the system <.5 (xo, T, D) is also a system <.5 (x, T, D) if xELl. From (41), (42), (44) we infer that

A<M[<.5(x,T,D)] if xELl. (45)

Since by (10) M[<.5(x, T, D)] ;;;;;'K(x, T, D),

(45) implies (43), and the proof is complete.

Remark 9. Given T and D as in 11.3.1, let ~: Vr+Rn be a sequence of continuous mappings (where ~-(Rn), and let Di ( Vi be a corresponding sequence of bounded domains. Suppose that the following condition holds: for every compact set F (D and every 13> 0 there exists an integer jo=jo(F, 13) such that (see 1.1.5, definition 5)

F ( D i ' e (T, ~,F) < 13 for j > j o· (46)

Definition. Under these conditions, we shall say that the mappings ~IDi converge uniformly to TID on compact subsets of D.

Theorem 4. Suppose that the mappings ~IDi converge uniformly to TID on compact sets in D, in the sense of the preceding definition. Take a point x and any sequence xi-+x. Then

K(x, T, D) ;;;;;, lim infK(xi' ~,Di)'

K+ (x, T, D);;;;;' lim inf K+ (Xi' ~,Di),

K- (x, T, D) ;;;;;, lim inf K- (Xi' ~,Di).

(47)

l48)

(49)

Proof. The argument being entirely similar in all three cases, we indicate the proof only for (48). Since (48) is obvious if K+ (x, T, D) = 0, we assume that K+ (x, T, D) > O. Consider any system <.5+ (x, T, D), and let D1 , .•. , Dm be the positive indicator domains that constitute



@5+ (x, T, D). Then [see (5)J m

M [@5+ (x, T, D)] = L,u (x, T, DkL k=l

,u(x, T, Dk ) > 0, k = 1, ... , m,

jrDkCDkCD, k=1, ... ,m, m

xE n C TlrDk • k=l

Choose ro to satisfy (see 1.1.4, definition 7)

(50)

(51 )

(52)

(53)

O<ro<te(x,TlrDkL k=1, ... ,m. (54)

In view of (53), such a choice of ro is possible by 1.1.4, exercise 3. Since xi~x, we shall have

IIXj - xii < ro for j > j~, (55)

for a properly chosen integer j~. Consider now the set

(56)

In view of (52), F is then a compact set in D. Hence we have an integer]o such that

(57)

Take j>j~+jo. For each k=1, ... ,m, we have then DkCDj , and (since Ir Dk C Dk CF)

e(T, ~,Ir D k ) < roo (58)

From (54), (55), (58) it follows that for k=1, ... ,m,

IIXj - xii + e(T, ~,Ir D k ) < e(x, T Ir D k ) for j > j~ + jo. (59)

By II.2.3, theorem 6, we conclude from (59) that Xj is (~, Dk)-admissible and

,u(xj,Tj,Dk)=,u(x,T,Dk ) for j>j~+jo. (60)

In view of (51), (56), (57), the relation (60) implies that the domains D1 , ... ,D", constitute a system @5+(xj' ~,Di) such that [see (50)]

By (11), applied to Tj and D j , it follows that

M [@5+ (x, T, D)] -::;. K+ (Xi' Tj, Di ) for j > j~ + jo.


By the definition of the inferior limit of a sequence, we conclude that

M[6+ (x, T, D)] ~lim infK+ (Xi' If, Di)'

Since this holds for every system 6+ (x, T, D), (48) follows.

Remark 10. Relevant special cases of the preceding theorems include the following.

(i) Given T and D as in IT.3.1, let {Di} be a sequence of domains such that (a) Di(D, (b) Di(Di+I for every j, and (c) for every compact set F(D there exists an integer jo=jo(F) such that F(Di for j>jo'

Under these conditions, we shall say that the domains Di fill up D from the interior. On choosing Ti = T, Xi = x, the assumptions of theorem 4 are clearly satisfied, and hence

K (x, T, D) :s:: lim inf K (x, T, Di ) . (61)

On the other hand, theorem 2 yields

K(x, T, D)~K(x, T, Di ) for every j.

In view of (61) it follows that if the domains Di fill up D from the interior, then

K (x, T, Di ) -*K(x, T, D).

An entirely similar argument shows that if the domains Di fill up D from the interior, then

K+ (x, T, Di ) -*K+ (x, T, D), K- (x, T, D j ) -*K'- (x, T, D).

(ii) Choosing If = T, D j = D, the assumptions of theorem 4 are clearly satisfied, and hence

K(x, T,D) :s:: lim infK(xi' T,D) if Xj-*x. (62)

Thus K (x, T, D) is a lower semi-continuous function of x, for fixed T and D. Applying the same argument to K+, K-, one sees that theorem 3 is implied by theorem 4.

Remark 11. The following observation will be useful later on. Let there be given a system 6 (x, T, D) such that

M[6(x, T,D)] =K(x, T,D). (63)

The positive and the negative indicator domains comprised in 6 (x, T, D) constitute a system 6+ (x, T, D) and a system 6- (x, T, D) respectively (one or both of which may be empty). We assert that (63) implies the relations

M [6+ (x, T, D)] = K+ (x, T, D),

M[6- (x, T, D)] =K- (x, T, D). 11*

(64)

(65)

164 Part II. Topological study of contiulJous transformations in Rn.

Indeed, if (64) and (65) do not both hold, then [see (11), (12), (63)] it would follow that

K(x, T, D) = M [t5 (x, T, D)] = M[t5+(x, T, D)] + + M [t5- (x, T, D)] < K+ (x, T, D) + K- (x, T, D),

in contradiction with theorem 1.

11.3.3. Essential maximal model continua. Definition 1. Given T and D as in 11.3.1, take a point xERft and consider an m.m.c. C for (x, T, D). Then C is termed an essential maximal model continuum (abbreviated to e.m.m.c.) for (x, T, D) provided that the following condition holds: for every open set 0 such that C (0 (D there exists an indicator domain D for (x, T,D) such that C(D andD(O. For given x, the class of all the essential maximal model continua for (x, T, D) will be denoted by (;l; (x, T, D). The number of elements of (;l; (x, T, D) will be denoted by k (x, T, D) (thus k may be equal to zero or (0).

Definition 2. A continuum C E (;l; (x, T, D) is said to be essentially isolated if the following condition holds: there exists an open set 0 such that (i) C(O,O(D, and (ii) C is the only element of (;l;(x, T,D) contained in O. An open set 0 with these properties (i), (ii) is then termed a characteristic neighborhood of C. For given x, the class of essentially isolated essential maximal model continua for (x, T, D) will be denoted by (;l;; (x, T, D).

Lemma 1. If CE (;l;i(X, T, D), then there exists a domain D such that the following holds: (a) C(D,D(D, and (b) if C't::(;l;(x, T,D) and C'nD=j=0, then C'=C.

Proof. Let 0 be a characteristic neighborhood of C. Since C is an e.m.m.c. for (x, T, D) and C (0, there exists (see definition 1) an indicator domain D for (x, T, D) such that C(D, D(O(D. Consider now any C'E (;l;(x, T, D) and suppose that C'nD=j= 0. By 11.3.1, lemma 5 it follows that C' (D and hence C' (0. Since C is the only element of (;l; (x, T, D) contained in 0, it follows that C' = C.

The following definition will be useful in locating elements of (;l;(x, T, D).

Definition 3. A determining sequence {Di} for (x, T, D) (see 11.3.1, definition 2) is termed essential if fl (x, T, Di) =j= 0 for every j [or equivalently, if every Di is an indicator domain for (x, T, D)J.

Lemma 2. If {Di} is an essential determining sequence for (x, T, D) then n Di = C E (;l; (x, T, D), (1 )

C (Di for every j. (2)


Prool. C is an m.m.c. and (2) holds by 11.3.1, lemma 6. Take now any open set 0 such that C(O(D. In view of the definition of C it follows from 1.1.3, exercise 24, that Dj(O for j sufficiently large. If j is so chosen, then C(Di , Dj(O, and Dj is an indicator domain for (x, T, D). Thus (1) follows in view of definition 1.

Lemma 3. If CE Q:(x, T, D), then there exists an essential determining sequence {Df} for (x, T, D) such that C= nDr

Prool. To begin with, by 11.3.1, lemma 8 there exists a determining sequence {Dj } for (x, T, D) such that

(3)

Then C(D1(D, and Dl is open. Since C is an e.m.m.c. for (x, T, D), there exists therefore an indicator domain Di for (x, T, D) such that C(Dt,Dt(D1. Then C(D2 nDt(D. Since D2 nDf is open and C is an e.m.m.c. for (x, T, D), there exists an indicator domain D: for (x, T, D) such that C (Dl, Dl (D2 n Dr Continuing this process, clearly one obtains a sequence {Dj} with the desired properties.

Lemma 4. If D is an indicator domain for (x, T, D), then D contains al least one e.m.m.c. for (x, T, D).

Prool. The assumption implies that

xtlTlrD, #(x,T,D)-=t-O. (4)

Let r be any real number such that (see 1.1.4, definition 7 and exercise 3)

o < r < e (x, T Ir D). (5)

Then the spherical neighborhood Ltr (x) (see 11.2.1) is contained in CTlrD. In view of (4) and 11.2.3, theorem 2, the set Dn T-1x is nonempty, and hence D n T-l Lt, (x) -=t- O. Set

0, = D n T-ILt,(X) -=t- 0,

and let srr denote the class of those components of Or that lie in D. By 11.3.1, lemma 3 [applied with G =L1r (x)] it follows that the elements of srr constitute an (x, T, D)-complete sequence. By 11.2.3, theorem 3 it follows that

In view of (4) we conclude that

# (x, T, Dr) -=t- 0 for some Dr E sr,. (6)

Now take a sequence

r1>···>rj>···>O, O<rj<e(x,TlrD), rj-+O.


Applying (6) with r=r1 , we obtain a domain Dr, such that

Dr, (D, I-" (x, T, Dr) =l= 0, Dr, E Sfr"

Replacing D by Dr, and r1 by r2 , the same argument yields a domain Dr. such that

Dr, (Dr" I-" (x, T, Dr,) =1= 0, Dr. E Sfr"

Continuation of this process yields a sequence of domains {Dri} such that

(7)

We verify now that {Dri} is a determining sequence for (x, T, D). Since I-" (x, T, D'i) =1= 0, by 11.2.3, theorem 2 it follows that

xE T Dr; (TDri .

Since L:1 r;+l(x) (Llr;(x), by 11.3.1, lemma 4 we have Dri+l (Dr;. As Drj ( T-IAr;(x), it follows that

15 T Dr;;;;;: (} T DrH~ 2rr;_1 -* 0.

Thus {Dr;} is a determining sequence for (x, T, D), and by (7) this determining sequence is essential. By lemma 2 it follows that

and the lemma is proved.

Remark. In the statement of the preceding lemma 4, let us add the assumption that D is a positive indicator domain [that is, I-" (x, T, D» 0]. Inspection of the proof reveals that (6) can be then stated in the form:

I-"(x, T, Dr) > ° for some D,E Sfr.

Similarly, it follows that the domains Dr; can be chosen to satisfy the condition I-" (x, T, Dr;) > 0. In summary: if D is a positive indicator domain, then the essential determining sequence {Dr;} can be so chosen that each Dr! is a positive indicator domain. Similarly, if D is a negative indicator domain, then each Dr; may be chosen as a negative indicator domain.

Lemma 5. If CE (2:i (x, T, D) and ° is a characteristic neighborhood of C, then the following holds.

(i) If D is any indicator domain for (x, T, D) such that D(O, then C(D.

(ii) If D1 , D2 are any two (x, T)-admissible domains such that C(D1(D2 (O, then I-"(x, T,D1)=I-"(x, T,D2)·


(iii) If D is any (x, T)-admissible domain such that C(D(O, then D is an indicator domain for (x, T, D).

(iv) 1£ DI , D2 are any two indicator domains for (x, T, D) such that C(DI(O, C(D2 (0, then p,(x, T, DI)=p,(x, T, D 2).

Proof. Given D as specified in (i), by lemma 4 there exists a continuum C'(D,C'E~(x, T,D). Since then C'(O, and ° is a characteristic neighborhood of C, we must have C=C'. Hence C(D, and (i) is proved. Consider next DI , D2 as specified in (ii) , and assume that p, (x, T, DI ) =1= p, (x, T, D 2). By remark 5 in 11.2.3 there follows the existence of an (x, T)-admissible domainD such that D(D2 , DnDI = 0, p, (x, T, D) =1= O. Then D would be an indicator domain for (x, T, D) contained in ° but not containing C, in contradiction with (i). Thus (ii) is proved. Consider now D as specified in (iii). Since C is an e.m.m.c. for (x, T, D) contained in the open set D, there exists an indicator domain D* for (x, T, D) such that C(D*(D. By (ii) it follows that p,(x, T, D) = p,(x, T, D*) =1=0, and thus D is also an indicator domain for (x, T, D). Finally, given DI , D2 as specified in (iv) , the inclusion C(Dl nD2 implies, since C is an e.m.m.c. for (x, T, D), the existence of an indicator domain D for (x, T, D) such that C(D(Dl nD2 • By (ii) it follows that

p,(x, T, DI ) =p,(x, T, D) =ft(x, T, D 2),

and (iv) follows.

Lemma 6. k (x, T, D) s;,K (x, T, D) [see definition 1 and 11.3.2 (10)].

Proof. The assertion is trivial if k (x, T, D) = O. So assume that k(x, T, D) >0. Let then CI , ..• , Cm be any finite set of elements of ~ (x, T, D). Since C1 , ••• , Cm are pair-wise disjoint continua in D, we can select open sets 01' ... , Om such that Cj(Oi(D, 0jnoi = ° for j =1= i. Since CjE ~ (x, T, D), we can select indicator domains Di for (x, T,D) such that Cj(Di ,15i (Oi,j=1, ... ,m. Then the domains Di constitute an indicator system S (x, T, D), and by 11.3.2 (7), (10) we conclude that

m = N[S(x, T, D)] s;, M[S(x, T, D)J :::;;;K(x, T, D).

Thus ~ (x, T, D) cannot contain more than K (x, T, D) elements, and the lemma is proved.

Lemma 7. k (x, T, D) is a lower semi-continuous function of x (for fixed T, D).

Proof. Take any point xO' If k(xo, T, D) =0, then clearly k is lower semi-continuous at xo, since k ~ O. So we can assume that k(xo, T, D) >0. Let any real number .it<k(xo, T, D) be assigned. Then the class ~(xo, T, D), which is now non-empty by assumption,


contains a set of distinct continua C1 , ••• , Cm , where m > A.. As in the proof of lemma 6, there follows the existence of pair-wise disjoint indicatordomainsD1 , ... , Dm for (xo, T, D) such thatCj(Dj,j = 1, ... , m. Then xoECTfrDj, j=1, ... , m. The set

m 0= n C T frD.

i=l 1

is then an open set containing xo. Hence 0 has a (unique) component Lt containing xo, and by 1.2.1, exercise 4 this component L1 is a domain. The lower semi-continuity of k at Xo will be established if we can show that

k(x,T,D»A. for xEL1. (8)

Now L1 is a connected subset of CTfrDj, j=1, ... , m, and hence (by 11.2.3, remark 1)

Il(X, T, Dj) =p(xo, T, Dj ) for x ELt. (9)

Now p(xo, T,DJ=t=-O, since Dj is an indicator domain for (xo, T,D). Thus, in view of (9), p(x, T, Dj)=!=O for xEL1. Accordingly, Dj is an indicator domain for (x, T, D) if xEL1. By lemma 4, there follows the existence of a continuum C,' E a: (x, T, D) such that C; (Dj • Since the domains Dl , "', Dm are pair-wise disjoint, the continua C~, ... , C:., are all distinct. In summary: for xELt, the class Cf(x, T, D) contains (at least) m> A. distinct elements, and (8) is verified.

11.3.4. The essential local index ie( C, T) and the algebraic mUltiplicity function (Le (;r, T, D). Given T and D as in 11.3.1, take a point xER" and consider a continuum CEC£';(x, T,D). Let 0 be a characteristic neighborhood of C. Since C is an e.m.m.c. for (x, T, D), there exists an indicator domain D for (x, T, D) such that C (D, D (0.

Definition 1. In terms of 0 and D as specified above, the essential local index ie (C, T) is defined by the formula

ie(C, T) =p(x, T,D). (1 )

Remark 1. To justify this definition, we have to verify, of course, that we obtain the same value for ie (C, T) if we select any other characteristic neighborhood 0' of C and any indicator domain D' for (x, T; D) such that C(D', D'(O'. Now since C(DnD', there exists an indicator domain D* for (x, T, D) such that C(D*, D* (DnD'. By 11.3.3, lemma 5 it follows that

p (x, T, D) = P (x, T, D*) = p (x, T, D').

Thus we get the same value for ie (C, T) whether we use 0 and D or 0' and D'.


Definition 2. For points xE Rn such that K (x, T, D) < 00, the algebraic multiplicity function p.. (x, T, D) is defined by the fonnula

p.e(x, T, D) = 2:>e(C, T), CE Q:,(x, T, D), (2)

with the understanding that p.. (x, T, D) = 0 if Q:; (x, T, D) is empty.

Remark 2. To clarify this definition, let us note that in view of 11.3.3, lemma 6 the assumption K (x, T, D) < 00 implies that the class Q:(x, T, D) is finite. It follows that Q:(x, T, D) = G:i(x, T, D). Indeed, this is obvious if @(x, T, D) is empty. On the other hand, if @(x, T, D) is non-empty and consists of the continua C1 , .•• , Cm ,

then these continua are pair-wise disjoint, and hence there exist pairwise disjoint open sets 01' ... , Om such that Cj(Oj(D. Clearly, each Cj is therefore essentially isolated. One sees now that the summation in (2) is finite (perhaps empty), and hence the definition (2) is certainly meaningful. One also sees that the condition CE@i(X, T, D) could be replaced by the condition CEG:(x, T, D).

Remark 3. If D is an indicator domain for (x, T, D), thenp.(x, T, D) is an integer different from zero. In view of (1) it follows that ie(C, T) is an integer different from zero. On the other hand, (2) may yield the value zero for P.e (x, T, D) as a result of cancellations or because (Ii (x, T, D) is empty.

Theorem 1. Given T and D as in 11.3.1, let D be a domain and ~

a point such that the following holds. (i) 15 (D. (ii) D is (x, T)admissible. (iii) D contains only a finite number m::? 0 of continua CE@(x, T, D). Then

p.(x, T,D) =2>e(C, T), CE@i(X, T,D), C(D, (3)

with the understanding that an empty summation has the value zero.

Proof. Case 1. m = O. Then the summation in (3) is empty, and we have to show that

p.(x, T, D) = O. (4)

If (4) is denied, then D should be an indicator domain for (x, T, D), and by 11.3.3, lemma 4 there would follow the existence of an e.m.m.c. for (x, T, D) contained in D, in contradiction with the assumption m=O.

Case 2. 0 < m < 00. Let then C1 , ... , Cm be the set of those elements of @(x, T, D) that lie in D. Since C1 , ... , Cm are pair-wise disjoint continua, there exist pair-wise disjoint open sets °1"", Om such that Cj(Oj(D, j = 1, ... , m. Clearly, each Cj is essentially isolated, and OJ is a characteristic neighborhood of Cj (see 11.3.3, definition 2). Thus

(5)


and there exists an indicator domain Di for (x, T, D) such that

Ci(Dj , Di(Oj' j=1, ... ,m. (6)

By definition 1, we have then

ie(Ci,T)=fl(x,T,Dj ), j=1, ... ,m,

and thus (3) will be established if we can show that m

fl (x, T, D) = L fl (x, T, D j ) • (7) i~l

Let (7) be denied. Then by II.2.3, theorem 4 there follows the existence of a domain D* such that D*(D, D*nDi = 0, j=1, ... , m, and fl (x, T, D*) =1= o. By II.3.3, lemma 4 there follows the existence of a continuum C*E~(x, T, D) contained in D*. Then C* would be an element of ~(x, T, D) contained in D and different from C1 , ... , Cm' in contradiction with the fact that C1 , ... , Cm are all the elements of ~ (x, T, D) lying in D. Thus (7) must hold, and the proof is complete.

Remark 4. In the preceding theorem, we required the inclusion D (D. Actually, in the proof we needed only the fact that T was defined and continuous on D. Accordingly, the theorem applies even if D = D, provided that T is defined and continuous on D. This observation yields the following statement.

Theorem 2. Given T and D as in IL3.1, assume that the inclusion D( U holds (thus T is now defined and continuous on D). Then

fle(x, T, D) =fl(x, T, D)

for all points x satisfying the conditions

xEf TlrD, K(x, T,D) <00.

(8)

(9)

Prool. Observe that the conditions (9) characterize precisely those points x where fle(x, T, D) and fl(x, T, D) are both defined. As noted in definition 2, the assumption K (x, T, D) < 00 implies that the class ~(x, T, D) is finite. Accordingly (see remark 3) we can apply theorem 1 with D = D, obtaining

fl(x, T,D)=Lie(C, T), CE~i(X,T,D), (10)

since now every element of ~i (x, T, D) is contained in D = D. Comparison of (10) and (2) yields (8).

Theorem 3. Given T and D as in 11.3.1, the following statements hold for the functions k (x, T, D), K (x, T, D).

(i) 0':;;;; k(x, T, D) ':;;;;K(x, T, D). (ii) k (x, T, D) = 0 if and only if K (x, T, D) = O.

(iii) k(x, T, D) =00 if and only if K(x, T, D) =00.

§ 1I.3. Multiplicity functions and index functions. 171

Proal. (i) is merely a re-statement of lemma 6 in II.3.3. In view of (i), K(x, T, D) = ° implies that k(x, T, D) = O. Conversely, if k(x, T, D) =0, then by II.3.3, lemma 4 it follows that there exists no indicator domain for (x, T, D), and hence K(x, T, D) =0 by 11.3.2, remark 2. Thus (ii) is also proved. In view of (i), k(x, T, D) =00

implies that K(x, T, D) =00. There remains to show that if k(x, T, D) is finite, then K (x, T, D) is also finite. If k (x, T, D) = 0, then this follows from (ii). So we assume that

0< k(x,. T, D) < 00. (11)

In this case, the class Q: (x, T, D) is finite and non-empty. Let C1 , ••. , Cm

be the continua that constitute ct(x, T,D), where m=k(x, T,D). As noted in remark 2, each Cj , j = 1, ... , m, is now essentially isolated, and

Q:(x, T, D) = (ri(X, T, D). (12)

Accordingly, ie(Ci , T) is defined, j=1, ... , m. Put m

L lie (Ci , T)I = N. (13) i=l

Then N is a (finite) positive integer. Consider now any indicator system 15(x, T, D), consisting of indicator domains D1 , • .. , D t • Then the assumptions of theorem 1 hold for each D k , k=1, ... , t, and we obtain the relation

(14)

Since the domains D1 , ... , Dt are pair-wise disjoint, no continuum Ci is utilized in (14) for two different values of k. Accordingly, (14) yields [in view of (13)]

t m

M[15(x, T, D)] = L Ifl(x, T, Dk)1 ~L lie (Ci , T)I = N. k=l f=l

Since the system 15(x, T, D) was arbitrary, in view of 11.3.2 (10) it follows that m

K(x, T, D) ~ L lie (Ci , T)I = N. (15 ) 1=1

Since N is finite, it follows that K (x, T, D) is finite, and the proof is complete.

Theorem 4. Given T and D as in II.3.t, let x be a point such that

K(x, T, D) < 00. (16)

Then the following formulas hold (with the understanding that an empty summation has the value zero).

K (x, T, D) = L lie (C, T) I, C E (ri(X, T, D), (17)

fle(x, T,D) =K+(x, T,D) -K-(x, T,D). (18)


Proof. Case 1. K(x, T, D) =0. By theorem 3 the class @3(x, T, D) and hence also the class @3,(x, T, D) is empty, and hence,u.(x, T, D) =0 by definition, and the summation in (17) has the value zero. Also, K+ and K- are both equal to zero by 11.3.2 (13), (14). Thus the theorem is trivial in this case.

Case 2. K(x, T, D»O. In view of (16) we have then

° <K(x, T, D) < 00.

By theorem 3 it follows that

0< k(x, T, D) < 00. (19)

Since (19) coincides with (11), the conclusions drawn from (11) in the course of the proof of theorem 3 are available. Hence, on denoting again by CI , ... , Cm the continua that constitute the class @3(x, T, D) = @3,(x, T,D), the inequality (15) holds again. Since the continua CI , ... , Cm are pair-wise disjoint, we have pair-wise disjoint open sets 01' ... , am such that Ci(O,(D, j=1, ... , m. Clearly each 0i is a characteristic neighborhood for the corresponding Ci , and we have indicator domains Di such that Ci (Di , Di (ai' j = 1, ... , m. By definition 1, we have then

ie(Ci , T) =fl(X, T, Di), j = 1, ... , m. (20)

The domains DI , ... , Dm constitute a system 15 (x, T, D), and (20) yields m

M [IS (x, T, D)] = L lie(Ci , T)I. (21) i~I

In view of 11.3.2 (10) it follows that m

K(x, T, D) ~ L Ii. (Ci , T)I· (22) i~I

From (15) and (22) we conclude that m

K(x, T,D) =L lie (Ci , T)I, (23) i~l

and (17) is proved. (21) and (23) yield

K(x, T, D) =M[15(x, T, D)J. (24)

In view 11.3.2, remark 11, we infer from (24) the relations

K+(x,T,D)=Lie(Ci,T), ie(Ci,T»O, (25)

K-(x, T,D) = -Lie(Ci , T), ie(Ci , T) <0, (26)

which correspond to (64), (65) in 11.3.2. Subtracting (26) from (25), in view of definition 2 we obtain (18), and the proof is complete.

§ I1.3. Multiplicity functions and index functions. 173

Remark 5. In view of part (iii) of theorem 3, the condition K(x, T, D) <00 can be replaced by the equivalent condition k(x, T, D) < 00 in definition 2, theorem 2 and theorem 4. Furthermore, in view of remark 2, in (17) the condition CE ~i(X, T, D) can be replaced by the condition CE ~(x, T, D).

11.3.5. Local properties. Given T and D as in 11.3.1, let x be a point in Rn and D a domain such that 15 (D. Then the property of D being (x, T)-admissible depends only upon the behavior of T on jr D, since D is (x, T)-admissible if and only if xEE TjrD. Furthermore, if D is (x, T)-admissible, then f-l (x, T, D) is defined and its value depends only upon the behavior of Ton jr D (see 11.2.3, remark 7). Accordingly, the property of D being a positive or negative indicator domain for (x, T, D) in the sense of 11.3.2 depends also only upon the behavior of T on jr D. The preceding observations yield examples of local properties, that is properties that depend only upon the behavior of T on certain subsets of D. To obtain precise statements of further relevant local properties, let us consider two transformations

(1 )

where 7;., T2 are continuous on the subsets U1 , U2 of Rn respectively. Let D1 , D2 be bounded domains such that Dl (U1 , D2 ( U2 • As in 11.3.1, we assume that T1 , T2 are bounded on D1 , D2 respectively.

Theorem. Let x be a point in R n and C a continuum in D1 nD2 •

Suppose that there exists an open set G such that

C(G, G(D1nD2 , T1IG=T2IG. (2)


(i) If C is an m.m.c. for (x, T1 , D1), then it is also an m.m.c. for (x, T2 , D2) (see 11.3.1, definition 1).

(ii) If C E (f (x, T1 , D1), then C E (} (x, T2 , D 2) (see 11.3.3, definition 1). (iii) If CE ~i(x, T1 , D 1), then CE (fi(x, T2 , D 2) (see 11.3.3, defini

tion 2). (iv) If ie(C, T1) is defined, then ie(C, T2) is also defined, and

i.(C, T;,)=ie(C, T2) (see 11.3.4, definition 1).

Pro oj oj (i). Suppose that C is an m.m.c. for (x, T1 , Dl)' Then T;, C = x, and hence by (2) also T2C = x. Thus

C(Gn T2-1 x (D 2 n T2-1 X •

As C is connected, it follows that C is contained in a component Cz of Dzn TZ-l x. Clearly (i) will be proved if we can show that Cz = C. Now since C is an m.m.c. for (x, 7;., D1), by 11.3.1, lemma 8 there


exists a determining sequence {Di} for (x, TI , DI ) such that

C = n Di ,

Di (G for every j.

(3)

(4)

Since TI !G=T2 !G and Di is (x, Til-admissible, we infer from (4) that Dj is also (x, T2)-admissible. As C2 is a component of D 2 n T2-I x, from 11.3.1, lemma 5 it follows that either C2 (Di or C2 (CDi . The second alternative is ruled out by the observation that c2 n Di=l= 0 since C2 )C and C(Di. Thus

C ( C2 (Dj (Di for every j. (5)


C (C2 (n Dj = C.

Hence C2 = C, and (i) is proved.

Proof of (ii). Suppose that CE 0: (x, 7;., DI). Then C is an m.m.c. for (x, T1 , DI ) and hence [by (i)] also for (x, T2 , D2). Let there be assigned any open set 0 such that C (0 ( D 2 • Clearly (ii) will be proved if we can exhibit an indicator domain D for (x, T2 , D 2) such that C(D,D(O. Put o*=onG. Then 0* is an open set, and C(0*(D1

by (2). Since C E 0: (x, TI , D1), there follows the existence of an indicator domain D for (x, T1 , DI) such that C(D, 15(0*. As O*(G and TI!G= T2! G, we have also I;. [15 = T2 [D. Hence (by the initial remarks in the present section) D is an indicator domain for (x, T2 , D 2) also, and C(D, 15(0*(0. Thus (ii) is proved.

Proof of (iii). Suppose that CE0:;(x, 7;., D1). Then CE0:(x, 7;., D I )

and hence also CE0:(x, T2 , D 2) by (ii). To prove (iii), there remains to exhibit a characteristic neighborhood of C (relative to T2 , D 2) in the sense of 11.3.3, definition 2. Now since CE 0:i (x, T I , D I ), there exists a characteristic neighborhood 0 1 of C relative to TI , D I . On setting O2 = 0 1 n G, it follows readily that O2 is a characteristic neighborhood of C relative to T2 , D 2 • Indeed, obviously C(02,02(G(D2 by (2). Thus we have to show yet that C is the only element of rJ (X,T2' D 2)

contained in 02. Now if C2 E 0: (x, T2 , Dz) and C2 (Oz, then also C2 (G, and on reversing in (ii) the roles of TI [D1 and T2 [D2 , it follows that C2E 0: (x, TI , Dl). As C2 (OZ(01 and 01 is a characteristic neighborhood of C (relative to TI , D I ), C2 must coincide with C, and thus it is established that C is the only element of 0:(x, Tz, D z) contained in 02.

Proof of (iv). Suppose that ie(C, TI ) is defined. Then CE rJi(x, 7;.,D I )

and hence CE 0:i (x, T2 , D z) by (iii). Thus ie (C, Tz) is defined. Let °1 , 02 be characteristic neighborhoods of C relative to I;. , Dl and


T2, D2 respectively. Put 0 =01 n 02n G. Then clearly 0 is a characteristic neighborhood of C, in view of (2), with respect to both 7;, D1 and T2 , D 2 • Let D be an indicator domain for (x, 7;, Dl ) such that C(D, D(O. Since ~iO= T2iO in view of (2) and D(O, it follows by the initial remarks in the present section that

(6)

On the other· hand, by definition 1 in 11.3.4,

ie(C, TI ) =f-l(x, Tl,D), ie(C, T2) =f-l(x, T2,D).

By (6) it follows that ie(C, TI ) =ie(C, T2), and the proof is complete.

Remark 1. Given T and D as in 11.3.1, consider a domain D*( D. Put DI = D*, D2 = D, 7; = T2 = T, and let x be a point in R" and C a continuum in D*. The following statements are then direct consequences of the preceding theorem.

(i) C is an m.m.c. for (x, T, D*) if and only if it is an m.m.c. for (x, T, D).

(ii) CEGl:(x, T,D*) if and only if CEGl:(x, T,D). (iii) CEGl:i(x, T,D*) if and only if CEGl:i(x, T,D). (iv) ie (C, T) is defined relative to T, D* if and only if it is defined

relative to T, D and has the same value in either case.

Remark 2. Given T, D, D* as in remark 1, take a point x in R". Then 11.3.3, definition 1 applies to T and D* and yields a multiplicity function k (x, T, D*). By part (ii) of remark 1 it follows that k (x, T, D*) is equal to the number of those continua CE Gl:(x, T, D) that lie in D*.

Remark 3. Given T,D,D*,x asinremark2,supposethatk(x, T, D*) < (Xl.

By 11.3.4, theorem 3 (applied to T, D*) we have then also K(x, T, D*) < (Xl,

and 11.3.4, definition 2 yields the algebraic multiplicity function

f-le (x, T, D*) = "L,ie (C, T), C E ~i(X, T, D*).

In view of remark 1 we have the equivalent formula

f-le (x, T, D*) = Lie (C, T), C E Gl:i (x, T, D), C (D*.

Similarly, remark 1 and 11.3.4, theorem 4 yield the formula

K(x, T, D*) = L iic(G, T)i, C E Gl:;(x, T, D), C (D*.

Remark 4. Given T and D as in 11.3.1, consider a domain D(D and a point xE R" such that

I5 ( D, x Ef T Ir D . (7)


Then fl (x, T, D) is defined (see 11.2.2). Assume that

fl(x, T, D) =f= O.

Then K(x, T,D);;;;;'1.

(8)

(9)

Indeed, (7) and (8) imply that D is an indicator domain for (x, T, D). By 11.3.3, lemma 4 it follows that D contains at least one e.m.m.c. C for (x, T, D). By remark 1, C is then also an e.m.m.c. for (x, T, D), and thus k(x, T, D) ;;;;;, 1. From lemma 6 in 11.3.3 (applied to D), we conclude that

K(x, T, D) ;;;;;'k(x, T, D) ~ 1,


11.3.6. Essential sets. Given TandD as in 11.3.1, denote by E (x, T, D) the union of all the continua CE (f (x, T, D) (see 11.3.3, definition 1). In symbols:

E(x, T, D) = U c, CE(f(x, T, D). ( 1)

Next, denote by E (T, D) the union of all the sets E (x, T, D) corresponding to all the points xE R". In symbols:

E(T, D) = U E(x, T, D), xE R". (2)

Let (fP (x, T, D) be the class of all those continua C E (f (x, T, D) which reduce to single points, and put

EP(x, T,D) = UC, CE(fP(x, T,D),

EP(T, D) = U EP(x, T, D), xE R". (3 ) (4)

Similarly, the class (fi (x, T, D) (see 11.3.3, definition 2) gives rise to the sets

Ei (x, T, D) = U C, C E (fi (x, T, D),

Ei(T, D) = U Ei(x, T, D), xE R".

(5) (6)

On denoting by (ft (x, T, D) the class of those continua C E (fi (x, T, D) which reduce to single points, we obtain [in analogy with (3), (4)] the sets

Et(x,T,D)=UC, CEcr't(x,T,D),

Et(T,D) = UEt(x, T,D), xER".

(7)

(8)

Some or all of the sets defined in (1) to (8) may be empty. The following relations are direct consequences of the definitions.

Ei(T, D) (E(T, D) (D,

Et(T, D) (EP(T, D) (D,

EP(T, D) (E(T, D),

Et(T, D) (E.(T, D),

Et(T, D) = EP(T, D) n Ej(T, D).

(9)

(10)

(11)

(12)

(13 )


Definition 1. The sets E(T, D), EP(T, D), E.(T, D), Ef(T, D) are termed the essential sets for (T, D).

Our first objective in this section is t9 show that these essential sets are BOREL sets (see 1.1.4, definition 11). In preparation, we shall presently introduce and discuss some auxiliary sets.

Definition 2. Let j and m be integers such that i;;;:;;'o, m>O. Then Sim (T, D) will denote the set of those points uED for which there exists a domain D (depending upon u, j, m) with the following properties.

(i) uED,15(D. (ii) D is an indicator domain for (Tu, T, D).

(iii) ~ T 15 < 11m. (iv) k(Tu, T, D»j (see 11.3.3, definition 1 and 11.3.5, remark 2).

Lemma 1. The set Sim (T, D) is open.

Proof. The assertion is obvious if Sim (T, D) = O. So assume that Sim(T, DH=O, and consider an arbitrary point uoESim(T, D). Then there exists a corresponding domain D with the properties (i) to (iv) stated in definition 2. Property (ii) implies that T U o ~ T fr D and p,(Tuo, T, D) =1=0. By 11.2.3, theorem 1 there follows the existence of an open set G1 such that

T~E~, 0~

p,(x, T, D) =p,(Tuo, T, D) =1= 0 for xE G1 • (15)

Since k (x, T, D) is a lower semi-continuous function of x (see 11.3.3, lemma 7), property (iv) implies the existence of an open set G2 such that

(16)

Then G1 nG2 is an open set containing Tuo. Since T is continuous, there exists an open set 0 such that

(17)

Clearly, the lemma will be proved if we can show that O(S;m(T, D). In turn, this inclusion will be established by showing that the domain D possesses the properties (i) to (iv) stated in definition 2 relative to every point uEO. For the properties (i) and (iii) this is obvious. As regards property (ii), we have TuE TO(G1 by (17), and hence p,(Tu, T, D) =1=0 by (15). Thus D is an indicator domain for (Tu, T, D), and property (ii) is verified. By (17) we have TuE TO(G2 , and hence k(Tu, T, D»j by (16). Thus property (iv) IS also verified, and the lemma is proved.


t 78 Part II. Topological study of continuous transformations in R".

Definition 3. If j is a non-negative integer, then the set Sj (T, D) is defined by the fonnula

00

Sj(T, D) = n Sjm(T, D). (18) tn=l

Lemma 2. Sj(T, D)) SfH (T, D).

Proof. Obviously Sim (T, D)) SjH,m (T, D), and thus the assertion is a direct consequence of (18).

Lemma 3. If uoESj(T, D), then there exists a continuum Co and a sequence of domains {Dm} such that the following holds.

(i) uoEDm' 15m CD. (ii) Each Dm is an indicator domain for (Tuo, T, D). (iii) ~ T 15m < 11m. (iv) k(Tuo, T, Dm»j. (v) ~toECoCDm' CoE~(Tuo, T, D).

Proof. In view of (18), the assumption implies that uoESjm(T, D) fcr every integer m;;:::;1. Accordingly, for every integer m;;:;;1 there follows, by definition 2, the existence of a domain D>n with the properties (i) to (iv). There remains to verify the existence of the continuum Co satisfying (v). Observe that the properties (i) to (iii) imply, by 11.3.1, lemma 9, the existence of a subsequence {Dm.} which is a detennining sequence for (Tuo, T, D). In view of property (ii), this detennining sequence is essential (see 11.3.3, definition 3). Hence by 11.3.3, lemma 2 the set

(19)

is an e.m.m.c. for (Tuo, T, D). Since U o ECo by property (i), we have verified property (v) except for the inclusion

Co C Dm , m = 1, 2, .... (20)

Now since Dm is (Tuo, T)-admissible by property (ii) , we conclude from 11.3.1, lemma 5 that either (20) holds or else CoCC15m • The second alternative is ruled out by the observation that Co n Dm =1= 0 since uoEConD",. Thus (20) holds, and the lemma is proved.

Lemma 4. So(T,D)=E(T,D). Proof. We first verify the inclusion

So(T, D) C E(T, D). (21)

If So(T, D) = 0, then (21) is obvious. So assume that So(T, D) =1= 0, and consider an arbitrary point uoE So(T, D). By part (v) of lemma 3 and in view of (1) and (2) we have

ttoEE(Tuo, T,D)CE(T,D),


and (21) follows. There remains to verify the inclusion

E(T, D) (So(T, D). (22)

If E (T, D) = 0, then (22) is obvious. So assume that E (T, D) =1= 0 and consider an arbitrary point uoE E (T, D). In view of (1) and (2) there exists a continuum Co such that

(23)

Take now any integer m ~ 1 and consider the spherical neighborhood LI(Tuo, 1/2 m) (see 1.1.4, definition 2). The set

0= Dn T-1L1 (Tuo, 1/2m) (24)

is then an open set containing Co. Since Co is an e.m.m.c. for (Tuo, T,D). there follows the existence of an indicator domain D for (Tuo, T, D) such that

Co(D, 15(0. (25)

From (24) and (25) we see that

TD (TO (LI (Tuo, 1/2m),

and hence t5 T 15 < 11m. Next we verify that

k(Tuo• T, D) > o. (26)

Indeed, D contains the continuum Co E <t(Tuo• T. D), and thus (26) holds by II.3.5, remark 2. In summary: the domain D satisfies, relalive to uo, the conditions (i) to (iv) in definition 2 for j = 0 and every integer m~1. Hence uoE So(T, D), and (22) follows.

Lemma 5. E, (T, D) n Sl (T, D) = 0.

Proof. We show that the assumption of the existence of a point

(27)

leads to a contradiction as follows. Since uoE Sl (T, D), from lemma 3 we infer the existence of a sequence of domains {D,J and of a continuum Co with the properties (i) to (v) stated there, with j = 1. In view of property (v), Co is then the unique m.m.c. containing uo, and since uoEE;(T, D) it follows [see (5), (6)] that Cot: ~,(Tuo, T, D). Accordingly (see 11.3.3, definition 2) there exists a characteristic neighborhood 0 of Co. From the definition of such a neighborhood it follows that if a domain D satisfies the conditions

(28) 12*


then Co is the only element of the class Q;(Tuo, T, D) that lies III D. Hence, by remark 2 in 11.3.5, (28) implies that

k(Tuo, T, D) = 1. (29)

As noted in the proof of lemma 3, the sequence {D",} contains a subsequence {DmT} which is a determining sequence for (Tuo, T, D) such that

()O)

Since CoCO, it follows by (v) in lemma 3 and 1.1.3, exercise 24 that

Co (DmT , 15m,( a for r large.

Let r be so chosen that (31) holds. Then by (28), (29), we have

k(Tuo, T, DmT) = 1,

while by property (iv) in lemma 3 we should have

k(Tuo, T, DmT) > 1,

(31)

()2)

(33)

since now j = 1. Thus (27) leads to the contradictory relations ()2) and (33), and the lemma is proved.

Lemma 6. E;(T,D)=So(T,D)-Sl(T,D).

Proof. By (9) and lemma 4 we have E,(T, D) (So(T, D), while lemma 5 yields E;(T, D)(CS1 (T, D). Hence

E;(T, D) (So(T, D) - Sl(T, D).

There remains to verify the inclusion

This inclusion will be proved by showing that the assumption

implies that U o E E;(T, D).

(34)

(35)

(36)

So assume that (35) holds. Then uoE So(T, D). By lemma 3, applied with j = 0, there follows the existence of a sequence of domains {Dm} and of a continuum Co with the properties (i) to (v) stated there. In particular

k(Tuo, T, Dm) > 0, m = 1, 2, .... (37)

We assert that k(Tuo, T, Dm) = 1 for some m. ()S)

Indeed, if (38) is denied, then in view of ()7) we should have k(Tuo, T, Dm) >1 for every integer m;;:;:1, and hence it would follow


that uoE SJ(T, D), in contradiction with (35). Thus (38) is verified. Ifm is an integer such that (38) holds, then by 11.3.5, remark 2 it follows that Co is the unique element of the class ~(Tuo, T, D) contained in Dm. As Dm is open, we conclude that CoE ~j(Tuo, T, D), and hence [see (5), (6)J

Thus (36) is established and the lemma is proved.

Definition 4. If m is a positive integer, then S! (T, D) denotes the set of all those points u ED for which there exists a domain D (depending upon m and u) with the following properties.

(i) uED,D(D. (ii) D is (Tu, T)-admissible (see 11.2.1). (iii) aD <11m (see 1.1.4, definition 4).

Lemma 7. The set S! (T, D) is open. The proof is similar to that of lemma 1.

Definition /). The set 5* (T, D) is defined by the formula

00

S*(T, D) = n S!(T, D). m=l

(39)

Lemma 8. If uoED, then uoE 5* (T, D) if and only if the point ~to by itself is an m.m.c. for (Tuo, T, D).

This is a ready consequence of the definitions involved.

Lemma 9. EP(T, D) =E(T, D) n 5* (T, D).

Lemma 10. EfCT,D)=Ei(T,D)nS*(T,D).

Since the proofs of these two lemmas are entirely similar, we indicate merely the proof of lemma 10. If uEEf(T, D), then ~t by itself is an m.m.c. (and indeed an e.m.m.c.) for (Tu, T, D), and hence uE 5* (T, D) by lemma 8. Thus Ef(T, D)(S*(T, D). In view of (12) it follows that

Ef(T, D) (Ej(T, D) n 5* (T, D). (40)

Next, assume that uEE,(T,D)nS*(T,D). Then uEEi(T,D), and hence there exists a continuum CE~i(Tu, T, D) such that u E C. But since also uE 5* (T, D), this continuum C must reduce to the single point u by lemma 8, and hence uEEf(T, D). Thus

Ei(T, D) n S*(T, D) (Ef(T, D),

and in view of (40) lemma 10 is proved.

Theorem 1. The essential sets E(T, D), Ei(T, D), EP(T, D), EfCT,D) are BOREL sets.


Proof. Since the set Sfm(T, D) is open by lemma 1, the set Sf(T, D), as a countable intersection of BOREL sets, is a BOREL set. Hence E(T, D) is a BOREL set by lemma 4, and E,(T, D) is a BOREL set by lemma 6. As the set S!(T, D) is open by lemma 7, the set S*(T, D), as a countable intersection of BOREL sets, is a BOREL set. In view of lemma 9 and lemma 10, EP(T, D) and Ef(T, D) are (finite) intersections of BOREL sets and hence these two sets are themselves BOREL sets.

Our next objective is to study certain auxiliary sets related to the local index ie (C, T).

Definition 6. If s is an integer different from zero and m is a positive integer, then Fsm (T, D) will denote the set of those points u E D for which there exists a domain D (depending upon u, s, m) with the following properties.

(i) uED,D(D. (ii) D is an indicator domain for (Tu, T, D). (iii) ~ T D < 11m. (iv) f-l(Tu, T, D) =S=f=O.

Lemma 11. The set F.", (T, D) is open. The proof is entirely similar to that of lemma 1.

Definition 7. If s is an integer different from zero, then the set Fs (T, D) is defined by the formula

00

Fs(T,D) = nFsm(T,D). m=l

(41)

Lemma 12. If uEFs(T, D), then there exists a continuum C and a sequence of domains {Dr} such that the following holds.

(i) u E C (Dr for every r. (ii) CE(f(Tu, T,D). (iii) {Dr} is a determining sequence for (T,u, T, D). (iv) C = n Dr. (v) f-l (T u, T, Dr) = s for every r.

Proof. The assumption uEFs(T, D) implies [see (41) and definition 6] that for every positive integer m there exists a domain Dm such that

uEDm,D",(D,

b TDm < 11m,

f-l(Tu, T,Dm) =s,

(42)

(43)

(44)

and each Dm is an indicator domain for (Tu, T, D). By 11.3.1, lemma 9 there follows the existence of a subsequence {Dmr} which is a determining sequence for (Tu, T, D). Since S=f=O, this determining sequence


is essential (see 11.3.3, definition 3). Hence, by lemma 2 in 11.3.3, the set C = n 15m, is an e.m.m.c. for Tu, T, D), and C (Dm, for every r. In view (42) and (44) we have

uEC, fJ,(Tu, T,Dm,)=s for every r.

Hence on setting D' = Dm" the conditions (i) to (v) hold, and the lemma is proved.

Definition 8. If s is an integer different from zero, then Fs* (T, D) denotes the union of all those continua C which satisfy the following conditions.

(i) C E (i;i (x, T, D) for some point x E R". (ii) i. (C, T) = s. Lemma 13. Fs*(T, D)= F.(T, D)nEi(T, D).

Proof. We assert the inclusion

F. (T, D) n Ei (T, 11) (Fs* (T, D), (45)

where we can assume that the set on the left in (45) is non-empty, since otherwise the inclusion is trivial. Consider any point

uEFs(T, D) n Ei(T, D).

Then uEF.(T, D), and hence there exists a continuum C and a sequence {Dr} with the properties stated in lemma 12. Since also uEEi(T, D), it follows that C E (i;i (Tu, T, D), and there remains to show (see definition 8) that

i.(C,T)=s. (46)

Let 0 be a characteristic neighborhood of C (see 11.3.3, definition 2). In view of (iii), (iv) in lemma 12, it follows by 1.1.3, exercise 24, that 15'(0 for r sufficiently large. If 15'(0, then (see 11.3.4, definition 1) i.(C, T) =fJ,(Tu, T, D'), and (46) follows by (v) in lemma 12. Thus (45) is verified. The complementary inclusion

Fs*(T, D) (F.(T, D) n Ei(T, D)

is established in a similar manner, and the lemma follows.

Theorem 2. The set Fs* (T, D) is a BOREL set.

Proof. Since Fsm(T, D) is open by lemma 11, the set Fs(T, D) is a countable intersection of open sets, and hence it is a BOREL set. As E;,{T, D) is a BOREL set by theorem 1, F.*(T, D) appears (in view of lemma 13) as the intersection of two BOREL sets and hence this set is also a BOREL set.


The following two lemmas are included in view of applications in the sequel.

Lemma 14. Let u be a point in D. Then uEEf(T, D) if and only if for every e > 0 there exists a domain Due (which depends upon u and s) such that the following holds.

(i) uEDue(D.

(ii) bDue<s.

(iii) 1:;;;:,K(Tu, T,Due)<oo. (iv) If D is any domain such that uED(Due , then

K(Tu, T,D) =K(Tu, T,Due)'

Proof. Assume first that for every e> 0 there exists a domain Due satisfying the conditions (i) to (iv). Consider the domain Dul (corresponding to e= 1). By (iii) and 11.3.4, theorem 3 (applied to Dul) it follows that k(Tu, T, Dul) is finite and different from zero. In view of remark 2 in 11.3.5, we conclude that DUI contains a finite, non-zero number of essential maximal model continua Cl , ... , Cm for (T u, T, D). Clearly (see 11.3.3, definition 2)

(47)

Consider now a domain DUe' In view of (ii), there exists a number rj > 0 such that

(48)

Assume that 0 < s < rj. In view of (iii) and 11.3.4, theorem 3 (applied to Due) we have

k(Tu, T, Due):;2; 1.

By remark 2 in 11.3.5, we conclude that Due contains at least one e.m.m.c. C for (Tu, T, D). In view of (48), it follows that C(Dul , and hence C must coincide with one of the continua Cl , ... , Cm . In other words, for arbitrarily small values of e > 0 at least one of the continua Cl , ... , C", must lie in Due' In view of (ii) this clearly implies that one of Cl , ... , Cm

contains u. Let the notations be so chosen that u E Cl . By the preceding remarks, we have then

uECl(Due for O<e<rj. (49)

In view of (ii), it is clear from (49) that Cl must reduce to the point u, and by (47) it follows then that uEEf(T, D) [see 11.3.6 (8)]. Suppose, conversely, that uEEf(T, D). Then u itself is an e.m.m.c. for (Tu, T,D), and there exists an open set ° such that uEO, 0 (D, and u itself is


the only e.m.m.c. for (Tu, T, D) contained in O. Assign now 8>0 arbitrarily. We can select a domain Due such that

(50)

We assert that Due satisfies the conditions (i) to (iv). For (i) and (ii) this is obvious by (50). Take now any domain D such that uED(Due ' Observe that since u itself is the only e.m.m.c. for (Tu, T, D) contained in Due and hence in D, we have

k(Tu, T, D) = k(Tu, T, Due) = 1.

By 11.3.5, remark 3 it follows that

K(Tu, T, D) = li,(u, T)I = K(Tu, T, Due)' (51)

Since i,(u, T) is finite and different from zero (see 11.3.4, remark 3), (51) implies (iii) and (iv) , and the lemma is proved.

Lemma 15. Let x be a point in Rn such that

K(x, T, D) < 00,

xE T[E(T, D) -Ef(T, D)].

(52)

(53)

Then there exists a number r;=r;(x»O such that the following holds: if Dl , ... , DN are pair-wise disjoint domains in D such that JDi<Tj, j=1, ... , N, then

N

L K (x, T, Dj ) < K (x, T, D). i=l

(54)

Proof. (53) implies that there exists at least one e.m.m.c. for (x, T, D). In view of (52) and 11.3.4, theorem 3 it follows that the number of essential maximal model continua for (x, T, D) is finite and different from zero. Let Cl , ... , Cm be these continua. At least one of these continua must have a positive diameter, since otherwise (53) could not hold. Let the notations be so chosen that

(55)

We assert that the number

(56)

satisfies the requirements of the lemma. Indeed, let Dl , ... , DN be pair-wise disjoint domains in D such that

(57)


By 11.3.4, theorem 4 and 11.3.5, remark 3 we have

In

K(x, T, D) = 2: lie(Ck , T)J, k=l

(58)

(59)

In view of (55), (56), (57) it is clear that the term lie(C1 , T)I does not occur for any value j = 1, ... , N in the formula (59). Thus (58) and (59) yield N

I K(x, T, D j ) :;;;;, K(x, T, D) -li.(CI , T)I. j=l

Since ie (C1 , T) =1= 0, the inequality (54) follows.

11.3.7. The local index iB ('U, T). This index function will be needed later on in connection with a theory of continuous transformations initiated by BANACH, and the purpose of this section is to present its definition and some of its properties. The bounded continuous transformation T:D_Rn is assumed to be given as in 11.3.1, where we now assume that U =D.

Definition 1. If u and u' are two distinct points in D such that Tu=Tu', then u and u' are termed relatives of each other.

Definition 2. The symbol I (T, D) denotes the set of those points uE D which possess some neighborhood clear of relatives. Thus uEI(T, D) if and only if there exists an open set Ou such that the conditions u'EOu,u'=I=u imply that Tu'=I=Tu.

Lemma 1. I(T, D) is a BOREL set.

Proof. We can assume that I(T, D) =1= 0, since otherwise the assertion is obvious. Given two positive integers m and n, and a point uED, let Rmn (u) denote the set of those points u' which satisfy the inequalities

_1_:;;;;, Ilu'-ull:;;;;,~. m+n m

Designate by Smn the set of those points uED for which the following holds. (i) Rmn (u) (D. (ii) Rmn (u) contains no relative of u. In other words, u'ERmn(u) implies that u'ED and Tu' =1= Tu. One sees readily, by means of elementary arguments, that Smn is open and

00 00

I(T, D) = U n Smn. m=l n=l

Thus I(T, D) is a BOREL set.

We now define the local index iB (u, T), for uED, as follows. We set

iB(u, T) = ° if u Ef I(T, D). (1 )


If uEI(T, D), then clearly u taken by itself is an m.m.c. for (Tu, T, D) (see 11.3.1, definition 1), and it is then either unessential or essential, in the sense of 11.3.3, definition 1. Using the essential sets introduced in 11.3.6, clearly a point uE I (T, D) is an e.m.m.c. if and only if uEEf(T, D). We now set

iB(u, T) = ° if uEI(T,D)-Ef(T,D),

iB(u, T) = ie(u, T) if u E Ef(T, D) n I(T, D).

(2)

(3)

As regards (3), note that if u EEf(T, D) then u by itself is an essentially isolated maximal model continuum, and hence the essential local index ie (u, T) is defined (see 11.3.4). Thus iB(u, T) is well defined for uED.

Lemma 2. iB (u, T) is BOREL measurable in D. Proof. Recall that I(T, D) is a BOREL set by lemma 1 and Ef(T, D)

is a BOREL set by 11.3.6, theorem 1. Since iB (u, T) vanishes on the BOREL set D-I(T, D), it suffices to show that iB(u, T) is BOREL

measurable on I(T, D). Let s be an integer, and denote by Is(T, D) the set of those points uEI(T,D) where iB(u, T)=s. As iB(u, T) is integral-valued, it is sufficient to show that the sets Is (T, D) are BOREL

sets. Now since the essential local index ie is always different from zero (see 11.3.4, remark 3), clearly

10(T, D) = I(T, D) - Ef(T, D),

and thus 10 (T, D) is a BOREL set. If s =l= 0, then consider the set Fs*(T, D) in 11.3.6, definition 8. Since F,*(T, D) is a BOREL set by 11.3.6, theorem 2, and clearly [in view of (2) and (3)J

Is(T, D) = I(T, D) nFs*(T, D) for s =l= 0,

it follows that Is (T, D) is a BOREL set, and the lemma is proved.

Remark. If D is any domain in D, then the preceding discussion applies to the transformation TID. Then I(T, D) denotes the set of those points uED which possess a neighborhood clear of relatives. Clearly

I(T, D) = D n I(T, D).

In view of the comments in 11.3.5 on local properties it is also clear that one has

iB (u, TID) = iB (u, T) for u ED.

II.3.S. Topologically similar transformations. In view of the topological character of the definition of the various multiplicity functions


and index functions that we have studied so far, it is clear that these functions should possess certain properties of topological invariance. The purpose of this section is to study this point in some detail.

Definition. Let there be given two bounded continuous transformations

T1:Dc,>-Rn,

Tz:Dz----7Rn,

(1 )

(2)

where D1 , Dz are bounded domains in RH. Then TI and Tz are termed topologically similar if there exists a homeomorphism T from DI onto Dz such that

(3) We shall write

(4)

to state that TI and T2 are topologically sirr..ilar. In case (4) holds, a homeomorphism T from DI onto Dz which satisfies (3) will be termed a connecting homeomorphism.

To justify the terminology just introduced, observe that if T is a homeomorphism from DI onto D2 for which (3) holds, then T-I is a homeomorphism from Dz onto DI such that Tz u=T1 T-1u for u EDz. Thus topological similarity is a symmetric relation. It is also evident that topological similarity is reflexive and transitive.

Let there be given two topologically similar transformations T1 , T2 and a connecting homeomorphism T as in (1), (2), (3). These transformations are thought of as fixed throughout this section.

Lemma 1. Let D be a domain in DI and x a point in Rn such that

l5 (D1 , x~.Tdr D.

Then T D is a domain in D2 such that

TD(D2, xEETzfrTD.

(5)

(6)

Proof. Since T is defined on l5 in view of (5), T D is a domain (see 11.2.4). By 11.2.4 (6) we have TD = TD, and hence

T D = T 15 ( T Dl = D 2 •

Also, in view of (3) and (7),

T2TD = Tz Tl5 = TID.

By 11.2.4 (7) we have T fr D = fr T D, and hence [in view of (3)]

T2 fr T D = Tz T fr D = TI fr D ,

and the relations (6) follow now from (7), (5) and (9).

(7)

(8)

(9)


Lemma 2. Let D be a domain in Dl and x a point in Rn such that (5) holds. Then (see 11.2.2 and 11.2.4 for notations)

(10)

Pro oj. Since (5) holds, (6) also holds by lemma 1. Accordingly, fl (x, 7;., D), fl (x, T2, T D) and l (T, D) are defined (see II.2.2 and II.2.4). Take now a frame [A, B, LlJ which is (x, T1 , D)-admissible (see 11.2.1, definition 3). Then we have the relations

x ELI,

TIID: (D, jr D) ~(A, B).

From (S), (9) and (12) it follows that

T2 T D = TI D ( A, T", jr T D = TI jr D ( B.

Hence we have the relation

T21T D: (TD, jr T D) ~(A, B).

(11)

(12)

( 13)

Thus the frame [A, B, LlJ is (x, T2 , T D)-admissible in view of (11) and (13). By the definition of the topological index there follow the relations [see 1I.2.20)J

(14)

(T2ITD)* 9 [A, B, LlJ =fl(x, T2 , T D) 9 [TD, jr T D, T DJ. (15)

Since TID is a homeomorphism from D onto T D, by II.2.4 (S), (10) there follow the relations

TID :(D, jr D) ~(TD, jr T D), (16)

(TID)* 9 [TD, jr T D, T DJ = l(T, D) 9 [D, jr D, DJ. (17)

In view of (3), (12), (13), (16), the vector law for induced homomorphisms yields (see 1.6.2)

(TIl 1))* = (TI D)* (T21 T D)*,

and (10) follows now from (is), (17), (15) and (14).

( 18)

Lemma 3. Let x be a point in Rn and C a continuum in DI which is an m.m.c. for (x, TI , D I ) (see II.3.1, definition 1). Then TC is an m.m.c. for (x, T2, D2)'

Prooj. Since C is a compact and connected subset of D I , T C is a compact and connected subset of D2 (see 1.1.5, exercise 8). Since Tl C =X, by (3) it follows that T2 TC = TI C =X. Thus TC is a continuum

190 Part III. Background in Analysis.

in D2 whose image under T2 is the point x. There remains to prove that T C is a component of the set T2-I x. As T C is connected, there exists a component C2 of the set T2- I X such that TC(C2. Then T-IC2 is a connected set in DI such that C(T-IC2 and TI T-IC2=T2C2=X. Since C is a component of TI-I x, it follows that C = T-I C2 and hence TC=C2. Thus TC is a component of T2-I X, and the lemma is proved.

The preceding lemmas 2 and 3 state invariance properties of the concepts of topological index and maximal model continuum respectively. Since all of our multiplicity functions and index functions, as well as the various essential sets, were defined in terms of these two concepts, the lemmas 3 and 2 yield invariance properties for all the basic concepts introduced in Part II. The following lemma 4 is an immediate consequence of this remark.

Lemma 4. Let there be given two topologically similar transformations ~, T2 and a connecting homeomorphism T as in (1), (2), (3). Then (see 11.3.2, 11.3.6 for terminology)

K(x, TI , DI ) =K(x, T2 , D2),

T E (TI' D I ) = E (T2' D 2), T E;(TI' DI ) = E;(T2' D 2),

TEP(T] , D I ) = £P(T2' D 2), TEf(TI , DI ) = Ef(T2' D2).

Part III. Background in Analysis. § lILt. Survey of functions of real variables.

II!.t.t. LEBESGUE measure and LEBESGUE integral in Euclidean n-space RH. If I: ai ~ Xi ~ bi , i = 1, ... , n, is an interval in Rn (see 1.2.2), then the elementary volume v (I) of I is defined by the formula

If E is a subset of R", then the exterior LEBESGUE measure L * E of E is defined by the formula

L * E = gr.l. b. L v (If) , i

where {Ii} is any (finite or infinite) sequence of intervals such that E ( U Ii' and the greatest lower bound is taken with respect to all such sequences {IJ Thus L * E may be infinite. If it is desirable to display the dimension n of Rn, then one writes L! E instead of L * E. However, n is thought of as fixed throughout this section, and hence the notation L * E is adequate.

§ III.1. Survey of functions of real variables. 191

A set E (Rn is tenned L-measurable (LEBESGUE measurable) if the relation L * (A U B) = L * A + L * B holds for every pair of sets in R" such that A(E, B(CE. For L-measurable subsets E of R" the L-measure (LEBESGUE measure) LE of E is defined by the fonnula LE = L * E. Thus LE is a non-negative set-function defined for those (and only those) sets E (R" which are L-measurable.

The class of L-measurable subsets of R" is a completely additive class (see 1.1.4, definition 10), and LE is countably additive on this class. That is, if {Ek} is a finite or infinite sequence of pair-wise disjoint L-measurable sets in Rn , then L (UEj ) = L LEi'

If a certain statement S(x), involving a variable point xER", is true for all points x of R" (or for all points x of a set E(R") with the exception of certain points x in a set F such that LF = 0, then we shall say that S(x) is true a.e. (almost everywhere) in R" (or in E).

Further concepts will now be listed, for convenient reference, in numbered definitions.

Definition 1. Let U o be a point of Rn. Consider an L-measurable set E (Rn. Then U o is termed a point of density of E if for every sequence {Qj} of oriented n-cubes in Rn (see 1.2.2), such that uoE Qj' r5 Qj--+O, one has

Definition 2. Let Xo = (x~, ... , x~) be a point of the L-measurable set E (R". Those real numbers h for which

(x~+h, xg, ... , x~)EE,

constitute a set of real numbers. Denote by e the set of these real numbers h. If e (as a subset of the real number-line) is L-measurable and if h = ° is a point of density of e (in the sense of definition 1 applied with n = 1), then Xo is termed a point of linear density of E in the direction of the xl-axis. Points of linear density of E in the direction of the axes X2, ••• , x" are defined in a similar manner.

Definition 3. Let E be an L-measurable subset of R". Then the parameter of regularity n (E) of E is defined by the fonnula

LE n(E) = l.u.b. -IQ'

where the least upper bound is taken with respect to all oriented n-cubes Q)E.

Definition 4. A sequence {Gm } of closed subsets of R" is tenned regular if there exists a number A>O such that n(Gm) >A, m=1, 2, ... (see definition 3).


Delinition 5. A real-valued function I (x), defined on an L-measurable set E(Rn , is termed L-measurable on E if for every real number r the set of those points xEE where I(x»r is L-measurable.

Consider an L-measurable set E (R", and let I (x) be a real-valued function which is defined and L-measurable on E - e, where e is a subset of E such that L e = 0. It one extends the definition of I (x) by assigning its values on e in an entirely arbitrary manner, then obviously the extended function is L-measurable on E as a consequence of the assumption that Le=O. On the basis of this observation, we adopt (as a matter of convenience) the following extension of definition 5. It E is an L-measurable set in Rn, and I(x) is a real-valued function defined and L-measurable on E - e, where e is a subset of E such that Le=O, then we shall say that I(x) is L-measurable on E.

We proceed to consider the LEBESGUE integral in Rfl. Let I(x) be a real-valued, non-negative L-measurable function defined on an L-measurable set E (Rfl. Let E1 , •.• , Em be any finite sequence of pair-wise disjoint L-measurable subsets of E such that E = U Ei , j = 1, ... , m, and denote by p.. (f, Ei ) the greatest lower bound of I (x) on E i , j = 1, ... , m. If the least upper bound of all the sums of the type

m

L: p.. (j, Ei ) LEi i=1

is finite, then I (x) is termed L-summable (LEBESGUE summable) on E, and the LEBESGUE integral of I(x) on E is defined by the formula

m

J I (x) d L = l.u.b. L: ,u (j, Ei ) LEi' E i=1

If g(x) is a real-valued, L-measurable function (of arbitrary sign) on E, then consider the auxiliary functions g+ (x), r (x) defined on E as follows.

() { g(x) if g(x);;;;:O,

g+ x = ° if g(x) < 0.

r (x) = {-g(X) ~f g(x):;;;: 0, ° If g(x) > 0.

Then g+ (x), g- (x) are non-negative, L-measurable functions on E. If g+ (x), g- (x) are both L-summable on E, then g (x) is termed Lsummable on E, and its LEBESGUE integral on E is defined by the formula

J g (x) d L = J g+ (x) d L - J g- (x) d L. E E E

§ !II.1. Survey of functions of real variables. 193

Let I (x) be a real-valued, L-measurable function in R", such that I (x) vanishes for every point xE CE, where E is a bounded L-measurable subset of R". Clearly I(x) is L-summable on R" if and only if it is L-summable on E. If I (x) is L-summable on R!' and F is any bounded L-measurable set in R" such that F )E, then clearly

fl(x)dL=fl(x)dL=fl(x)dL if I(x)=o for xECE. R" F E

In the situation just described, we shall write f I (x) d L, instead of fl(x)dL, for the integral of !(x) over R!'.

R" Delinition 6. Let F be a family of L-summable functions on an

L-measurable set E (R!'. Then the family F is said to satisfy the condition (V) (the VITALI condition) on E if for every e>O there exists an 'fj='fj(e»O such that the following holds: If S is any L-measurable subset of E such that LS <'fj, then

fi/(x)i d L < e for every IE F. s

Delinition 7. Let A be a subset of Rn. A family F of closed subsets of R" is said to cover A in the VITALI sense if A (UF, FE F, and for every point xE A there exists a regular sequence {Fj} of sets in F (see definition 4) such that xE Fj for every j and {) Fj --+ 0 for i --+ 00.

We shall now list, in the form of lemmas, a series of facts needed later on. The proofs are either contained in the SAKS treatise or the MAYRHOFER treatise (see the Bibliography) or can be readily supplied in view of the definitions involved.

Lemma 1. If E is an L-measurable subset of R" such that LE>O, then E contains a set which is not L-measurable.

Lemma 2. Let El' E2 be any two L-measurable subsets of R". Then

Lemma 3. Let E be a bounded L-measurable subset of Rn. Then LE<oo, and for every open set 0 containing E and for every e > 0 there exists an open set G such that

ECGCO, LG<LE+e.

Lemma 4. Let {Ej } be a sequence of L-measurable subsets of R n such that El CE2 ( ... (S, where S is a bounded subset of R". Then E = UEi is L-measurable, and L (E - E;) -4- 0 for i -4- 00.

Lemma o. BOREL sets and analytic sets in R" are L-measurable (see 1.1.4, definitions 11 and 12).

Lemma 6. (The VITALI covering theorem). Let F be a family of closed sets in Rn which covers an L-measurable subset A of Rn in the VITALI sense (see definition 7). Then there exists in F a (finite or infinite) sequence {Ff} of pair-wise disjoint sets such that L (A - UFf) = o.



Lemma 7. Let S be a subset of R n such that L S = o. Then there exists a BOREL set B (R" such that S (B, ~ B = o·

Lemma 8. Let E be an L-measurable subset of Rn. Then there exists a decomposition E = B U V, such that B is a BOREL set, and LV = 0, B n V = 0.

Lemma 9. If E is an L-measurable subset of R n, then almost every point of E is a point of density of E (that is, those points of E which are not points of density of E constitute a set of L-measure zero).

Lemma 10. If E is an L-measurable subset of R n, then almost every point of E is a point of linear density of E in the direction of each one of the axes Xl, ... , xn (see definition 2).

Lemma 11. If a point x is a point of density of each one of the L-measurable subsets El> ... , Em of R n, then x is also a point of density of the intersection of these sets.

Lemma 12. Let c be a real number, and let i be one of the integers 1, ... , n. Then the set of those points x = (Xl, .•. , x") of R n for which x j = c is of L-measure zero.

Lemma 13. Let {Qj} be a (finite or infinite) sequence of oriented n-cubes in R n, such that

int Qj nint Qk = 0 for i 4" k.

Then LUQi=2:;LQj. Lemma 14. Let F be a bounded, L-measurable subset of R", and let 0 be an

open subset of Rn such that F (0. Then for every e > 0 there exists a (finite or infinite) sequence {Qi} of oriented n-cubes such that F ( U Qj ( 0, 2:; L Qi< LF + E, and int Qj n int Qk = 0 for i =F k.

Lemma 15. Given two distinct points ul , u2 of RI!, consider the open spherical neighborhoods (see 1.1.4, definition 2)

,11=t1(u1,liu2-Ullll, t1 2 =t1(u2 ,lu2 -u1 1).

Then the number

satisfies the inequalities 0<0:< 1. and has the same value for every choice of the distinct points u1 ' u2 in Rn.

Lemma 16. Let P be a parallelotope in R" with edge-vectors Vi = (vi, ... , vf'), i = 1. ... , n (see 1.2.2). Then LP = idet(vj)i·

Lemma 17. Let E be a bounded, L-measurable subset of R2. For each real number r, denote by Er the set of those real numbers Xl for which (Xl, r) EE. Assume that there exists a set e of real numbers of (linear) L-measure zero such that for r (jOe the set Er is of (linear) L-measure zero. Then LE = 0 (if E is not L-measurable, then this statement is false).

Lemma 18. If f1' f2 are L-measurable functions on an L-measurable set E (RN, then 11 + 12 ,/112 are also L-measurable on E. If {Ii} is a sequence of L-measurable functions on an L-measurable set E (Rn, then lim sup Ii' lim inf Ij are also Lmeasurable on E, and if the sequence converges a.e. on E then lim Ii is L-measurable on E (see the comments following definition 5).

Definition 8. A real-valued function f (x), defined on a BOREL set B(Rn, is termed BOREL measurable on B if for every real number r the set of those points xE B where f(x»r is a BOREL set.

§ III.1. Survey of functions of real variables. 195

Lemma 19. If the real-valued function I(x) is BOREL measurable on the BOREL set B (R", then 1 (x) is L-measurable on B.

Lemma 20. If the real-valued function I (x) is lower semi-continuous (see 1.1.3, definition 28) on the BOREL set B (R", then I (x) is BOREL measurable and hence also L-measurable on B.

Lemma 21. Let 11 (x), 12 (x) be two real-valued functions on the L-measurable set E(Rn. If Idx) is L-measurable on E and 11 (x) =/2(x) a.e. on E, then 12(x) is also L-measurable on E.

Lemma 22. (LUSIN'S Theorem.) If the real-valued function I(x) is L-measurable and a.e. finite on the L-measurable set E (R", then for every e> 0 there exists a closed set E. (E such that L (E - Ee) < e and I (x) is continuous on E. (relative to Ee)'

Lemma 23. If H (x) is a real-valued, finite-valued, L-measurable function in R n, then there exists a finite-valued BOREL measurable function H*(x) in R" such that H*(x) = H (x) a.e. in Rn.

Lemma 24. If I (x) is a non-negative, real-valued, L-measurable function on a BOREL set B (Rn, then there exists a BOREL measurable function 1* (x) on B such that o::;;;/*(x):;>;/(x) and 1*(x)=/(x) a.e. on B.

Lemma 25. Let Idx),/2(x) be two real-valued, L-measurable functions on the L-measurable set E (R". If 11 (x) is L-summable on E and 11 (x) = 12 (x) a.e. on E, then 12 (x) is also L-summable on E and

f 12 (x) d L = f 11 (x) d L. E E

Lemma 26. If the real-valued, L-measurable function I (x) is L-summable on the L-measurable set E (Rn, then II (x) I is also L-measurable and. L-summable on E.

Lemma 27. Let {1m} be a sequence of real-valued, non-negative, L-measurable functions on the L-measurable set E (Rn, and let I (x) be a real-valued, nonnegative function on E such that

00

f(x) = ~/m(x) a.e. on E. m=l


(i) I(x) is L-measurable on E.

(ii) If I (x) is L-summable on E, then 1m (x) is also L-summable on E, In = 1, 2, ... , and

00

f I (x) d L = ~ f 1m (x) d L. E m=lE

(iii) Conversely, if 1m (x) is L-summable on E, m = 1,2, ... , and the series ~ JIm (x) d L is convergent, then I (x) is L-summable on E [and hence (ii) is

E applicable].

Lemma 28. (MINKOWSKI'S inequality). Given a real number p;;;;; 1. let A (x), ... , 1m (x) be real-valued. L-measurable functions on the L-measurable set E (R" such that 1/1IP, ... , IlmlP are L-summable on E. Then (1/11 + ... + IlmllP is also L-summable on E and

13*


Lemma 29. (HOLDER'S inequality). Let p, q be two real numbers such that

p> 1, q> 1, 1 1 -+-=1-P q

If I(x), g(x) are real-valued, L-measurable fuuctions on an L-measurable set E (RH such that \lIP and Iglq are L-summable on E, then fg is L-summable on E and

11 f (x) g(x) d L I ;;;;:; [111 (x)IP d LjllP [l lg (x)l q d Ljl/q.

Lemma 30. (FATOU'S theorem or lemma.) Let f(x), I; (x), i= 1, 2, •.. , be realvalued, non-negative, L-measurable functions on an L-measurable set E (RH ,

such that f;(x) is L-summable on E, i= 1,2, ... , and

I(x);;;;:;liminflj(x) a.e. on E, liminfflj(x)dL<oo. E

Then I(x) is L-summable on E, and

f/(x) dL;;;;:; lim inf ffi (x) dL. E E

Lemma 31. Let I(x) be a real-valued, non-negative, L-summable function on an L-measurable set E (RH • Then the set of those points x EE where I(x) = 00

is of L-measure zero.

Lemma 32. If I (x) is a real-valued, non-negative, L-summable function in R n, then there exists a BOREL set X (Rn such that LX = 0 and I (x) < 00 for x E ex.

Lemma 33. Let f(x) be a real-valued, L-summable function on an L-measurable set E (RH. Then for every s> 0 there exists an 7] = 7] (s) > 0 such that

fl/(x)1 dL < s 5

whenever 5 is an L-measurable subset of E such that L 5 < 7].

Lemma 34. Let I(x) be a real-valued, L-summable function on an open set G (Rn. Let E be an L-measurable set in Rn such that E (G. Then there exists for every s>o an open set 0 = 0 (s) such that

E(O(G, fl/(x)ldL<fl/(x)ldL+s. o E

Lemma 35. Let I(x) be a real-valued, non-negative, L-summable function on an L-measurable set E (Rn such that ff (x) d L = O. Then I (x) = 0 a.e. on E.

E

Lemma 36. Let I, Ii' i = 1, 2, ... , be real-valued, L-summable functions on an L-measurable set E (Rn, such that Ir+- f a.e. on E and the sequence {Ii} satisfies the condition (V) on E (see definition 6). Then

flj(x) d L ..... fl(x) d L. E E

Lemma 37. If I (x) is a real-valued, L-summable function on an L-measurable set E (Rn and {E;} is a sequence of pair-wise disjoint L-measurable subsets of E such that E = U E;, then

fl(x) dL = "L,ff(x) dL. E E;

Lemma 38. Let I, I;, i = 1, 2, ... , be real-valued, L-summable functions on an L-measurable set E (Rn, such that Ii ..... I a.e. on E. If there exists a non-negative,

§ III.i. Survey of functions of real variables.

L-summable function F on E such that Ilil:::;;F a.e. on E.i=1. 2 ••••• then

ff,(x) d L -+- I I (x) d L. E E

197

Lemma 39. Let I (x) be a non-negative. L-measurable function on an L-measurable set E(R". Let {E,} be a sequence of pair-wise disjoint L-measurable subsets of E such that E= UEj • If I(x) is L-summable on E j • i= 1.2 •...• and the series

I:. II (x) dL Et

is convergent. then I(x) is L-summable on E and

I I (x) d l = I:. I I (x) d L. E Ef

Lemma 40. Let {/j} be a sequence of real-valued. non-negative. L-summable functions on an L-measurable set E (R". such that the series

I:. I I, (x) dL E

is convergent. Then the series I:. Ii (x) is convergent a.e. on E. the sum of this series is l-summable on E. and

I (I:. Ij(x» d l = I:. II, (x) d L. E E

Lemma 41. Let 1.1" i = 1, 2 •... , be real-valued, L-summable functions on an l-measurable set E (R", such that

I It (x) - Ii (x) I d l-+- O. E

Then I II, (x)! dL-+- I II (x)! dl. E E

Lemma 42. Let {Qj} be a sequence of oriented n-cubes in R" such that int Qj n int Qk = 0 for i =1= k, and let I be a real-valued function which is lsummable on the set E = U Q" Then

II (x) dL = I:.II(x) dL. E Qf

Lemma 43. Let I be a real-valued, L-summable function on an open set 0 (R". Then there exists a subset e of 0 of L-measure zero such that the following holds for every point uEO-e.

(i) 1 (u) is finite. (ii) If {Gm} is any regular sequence of closed subsets of 0 (see definition 4)

such that UEGm and "Gm-+-O. then

I(u) = lim L~ II(x)dL. m Gm

Lemma 44. Let 11 , la be real-valued, l-summable functions on an open set o (R", such that

I 11 (x) dL = I la(x) dL int Q int Q

for every oriented n-cube Q (0. Then 11 = 12 a.e. in O.


III. 1.2. Finite measures on BOREL sets in B". We shall need only a few basic concepts and· facts from the very comprehensive theory of general measures. In this section we collect this minimum amount of information for convenient reference (proofs and further details may be found in the SAKS treatise listed in the Bibliography).

A real-valued function v(B), defined for every BOREL set B in a fixed BOREL set BO(Rn, is termed a finite measure on BOREL sets in BO if the following holds. (i) 0 ~ 'I' (B) < 00 for every BOREL set B (Bo. (ii) 'I' is countably additive [that is, if {B j } is a finite or infinite sequence of pair-wise disjoint BOREL sets in BO, then 'I' (U B j ) = LV (B j ) J.

A finite measure 'I' on BOREL sets in BO is termed absolutely continuous (relative to L-measure in Rn) if for every BOREL set B (BO such that L B = ° one has 'I' (B) = 0. This condition is equivalent to the following one: for every 8>0 there exists an 1]=1](8»0 such that V(B)<8 whenever B is a BOREL set in BO such that L B <YJ.

A finite measure 'I' on BOREL sets in BO is termed singular (relative to L-measure in Rn) if there exists a BOREL set b (BO such that (i) L b = ° and (il) 'I' (B) = ° for every BOREL set B (BO - b. This condition is equivalen t to the following one: for every BOREL set B (BO one has v(B)=v(Bnb), where b is a BOREL set in BO such that Lb=O.

Let 'I' be a finite measure on BOREL sets in BO. Then 'I' gives rise to a v-integral on BOREL sets in BO as follows (note the analogy with the L-integral considered in 111.1.1). Let j (x) be a real-valued, nonnegative, BOREL measurable function on a BOREL set B(Bo. Let B1 , •.• , Bm be any finite system of pair-wise disjoint BOREL sets such that B = U B j , j = 1, ... ,m. Denote by f-l (f, B j ) the greatest lower bound of j (x) on Bj , j = 1, ... , m. Then j (x) is termed v-summable on B if the least upper bound of all the sums of the type Lf-l(f,Bj ) v(B j )

is finite, and if this is the case, then the v-integral of j(x) on B is defined by the formula

If g(x) is a real-valued, BOREL measurable function (of arbitrary sign) defined on a BOREL set B (BO, then the functions

if g(x);;;;: 0,

if g(x) < 0, g_ (x) = {-g(X) ~f g(x) ~ 0, ° If g(x) > 0,

are again BOREL measurable (and non-negative) on B. If g+ (x), g- (x) are both v-summable on B, then g (x) is termed v-summable on B, and one defines:

Jg(x) dv=Jg+(x) dv-Jg-(x)dv. B B B

§ III.i. Survey of functions of real variables. 199

Let v be a finite measure on BOREL sets B in an open set G (RH. Given a BOREL set B(G and a real number 8>0, there exist two sets F,O in G such that F is closed, 0 is open, F(B(O, and 1'(0) < v(B) +8, v (B) < v (F) + 8.

Let v be a finite measure on BOREL sets B (Bo. A real-valued, BOREL measurable function I(x) defined on a BOREL set B(BO is v-summable on B if and only if II (x) I is v-summable on B. If 11 (x), ... , 1m (X) are real-valued, BOREL measurable functions, defined and 1'summable on the BOREL set B(BO, and CI , ••• , Cm are real constants, then

Let I (x) be a real-valued, non-negative, BOREL measurable function on a BOREL set B(BO, and let {B j } be a (finite or infinite) sequence of pair-wise disjoint BOREL sets such that B = UBi and I (x) is 1'summable on B j , j=1,2, .... Then I(x) is v-summable on B if and only if the series

"L Jt(x) dv BJ

is convergent; and if this is the case, then

JI(x) dv="LJI(x)d1'. B Bi

If v is a finite measure on BOREL sets B (BO, then there exists a unique decomposition of the form 1'=va+1's, where Va' Vs are finite measures on BOREL sets in BO such that Va is absolutely continuous and Vs is singular (relative to L-measure). This unique decomposition is termed the LEBESGUE decomposition of v. Furthermore, there exists a real-valued, non-negative, BOREL measurable, L-summable function F(x) on BO such that Va (B) = JF(x)dL for every BOREL set B(Bo.

B

Thus the LEBESGUE decomposition of v can be written in the form

v (B) = J F(x) dL + 1's(B) , B

where B is any BOREL set in BO. The uniqueness of the LEBESGUE decomposition implies that v is absolutely continuous on BO if and only if vs==O, and similarly v is singular on BO if and only if 1',==0. If f (x) is a real-valued, BOREL measurable function on a BOREL set B (BO, then I (x) is 1'-summable on B if and only if it is both va-summable and vs-summable on B. Furthermore, if I(x) is v-summable on B, then

J I(x) dv = J l(x)F(x) dL + J I(x) dvs' B B B


where F(x) is the function occurring in the LEBESGUE decomposition of v. The preceding formula, which expresses the relation between the v-integral and the L-integral, is referred to as the theorem on change of measure. An important special case arises if v is absolutely continuous. One has then

J f(x) dv = J f(x) F(x) dL. B B

In fact, this formula holds as soon as one of the two integrals involved exists (assuming, of course, that v is absolutely continuous and f(x) is BOREL measurable on B).

111.1.3. Derivatives and differentials in RH. We shall collect in this section, for convenient reference, a few definitions and facts concerning derivatives and differentials of real-valued functions of several real variables, in the form best suited for applications later on. Proofs and further details may be found in the SAKS treatise listed in the Bibliography.

In an open set O(R'" let f(x) =f(xl , ... , x") be a real-valued continuous function of the point x = (Xl, ... , x"). The first partial derivative of f with respect to xi will be denoted by offoxi (assuming that it exists at a certain point in 0). If offoxi exists a.e. in 0, then it is L-measurable there (see the comments following definition 5 in IIL1.1). Let xo= (x~, ... , x~) be a point of 0 such that offoxl , ... , offox" exist at Xo. Then the expression

where x = (xl, ... , x") is a variable point in 0, is termed the differential of f at xo. If

" 1 If(x) - f(x) - ~ o!(x.o) (xi - xi)I""""o Ix-x 1 0 L.J ox1 0

o i~l

for x=j=xo' x......,.xo, then f is said to possess a total differential at Xo. If for every s> 0 the point Xo is a point of density of the set

n

E. = {xix EO, If(x) - f(xo) - ~ O~~o) (xi - x~)1 s sllx - xo"}, 1~1

then f is said to possess an approximate total differential at Xo. Actually, this concept can be (and usually is) formulated in a more general manner. However, the preceding definition is adequate for our purposes.

§ III.2. Functions of open intervals in R". 201

A real-valued function I(x), defined on a set E(R", is said to be Lipschitzian on E if there exists a (finite) constant M;;;;;O such that

for every pair of points Xl' x2 in E. In terms of this concept, we have the following theorem: if I(x) is a real-valued function on an open set 0 (R" which is Lipschitzian on 0, then 1 (x) possesses a total differential a.e. on O.

More generally, let I(x) be a real-valued, continuous function on an' open set O(R", such that

lim sup If(x) -:::1 (xo)1 x-+x, Ilx- xoll <00

at all those points Xo E 0 which do not belong to a certain subset e of 0 such that Le=O. Then I(x) possesses a total differential a.e. in 0 (theorem 01 RADEMACHER).

Let I(x) be a real-valued, continuous function in an open set o (R", such that the first partial deri va ti ves 81/8 Xl, ... , 81/8 x" exist a. e. in O. Then I(x) possesses an approximate total differential a.e. in 0 (this is a special case of a theorem of STEPANOFF).

§ III.2. Functions of open intervals in R" 1.

111.2.1. Bounded variation. An interval I in R n is a point set determined by inequalities of the form a,::S:: ui :s:: bi , where ai < bi ,

i = 1, ... ,n. An interval I is a compact set (see 1.2.2, exercise 3) and clearly its interior int I is given by the inequalities ai < ui < bi ,

i = 1, ... , n. The set int I is termed an open interval in Rn.

Definition 1. Let 0 be a non-empty open set in Rn. A (finite) system of intervals II' ... , 1m will be termed admissible for 0, and will be denoted generically by a(O), if intIk(O, k=1, ... ,m, and int Ii n inti k = 0 for j oF k.

Delinition 2. Let 0 be an open set in R". If with every open interval intI(O there is associated a (finite) real number <p(intI), then <p is termed a (real-valued) function of open intervals in O.

Of course, one can also consider functions 1p (I) of closed intervals I. If <p (int I) is a function of open intervals, then on setting 1p (I) =<p (int I) one obtains a function 1p of closed intervals. Conversely, if 1p (I) is a function of closed intervals, then on setting <p (int I) = 1p (I) one obtains

1 For historical comments and further details, see the SAKS treatise listed in the Bibliography.


a function rp of open intervals. For our purposes it is more convenient to operate with functions of open intervals.

To simplify formulas, let us introduce the following notations. Given a(O) and rp as specified in definitions 1 and 2, put

L [rp, a (O)J = L IP(int I), IE a(O),

L[a(O)J=LLI, IEa(O).

(1 )

(2)

Throughout this section, rp will denote a function of open intervals in an open set 0 ( Rn.

Definition 3. rp is termed bounded in 0 if there exists a (finite) constant IC;;;;;.O such that Irp(intI)I;;;;:K for intI(O.

Definition 4. The least upper bound of the sums L[lrpi,a(O)] is termed the total variation V(IP, 0) of IP in O. In symbols:

V(rp, 0) = l.u.b. L [irp/, a(O)J. . a(O)

Note that V(q;, 0) may be infinite. Clearly V(rp, 0):;:;; o. Definition 5. rp is said to be of bounded variation in 0 if there exists

a (finite) constant M:;;;; 0 such that

L [IIPI, a(O)J ~ M (4)

for every system a(O).

The following two lemmas are direct consequences of the definitions involved.

Lemma 1. rp is of bounded variation in 0 if and only if V(rp, 0) < 00.

Lemma 2. If rp is of bounded variation in 0, then it is also of bounded variation in every open set 0*(0, and V(rp, 0*) ~V(rp, 0).

III.2.2. Absolute continuity. The notations and definitions introduced in 111.2.1 will be used in this section.

Definition. rp is said to be absolutely continuous in 0 if for every e> 0 there exists an 'rj (e) > 0 such that

L [lrpJ, a (0)] < e if L[a(O)J < ,](e). (1)

Lemma 1. If rp is absolutely continuous in 0, then it is also absolutely continuous in every open set 0* (0.

This is a direct consequence of the definition.

Lemma 2. Assume that rp is absolutely continuous in O. Take a positive number c:, and let 'rj(C:) correspond to c: in the sense of (1). Let 0* be any (non-empty) open set in 0 such that

LO* < 'rj(C:). (2)

§ III.2. Functions of open intervals in Rn. 203

Then V(cp, 0*)::;;; 8.

Proof. Consider any system 0'(0*). In view of (2), clearly L[a(0*)]<1](8). Since 0'(0*) is also a system 0'(0), by (1) we conclude that

L.: [Icpl, 0' (O*)J < 8. (4)

As '0'(0*) was arbitrary, (4) implies (3). Lemma 3. Assume that cp is absolutely continuous and bounded

in the bounded open set O. Then cp is of bounded variation in O.

Proof. As 0 is bounded, clearly

LO < 00.

Since cp is bounded in 0, we have a constant K such that

Icp(intI)1 <K for intI(O.

On choosing 8 = 1, (1) yields

L.:[lcpl,a(O)J<1 if L[a(O)] <1'}(1).

Let us put A = 17(1)/2.

Consider now a system 0'0 (0), consisting of intervals II' ... , 1111 ,

Case 1. Suppose that

(5)

(6)

(7)

(8)

L1i<A, j=1, ... ,m. (9)

Let us say that a grouping of the intervals II' ... , 1m into mutually exclusive systems 0'1(0), ... , ak(O) is a A-grouping if

L[as(O)] < 2A, s = 1, ... , k. (10)

In view of (9), we have at least one A-grouping [namely, the one obtained by considering each individual interval as constituting a system a(O)J. Clearly, there exists a A-grouping 0'1 (0), ... , ak (0) for which the number k has a minimum value. Let 0'1 (0), ... , ak (0) be a A-grouping with this minimal property. Then the inequality L [as (0)] < A holds for at most one of the integers s = 1, " ., k. Otherwise two of the systems as (0) could be combined into one, and we would obtain a A-grouping consisting of k-1 systems 0'(0), in contradiction with the minimal property of the A-grouping 0'1(0), ... , ak(O). Thus the sum L[as(O)] has a value ~ A for at least k -1 of the integers s = 1, ... , k. Hence

(k - 1) A;;;; L[ao(O)] ;;;; LO, and thus

k~1+\O. (11)


Observe now that k

L [I!pI, (To (O)J = L L [I !pI, (Ts(O)J, s=1

and [by (10), (8) and (7)J

L [I!p!, (Ts(O)J < 1, s = 1, ... , k.

From (11) to (13) it follows that

LO L [I!pl, (To (O)J < 1 + T'

Case 2. Suppose that Llj;;;;;A, j=1, ... , m. Then clearly

and thus m::S;; LO/A. By (6) we conclude that

KLO L [I!p!, (To (O)J ;;;;:-;.-.

(12)

(13 )

(14)

( 15)

In the general case, (10(0) can be split into two systems (1~(0) and (1~ (0), one presenting Case 1 and the other presenting Case 2. By (14) and (15), applied to (T~ (0) and (T~ (0) respectively, it follows that

L [I!pI, (10 (O)J < 1 + Lf + K~O . (16)

Since (To (0) was arbitrary, (16) shows that !p is of bounded variation in O.

1II.2.3. Derivatives. The notations and definitions introduced in 111.2.1 and 111.2.2 will be used in this section.

Definition 1. Let u be a point of O. The upper derivative 15 (u,!p) of!p at u is the least upper bound ot all those numbers A (including ± 00) for which there exists a sequence of oriented n-cubes {Qk} such that

u E int Qk ( 0, L Qk -7 O,IP(int Qk) -7 A. LQk

The lower derivative 12 (u,!p) is the greatest lower bound of these same numbers A.

Definition 2. If 15 (u,!p) and 12 (u,!p) are finite and equal to each other at a point uEO, then their common value is termed the derivative D (u,!p) of!p at u.

Theorem 1. 15 (u,!p) and 12 (u,!p) are BOREL measurable in O.

Proof. Since the argument is entirely similar in both cases, we present it only for 15 (u, !pl. Assign any real number Cl. Let Q be a generic notation for an oriented n-cube. For each pair of positive

§ II1.2. Functions of open intervals in RH. 205

integers m, j define a set Ea.mi by the fonnula

Ea.mi = U int Q, int Q (0, L Q < -~, !p(int Q) > ex +~ . 1 LQ m

If Ea. denotes the set of those points uE 0 where 15 (u, rp) > ex, then it follows readily from the definition of 15 (u, rp) that

Ea. = U (1 Ea.mi· m, (1 )

Since Ea.mi is open, (1) shows that Ea. is a BOREL set, and hence 15(u, rp) is BOREL measurable.

Theorem 2. Let E be the set of those points uEO where D(u,rp) exists (see definition 2). Then E is a BOREL set, and D (u, rp) is BOREL

measurable on E.

Proof. Clearly, E consists of those points uE 0 where the following conditions hold simultaneously.

- 00 < 15(u, rp) < 00, - 00 <!2 (u, rp) < 00, 15(u, rp) =!2 (u, rp).

In view of theorem 1 and 111.1.1, definition 8, it follows that E is the intersection of three BOREL sets, and hence E is a BOREL set. As D (u, rp) = 15 (u, rp) on E, by theorem 1 one sees that D (u, rp) is BOREL

measurable on E.

Theorem 3. If D (u, rp) exists a.e. in 0, then it is L-measurable in O.

Proof. Denoting again by E the set of those points uEO where D(u,rp) exists, the set O-E is now of L-measure zero by assumption. In view of the conventions stated in 111.1.1, the present theorem is therefore a consequence of theorem 2, since BOREL measurability implies L-measurability (see 111.1.1, lemma 19).

Theorem 4. Assume that the set 0 is bounded, rp is of bounded variation in 0, and D (u, rp) exists a.e. in O. Then D (u, rp) is L-summable in 0, and (see 111.2.1, definition 4)

JID(u,rp)1 dL:;;;;V(rp,O). (2) o

Proof. Take an oriented n-cube Q such that O(Q. We shall make use now of the subdivision Dp; of Q (see 1.2.3, definition 3). For each positive integer j, let Qi be a generic notation for a cube of D p1 for which int Q i (0, and let OJ be the class of all such cubes Q j' Since 0 is bounded, clearly OJ is a finite class. For each positive integer j, define in 0 a function fi(u) as follows: If there exists a cube Qi such that uE int Qi' then

(3 )


while Ii (u) = 0 if no such cube Qi exists. Clearly, Ii (u) is BOREL measurable in 0, and

(4)

If u is any point in 0, then by 1.2.3, exercise 2 this point will lie for j sufficiently large in the interior of some cube Qi' and thus (3) will hold. Hence, if D(u,r) exists at u, then li(u)-+D(u,r). Thus

.lim I/i(u)! = !D(u,r)! a.e. in O. 1 --+ 00

(5)

Since OJ is clearly a system a(O), we infer from (4) that (see III.2.1, definition 4)

(6)

By the theorem of FATou (see 111.1.1, lemma 30), (5) and (6) imply both the L-summability of ID (u, r)! in 0 and the inequality (2).

Lemma. Assume that 0 is bounded and D (u, r) exists a.e. in O. Using the notations Qi' OJ as in the proof of theorem 4, assume further that

(7)

Then D(u,r) = 0 a.e. in O. (8)

Prool. Note that we do not assume now that r is of bounded variation in O. Still, we can introduce the functions fj(u) as in the proof of theorem 4, and the relations (4) and (5) are available as before. From (4) and (7) we see that

.lim J !fi(u)1 dL = o. 1--+ 00 0

(9)

By the theorem of FATOU (see III.1.t, lemma 30), (5) and (9) imply that

J ID(u, r)1 dL = 0, o

and (8) follows by III.1.t, lemma 35. Delinition 3. r is termed sub-additive in 0 if

m

L:r(int Ii) ~r(int I), i~1

(10)

for every choice of the intervals II' ... , 1m , I subject to the conditions

intIi(intI(O, j=1, ... ,m,

intIi n intIk = 0 for j =f= k.

(11)

( 12)

§ III.2. Functions of open intervals in R". 207

Theorem 5. If q; is non-negative and sub-additive in 0, then D (u, q;) exists a.e. in 0.

By 1.2.2, exercise 8, ° can be covered by a countable set of open intervals. Hence it is sufficient to show that if I is any interval such that int 1(0, then D (u, q;) exists a.e. in int I. For clarity, we subdivide the proof into several steps.

(a) Let I be an interval such that int 1(0, and let IX be a positive number. Denote by E" the set of those points uE int I where 15 (u, q;) > IX.

Then (13 )

Proof. Let F be the family of those oriented n-cubes Q which satisfy the conditions

int Q (intI, q> (int Q) ~>IX.

Clearly, the cubes QE F cover ErJ. in the sense required in the VITALI

covering theorem (see III.1.t, lemma 6). Hence F contains a (finite or infinite) sequence of cubes {Qj} such that Qin Qi = 0 for i =1= j and

(14)

Since q; is sub-additive, there follows for every positive integer k the inequality k k

IX L: L Qi < L:q;(int Qi) ~q;(int 1). ;=1 ;=1

For k-+ 00 we obtain, in view of (14), the inequality (13)· If {Qj} is a finite sequence, then the step k~ 00 is not needed in the proof.

(b) Let I be an interval such that int 1(0. Denote by E* the set of those points u E int I where 15 (u, q;) = 00. Then

LE* = o. ( 15)

Proof. Clearly E*:, EfL for every positive number IX, and thus (15) follows from (13).

(c) Let I be an interval such that intI(O. Then15(u,q;)=Q(u,q;) a.e. in int I.

Proof. If Q (u, q;) < 15 (u, q;) at a point uE int I, then there exist two rational numbers rand s such that

Q (u, q;) < r < s < 15 (u, q;). ( 16)

Denote by E TS the set of those points uE int I where (16) holds for a given pair of rational numbers r, s such that 0 < r< s. Then the set of those points uEintI where Q(u,q;)=I=15(u,q;) is covered by the family of all the sets E, S' corresponding to all possible pairs of rational


numbers r, s. Since this family is countable, it is sufficient to show that each set E rs is of L-measure zero. Deny this, and suppose that

L Ers > 0, 0 < r < s, (17)

for some pair r, s. Take a positive number e. Since E,s is BOREL measurable by theorem 1, we have (see 111.1.1, lemma 3) an open set G such that

(18)

Let now F denote the family of those oriented cubes Q which satisfy the conditions

intQ(G, 'P (int Q) - LQ < r. (19)

Clearly, the cubes QE F cover E,s in the manner required in the VITALI covering theorem (see III. 1. 1, lemma 6). Hence F contains a (finite or infinite) sequence of cubes {Qi} such that Q;n Qi = 0 for i=f=.j and

L(Ers - U Qi) = O. (20)

There follow [by (19) and (18)] the inequalities

L<p(int Qj) < r L L Qi ~r L G < r(LE,s + e). (21) i i

Observe now that E rs n int Qi is a subset of the set of those points u E int Qj where I5 (u, <p) > s. In view of (a) above there follows the inequality

By (20) we have L L (ETS n int Qi) = L E rs · i

From (23), (22), and (21) we obtain the inequality

s LErs < r(LErs + e).

Since e was arbitrary, it follows that

(22)

(23)

and in VIew of (17) this is a contradiction. Thus (c) is established.

Combining (b) and (c), we see that I5 (u, <p) and 12 (u, <p) are equal to each other and are finite a.e. in int I. Hence (see definition 2) D (u, <p) exists a.e. in int I, and the proof is complete.

111.2.4. Applications. In this section, D denotes a bounded domain in Rn, and F denotes a family of subsets of D such that F contains, in particular, every domain D (D.

§ III.2. Functions of open intervals in Rn. 209

Definition 1. If with every set FE F there is associated a (finite) real number (/) (F), then (/) is termed a (real-valued) function of sets in F.

If I is an interval such that int I (D, then (since int I is a domain by 1.2.2, exercise 3) (/) (int I) is defined. Thus (/) gives rise to a function (/)* of open intervals in D, by means of the formula

(/)* (int I) = (/) (int I) , int I ( D. (1 )

Accordingly, we can define the derivative D (u, (/») of (/) by the formula (see 111.2.3, definition 2)

D (u, (/») = D (u, (/)*) . (2)

By 111.2.3, theorem 2, the set where D (u, (/») exists is a BOREL set, and D (u, (/») is BOREL measurable on that set.

Definition 2. A function (/) of sets FE F is termed sub-additive on F if

whenever Fl , ... , Fm , F are sets of the family F such that Fk(F, k = 1, ... ,m, and Fi n F j = 0 for i =F j.

Theorem 1. Assume that (/) is a non-negative, sub-additive function of sets FE F. Then the derivative D (u, (/») exists a.e. in D and is Lsummable in D. Furthermore, if 0 is any open set in D such that OE F, then

f D(u, (/») dL $;(/)(0). (4) o

Proof. Clearly, the associated function (/)* of open intervals [see (1) J is non-negative and sub-additive. Accordingly, in view of (2) and 111.2.3, theorems 3 and 5, D (u, (/») exists a.e. in D and is L-measurable in D. Let now 0 be a (non-empty) open set in D such that 0 E F. Consider any system a(O) (see 111.2.1, definition 1). Using the notations of 111.2.1, we have by (3) and (1) the inequality

L [(/)*, a(O)J $;(/)(0) < 00.

Since a(O) was arbitrary, it follows that

V((/)*, 0) $;(/)(0) < 00. (5)

Hence, by 111.2.3, theorem 4, D(u, (/)*) is L-summable in 0 and

J D (u, (/)*) d L $; V((/)*, 0). (6) o

Now (4) follows from (6), (5) and (2). The L-summability of D(u, (/») in D follows if one chooses 0 = D in the preceding argument.



Let us now consider a finite measure v(B) on BOREL sets in D (see IIT.1.2). The family of BOREL sets in D contains all open sets in D and hence also all domains in D. If B 1 , ••• , B m , B are BOREL

sets in D such that Bk(B, k= 1, ... , m, and BinBf = 0 for i=l= j, then m m

v(B) = 2: V (Bk) + v(B - U B). k=l k=l

Since v is non-negative, it follows that m

2: v (Bk) ;;;;: v(B). k=l

Thus v is non-negative and sub-additive on BOREL sets in D. The assumptions of theorem 1 are therefore satisfied (the family of BOREL

sets in D playing now the role of the family F), and hence we have the following statement.

Lemma 1. If v (B) is a finite measure on BOREL sets in D, then the derivative D (u, v) of v exists a.e. in D and is L-summable in D. Furthermore, if 0 is any open set in D, then

JD(u, v) dL ;;;;;v(O). (7) o

Lemma 2. Let v(B) be a finite measure on BOREL sets in D. If B is a BOREL set in D such that

v(B) = 0, then

D(u, v) = ° a.e. on B.

(8)

(9)

Proof. Assign e>O. In view of ITL1.t, lemma 3, (8) implies the existence of an open set 0 such that

v(O) < e, B (0 (D.

Since D(u, v) ;;;;;0, from (10) and (7) we obtain

° :;;;.JD(u, v) dL;;;;;J D(u, v) dL ;;;;:v(O) < e. B 0

As e was arbitrary, it follows that

JD(u,v)dL=O, B

and (9) follows by IIT.1.t, lemma 35.

(10)

Lemma 3. Let l' (B) be a finite measure on BOREL sets in D. If 'V

is singular relative to L-measure, then

D(u, v) = ° a.e. in D. (11)

§ III.2. Functions of open intervals in Rff. 211

Proof. Since v is singular relative to L-measure, there exists (see 111.1.2) a BOREL set BoCD such that LBo=O' v(D-Bo)=O. By lemma 2 we conclude that D (1-', v) =0 a.e. on D - Bo. Since LBo = 0, the relation (11) follows.

Given a finite measure v (B) on BOREL sets in D, let us recall that we have at our disposal the LEBESGUE decomposition (see 111.1.2)

v(B) = f I(u) dL + vs(B) , (12) B

where I (u) is non-negative and L-summable in D, VS is a finite singular measure on BOREL sets in D, and B is any BOREL set in D. In view of lemma 3 and 111.1.1, lemma 43, we obtain from (12) the relation

I(u) =D(u, v) a.e.inD. (13)

Accordingly, (12) may be re-written in the form

v(B) = f D(u, v) dL + vs(B). ( 14) B

Theorem 2. Let v(B) be a finite measure on BOREL sets in D which is absolutely continuous with respect to L-measure. Then

v(B) = f D(u, v) dL ( 15) B

for every BOREL set BCD.

Indeed, in this case Vs = 0 by 111.1.2, and thus (15) follows from (14).

Theorem 3. Let v(B) be a finite measure on BOREL sets in D. Then the derivative D (u, v) exists a.e. in D and is L-summable in D. Further-more

f D(u, v) dL ~ v(D), D

(16)

where the sign of equality holds if and only if v is absolutely continuous in D with respect to L-measure.

Prool. In view of lemma 1 and theorem 2, there remains to show that v is absolutely continuous with respect to L-measure if

f D(u, v) dL = v(D). (17) D

Now (17) implies, in view of (14), that Ys(D) =0. Since O~ vs(B);:;;: vs(D) for every BOREL set BCD, it follows that Vs = 0, and hence v is absolutely continuous with respect to L-measure (see 111.1.2).

Theorem 4. Let v (B) be a finite measure on BOREL sets in D which is absolutely continuous with respect to L-measure, and let h(u) be

14*

212 Part IV. Bounded variation and absolute continuity in R".

a BOREL measurable function in D. If B is any BOREl_ set in D. then

J h(u) dv = J h(u) D(u. v) dL, (18) B B

as soon as one of the two integrals involved exists.

In view of (13), this statement is a direct consequence of thegeneral theorem on change of measure in 111.1.2.

Part IV. Bounded variation and absolute continuity in R UI •

§ IV.1. Measurability questions 2 •

IV.t.t. Basic measurability theorems. From this point on, our general objective is the study of continuous transformations in Euclidean n-space R" from the point of view of Analysis. As before, we shall deal with transformations given in the form

T:D-?R", (1 )

where D is a bounded domain in R" and T is continuous and bounded in D. The notations and the terminology adopted III 11.3.1 will be followed.

Theorem 1. Given a continuous transformation T as III (1) and a BOREL set B (R", the set T-l B is again a BOREL set.

Proof. Let ® denote the class of those sets S (R" for which the set T-1 S is a BOREL set. The class ® possesses the following properties

(i) If 0 is an open set in Rn, then OE ®. Indeed, since T is continuous, the set T-1 0 is open and hence it

IS a BOREL set. (ii) If SE e, then C S F: e. Indeed, T-l C S;:= D - T-l S. Since D and T-1 S are BOREL sets,

it follows that T-IC S is a BOREL set. (iii) If Sl' ... , S"" ... is a (finite or infinite) sequence of sets of the

class ®, then USmEe.

1 Part IV and Part V constitute an elaboration (induding various improvements) of the note by T. RAno and P. V. REICHELDERFER, On n-dimensional concepts of bounded variation, absolute continuity, and generalized Jacobian [Froc. Nat. Acad. Sci. U.S.A. 35, 678-681 (1949)J.

2 For historical comments and further details concerning the role of B~REL sets and analytic sets in connection with transformations, see the HAHN treatise listed in the Bibliography. A paper by E. J. MICKLE, Metric foundations of continuous transformations [Trans. Amer. Math. Soc. 63, 368-391 (1948)], contains a great deal of instructive material.

§ IV.1. Measurability questions. 213

Indeed, T-l U Sm = U T-l Sm' and hence T-l U Sm is a BOREL set, since the sets T-lSm are BOREL sets.

The properties (i), (ii), (iii) mean that S is a completely additive class of sets in R", and hence @3 contains all BOREL sets in R!' (see 1.1.4, definitions 10 and 11). Thus if B is any BOREL set in R", then T-IB is a BOREL set.

Theorem 2. If H (x) is a (real-valued) BOREL measurable function in R", then the function H (Tu) is BOREL measurable in D (see 111.1.1, definition 8).

Proof. Let c be any real number, and denote by Sc the set of those points x E R" where H (x) > c, and by 5: the set of those points uED where H(Tu»c. Clearly 5: =T-ISc. Since Sc is a BOREL set [since H(x) is BOREL measurable], it follows by theorem 1 that 5: is also a BOREL set. As c was arbitrary, we conclude that H(Tu) is BOREL

measurable. There exist examples showing that the continuous image of a BOREL

set need not be a BOREL set. Due to this discrepancy, one has to utilize analytic sets (see 1.1.4, definition 12) in deriving certain measurability theorems needed in the sequel.

Theorem 3. If A is an analytic set in D, then T A is an analytic set in Rn.

Proof. Since the assertion is trivial if A is empty, we assume that A is non-empty. In view of 1.1.4, exercise 7, we can further assume that A is given in terms of a regular determining system ~ of closed sets A (ml' ... , mk ) such that

lim (j A (ml , ... , mk) = o. k--..oo

(2)

Consider the system ~* of the sets

(3)

Clearly, ~* is a regular determining system of closed sets. We first verify that (see 1.1.4, definition 12)

T A (A(~*). (4)

Indeed, consider any point xE TA. Then there is a point uEA such that x = T u. Since uEA = A (~), there exists an infinite sequence ml , m2 , ... of integers such that

uEDnA(ml, ... ,mk ), k=1,2, ....

In view of (3) it follows that

x = T u E A * (ml' ... , mk), k = 1, 2, ... ,

214 Part IV. Bounded variation and absolute continuity in RO>.

and hence xE A (\ll*). Thus (4) is established. Next we verify that

A (\ll*) ( T A . (5)

Indeed, take any point xE A (\ll*). Then there exists an infinite sequence of integers ml , m2 , ••• , such that

xEA*(ml, ... ,mk)' k=1,2, ....

In view of (3), for each integer k there is a point Xk such that

(6)

The inclusion (6) implies the existence of a point Uk such that

ukEDnA(ml,··.,mk ), TUk=Xk' (7)

Since the determining system \ll is regular, clearly

(8)

In view of (2) it follows that the points Uv u 2 , ••• constitute a CAUCHY

sequence (see 1.1.4, definition 8), and hence they constitute a convergent sequence. Put

u = lim Uk'

Since the set A (ml' ... , mk ) is closed, it follows from (8) that

u E A (ml , ... , mk ) , k = 1, 2, ... ,

and hence [since A = A (\ll)] uE A (D.

Since T is continuous, (10), (9), (7), (6) imply that

T u = lim T Uk = lim Xk = x.

(9)

(10)

By (10) it follows that xET A, and (5) is established. From (4) and (5) it follows that T A = A (\ll*), and the theorem is proved.

Theorem 4. If B is a BOREL set in D, then T B is L-measurable.

Proof. Since BOREL sets are analytic (see 1.1.4, exercise 7), TB is an analytic set in R" by theorem 3. But analytic sets in R" are L-measurable (see III.1.1, lemma 5), and the theorem follows.

IV.1.2. The crude multiplicity function. Given T as in IV.1.1, let 5 be a subset of D and x a point in R". The crude multiplicity function N (x, T, 5) is then equal to the number (perhaps 00) of the points u in the set 5 n T-I x. Some elementary properties of N (x, T, 5), needed in the sequel, will now be verified.


Since D is a bounded domain in R!', we have an oriented n-cube Q containing D. For each positive integer j, let Pi denote the j-th positive prime, and consider the subdivision of Q into Pi congruent oriented n-cubes (see 1.2.3, definition 3). Let Qi be a generic notation for a cube occurring in this subdivision. For each j, designate by 0i the class of those cubes Qj which satisfy the condition Qi(D.

If G is a set in Rn, then denote by c(x, G) the characteristic function of G, defined as follows:

c (x, G) = { 1 ~f x E G, o If x Ef G.

Then clearly G is L-measurable if and only if c (x, G) is L-measurable, and G is a BOREL set if and only if c (x, G) is BOREL measurable (see 111.1.1, definitions 5 and 8).

We proceed to verify the following relations, in which 5 stands for an arbitrary subset of D.

N (x, T, 5) ;;;;::; L:; N (x, T, 5 n int Qj), Qi E 0i.

N(x, T, 5) = .1imL:; N(x, T, 5 n int Qi)' QiE 0i. 1_00

N(x, T, 5) ;;:;: L:; c(x, T(5 n int Qi)) ,

N(x, T, 5) = .1imL:; c(x, T(5 n int Qj)) , 1-00

QiE OJ.

QiE OJ.

(1 )

(2)

(3)

(4)

Note that for fixed j the sets 5n int Qi (corresponding to the various cubes Qj E Oil are pair-wise disjoint subsets of 5, and thus (1) is obvious. If F is any subset of D, then clearly

N(x, T,F) ::2:c(x, TF) ::2:0, (5 )

and thus (3) follows from (1). In view of (5) it is evident that (2) is a consequence of (1) and (4) and thus there remains to show that (4) is valid. Fix a point xE Rn. If N (x, T, 5) = 0, then (4) is obvious, since then every term in the summation appearing in (4) vanishes. So assume that N(x, T, 5»0. Let m be any positive integer such that

N(x,T,5)::2:m. (6)

The set 5n T-I x contains then at least m distinct points u1 , ... , um .

By 1.2.3, exercise 2, for j sufficiently large each of these points is interior to a cube Qj. Let Q}, ... , Qi be the cubes Qi for which

u1 E int Qt, ... , u'" E int Qi. Since the diameters of the cubes Qi converge to zero for j -700 and since U 1 , ... , Um are distinct points of the domain D, it is clear that


for j sufficiently large the cubes Ql' ... , Q,!, are distinct and are contained in D, and hence belong to the class OJ. Thus, for j sufficiently large, we have

m

m ;;:;; l: c ( x, T (S n in t Q;)) ;;:;; l: c ( x, T (S n in t Q j)) , Q i E OJ. ;=1

Since m is any integer satisfying (6), it follows that

N(x,T,S);;:;;liminfl:c(x,T(SnintQj))' QjEOj • (7) 1-->-00

Clearly, (7) and (3) imply (4). Theorem. If S is a BOREL set in D, then N (x, T, S) is L-measurable

in Rn.

Prool. Consider the expression (4) for N (x, T, S). Since now S n int Qi is a BOREL set, the image set T(S n int Qj) is L-measurable by IV.1.t, theorem 4. Hence the characteristic functions c (x, T(S n int Qj)) are also L-measurable, and thus (4) shows that N(x, T, S) is L-measurable (see III.1.t, lemma 18).

Let now ° be an open set in D. For each positive integer j, denote by Of the class of those cubes Qj for which int Qj( 0. An argument entirely analogous to that used in verifying (1), (2), (3), (4) yields the following relations.

N (x, T,O) ::::: l: N (x, T, int Qi)' Qi E Dj. (8)

N(x,T,O) = lim LN(x, T,intQj)' QjEOj. (9) J -->-00

N(x,T,O):::::l:c(x,TintQj)' QjEOj. (10)

N(x, T,O) = lim LC(X, TintQj), 1-->-00

(11 )

IV.1.3. Summatory functions. Given the transformation T as in IV.1.t, let flu) be a real-valued, non-negative function in D (see 1.1.2). If S is a subset of D and x is point in Rn , put

W(x, T, S, f) = l: f (u), u E S n T-l X. (1 )

Note that according to the conventions adopted in 1.1.2, the nonnegative function I(u) is permitted to take on the value 00. Furthermore, the summation in (1) is generally non-countable, since the set S n T-l x need not be countable. The general comments in 1.1.2 concerning non-countable summations apply to the present situation. In terms of the notations used there, the defining formula (1) appears in the form

W(X, T, s, I) = l: (S n T-l x, f), (2)

which yields the precise meaning of the summation in (1).


Definition 1. The quantity W(x, T, 5, I) is termed the summatory function induced by T, 5, /.

Thus W(x, T, 5, I) is a non-negative function defined for all points xE Rn. If the transformation T is thought of as fixed, then the simpler notation W(x, 5, I) may be used. In cases where the function f(u} is also fixed, we shall write merely W(x, 5), omitting both T and I. In view of (2), the various elementary properties of non-countable summations (listed in 1.1.2) are available in dealing with the summatory function W. When using such properties in the sequel, the reader will be referred to the general comments in 1.1.2, since the application to W will be immediate.

Lemma. Let 5 be a subset of D, and consider the characteristic function

Is (u) = {1 ~f u E 5, o If uED-5,

of 5 in D. If F is any set in D, then

W(x, T, F, Is} = N(x, T, F n 5).

This statement is an obvious consequence of the definitions involved.

Theorem. If f (u) is a non-negative, BOREL measurable function in D and B is any BOREL set in D, then the summatory function W(x, T, B, I) is L-measurable in R".

Proof. Since T is fixed in the course of the argument, we shall use the simplified notation W(x, B, f). For convenience, the proof is given in three steps.

Case 1. Assume that I is the characteristic function in D of a BOREL

set B* (D. That IS,

I iu) = f 1 if u E B*, . to if u E D - B* .

If B is any BOREL set in D, then (see the preceding lemma)

W(x, B, I) = N(x, T, B n B*). (3)

Since B nB* is a BOREL set, the crude multiplicity function appearing in (3) is L-measurable by the theorem in IV.1.2, and thus (3) shows that W(x, B, f) is L-measurable.

Case 2. Assume that I (u) is non-negative, BOREL measurable, and bounded in D. Then we have a constant M>O such that O:;;;.,/(u):;;;.,M in D. Take a positive integer m. For each integer j such that O:;;;.,j :;;;"2"' denote by Bmi the set of those points uED where

_1_' M-::;:'f(u)<j+1 M . 2m - 2m

218 Part IV. Bounded variation and absolute continuity in Rn.

Then the sets Bmi , j=O, 1, ... , 2m, are pair-wise disjoint BOREL sets whose union is D. Let Imi(u) be the characteristic function in D of the set Bmi . That is,

{ 1 if U E Bmi , I (4) Imi(u) = 0 l'f uE D-Bmi' f

Introduce the auxiliary function

Clearly

and hence

0:;:;,/1:;:;,/2:;:;"",

JvI 0:;:;,1- 1m <2""

limj", =1.

(4*)

(5)

(6)

(7)

Let B be any BOREL set in D. The relations (5) and (7) imply (see 1.1.2) that

lim W(x, B,lm) = W(x, B, I). (8)

In view of (4), (4*) and Case 1, W(x, B, 1m) is L-measurable, and thus (8) shows that W(x, B, f) is also L-measurable (see I1l.l.l, lemma 18).

Case 3. Let finally I (u) be an arbitrary non-negative BOREL measurable function in D. For each positive integer m, put

gm(U)={/(U) ~f O:;:;'f(u):;:;,m,} m If I(u) > m.

Then clearly gm (u) is BOREL measurable, and

0:;:;' gl:;:;' g2:;:;"" ,

limgm = I.

In view of 1.1.2, the relations (10) and (11) imply that

lim W(x, B, gm) = W(x, B, I),

(9)

( 10)

(11)

( 12)

where B is any BOREL set in D. Since gm is non-negative, BOREL measurable, and bounded, it follows by Case 2 that W(x, B, gm) is L-measurable, and thus the L-measurability of W(x, B, t) follows in view of (12).

IV.t.4. The condition (N). Again, T:D -+R" is a bounded continuous transformation, as in IV.1.l.

Delinition. If U is a subset of D, then the transformation T is said to satisfy condition (N) on U if every subset of U of LEBESGUE


measure zero is transformed by T into a set of LEBESGUE measure zero. In symbols: V( U and LV =0 imply LTV =0.

Lemma 1. (i) If T satisfies condition (N) on U, then it also satisfies condition (N) on every subset of U. (ii) If U is a subset of D such that LT U = 0, then T satisfies condition (N) on U. (iii) If rUm} is a sequence of sets in D and T satisfies condition (N) on every Um , then it also satisfies condition (N) on U Um .

These statements are obvious consequences of the definitions involved.

Lemma 2. Let I be a non-negative BOREL measurable function on D, and L an L-measurable subset of D such that T satisfies condition (N) on the set L, consisting of those points u E L where I (u) > O. Then the summatory function W(x, L, f) is L-measurable in Rn.

Proof. The set Lt is L-measurable and hence it admits of a decomposition (see 111.1.1, lemma 8)

Lt = B U V, B n V = 0, L V = 0, (1)

where B is a BOREL set. Since T satisfies condition (N) on L t , it follows from (1) that

LT V = o. (2)

From (1) one infers, by 1.1.2, that

W(x, L, f) = W(x, Lt , f) = W(x, B, I) + W(x, V, f)· (3)

Clearly W(x, V, f) =0 for xECTV, so (2) and (3) imply that

W(x, L, f) = W(x, B, f) a.e. in R". (4)

Since W(x,B, f) is L-measurable in R" by the theorem in IV.1.3, it follows from (4) that W(x, L, f) is also L-measurable in R", and the lemma is proved.

Corollary 1. If L is an L-measurable subset of D such that T satisfies condition (N) on L, then the crude multiplicity function N(x, T, L) is L-measurable in Rn.

Proof. The characteristic function fn of the domain D is clearly BOREL measurable in D, and by the lemma in IV.1.3 one has

W(x, L, In) = N(x, T, L).

Thus corollary 1 is an immediate consequence of lemma 2.

Corollary 2. If L is an L-measurable subset of D such that T satisfies condition (N) on L, then T L is L-measurable.

Proof. Since T L coincides with the set of points x in R" where N(x, T, L):2: 1, corollary 2 is a direct consequence of corollary 1.


Theorem. Let I be a non-negative L-measurable function on D such that T satisfies condition (N) on the set Dt of points u E D where I (u) > O. Then for every L-measurable set L in D the summatory function W(x, L, f) is L-measurable in R".

Proal. By 111.1.1, lemma 24 there exists a non-negative BOREL measurable function 1* on D such that

(5)

1* (u) = I(u) a.e. in D. (6)

Let }'denote the set of points uED where 1*(u)=t=/(u). From (5) and (6) it follows that V (D, and LV = O. Since T satisfies condition (N) on D" one concludes that

LT V = o. (7)

If xECTV, it is clear that 1*(u)=/(u) foruELnT-1x, and consequently W(x, L, 1*) = W(x, L, /) if x E C TV. Using (7) one concludes that

IfT(X, L, f*) = W(x, L, f) a.e. in Rn. (8)

In view of (5) the set Lt. of points u E L where 1* (u) > 0 is a subset of Df . Since T satisfies condition (N) on Dt , it follows that it also satisfies condition (N) on Lt •. Thus the summatory function W(x, L, f*) is L-measurable in R" by lemma 2, and the conclusion of the theorem now follows from (8).

Lemma 3. Let ~ be a family of connected sets C in D, such that the following conditions are satisfied. (i) }l" 0 set C E j"j reduces to a single point. (ii) T is constant on each set C E ~ (that is, T C is a single point). (iii) T satisfies the condition (N) on the set

(9) Then

LT S = o. (10)

Proal. Let k be an integer, In a positive integer, and let j be one of the integers 1, ... , n. Denote by Gk '" i the set of those points u=(u1, ... , un)ED for which Hi = kj2"'. By 111.1.1, lemma 12 we have then L Gk 111 i = O. In view of condition (iii) it follows that

Since the collection of the sets Gklllj (corresponding to all admissible choices for k, m, j) is countable, it follows that on setting

(\1 )

§ IV.2. Absolute continuity with respect to a base-function. 221

we have LT(SnG)=o. (12)

We assert that C n G =F 0 if C E ~. (13 )

Indeed, by condition (i), the set C contains at least two distinct points ul=(ui, ... ,U~),U2=(U~, ... ,u;). For some integer i=1, ... ,n we must have then u~ =F u~, say ui < u~. It follows that for some integer k and some positive integer m we shall have the inequalities

ui < k121n < u~.

Since C is connected, by 1.1.3, exercise 37 there follows the existence of a point u = (u1, ... , ui , ... , un) E C such that 1~i = kl2m. For this point we have then

u E C n Gkmi ( C n G,

and (13) IS verified. Next we assert that

TC=T(CnG) if CElY. (14)

Indeed, by (13) there exists a point uE C n G. Since T is constant on C, we conclude that TC=TuET(CnG), and (14) follows, since obviously T(Cn.G) (TC. From (14) and (9) we conclude that TS=T(SnG), and (10) follows in view of (12).

§ IV.2. Bounded variation and absolute continuity with respect to a base-function.

IV.2.t. Base-functions. Let there be given a bounded continuous transformation T:D-+Rn as in IV.1.t.

Definition 1. A function b (u), defined in D, is termed a base-function in D if it possesses the following properties. (i) At each point uE D, b (u) is either a non-negative integer or + 00. (ii) b (u) is BOREL measurable in D.

The base-functions actually utilized in the sequel will be selected on the basis of topological considerations. Our present objective is the development of a metric theory of base-functions which is of sufficient generality to cover the situations actually arising in the sequel. This theory could be generalized in various ways. For example, the requirement that the finite values of a base-function should be integers could be dropped, as far as the topics discussed in § IV.2 are concerned.

Definition 2. Given a base-function b (u) in D, the set of those points uED where b(u) >0 will be denoted by Db and will be termed


the base-set for b. The set of those points uED where 0 < b (u) < 00

will be denoted by Dr, and will be termed the reduced base-set for b. Since b(u) is BOREL measurable in D, it is clear that Db and Db

are BOREL sets. Obviously (1 )

If S is any subset of D, then there arises the summatory function

W(x, T, S, b) = L b (u) , u E 5 n T-l x, (2)

in the sense of IV.1.3. As T and b will be kept fixed throughout the rest of § IV.2, we shall write W(x, S) instead of W(x, T, S, b). Since b (u) is BOREL measurable, by the theorem in IV.1.3 we have the following statement (which is formulated as a lemma merely for the sake of convenient reference).

Lemma 1. If B is a BOREL set in D, then the summatory function W~x, B) is L-measurable (as a function of x in Rn).

Let S be a subset of D. If xEJ: T S, then S n T-l x = 0, and hence clearly

W(x,S)=O if xECTS. (3)

The following lemmas are concerned with the case when x E T S.

Lemma 2. If SeD-Db' then W(x, 5) =0 even for xET S.

Proof. Since now SnT-1x(D-Db, and b(u)=O for uED-Db' the assertion is obvious in view of (2).

Lemma 3. If S (Db - Db, then W(x, S) = + 00, and W(x, D) = + 00

for xET S.

Proof. The assertion is obvious, since b (u) = + 00 for uE Db - D~

and W(x, D):;;: W(x, S).

Lemma 4. If SeD'/"~ then W(x, S):;;: 1 for xET S. Proof. If xE T S, then there exists at least one point uoE S (Db

such that Tuo=x. Then uoESnT-lx and b(uo)::2::1, and hence

W(x, S) :;;: b (uo) :;;: 1.

IV.2.2. Bounded variation with respect to a base-function.

Definition 1. The transformation T, given as in IV.1.t, is said to be of bounded variation in D with respect to the base-function b (briefly, B Vb in D) if the corresponding summatory function W(x, D) is Lsummable in Rn (see IV.2.t).

Observe that W(x, D) is L-measurable by IV.2.t, lemma 1, and hence the preceding definition is meaningful. If B is any BOREL set in D, then clearly

0;;;;: W(x, B) ;;;;: W(x, D). (1 )


Thus the L-summability of W(x, D) implies the L-summability of W(x, B). This remark yields the following statement.

Lemma 1. If T is B V b in D, then it is also B V b in every domain D(D.

Assume now that T is B V b in D. If B is any BOREL set in D, then W(x,B) is L-summable in R n as a consequence of (1). On setting

v (B) = f W(x, B) dL, (2)

we obtain therefore a finite-valued, non-negative function v of BOREL

sets in D. As regards the definition of v (B), observe that W(x, B) = 0 for

xECTBbyIV.2.1(3)' and hence (since Tisbounded) W(x, B) vanishes outside of a properly selected n-cube in Rn. The integration in (2) is taken over R", as explained in 111.1.1. Note that v depends also upon T and b, and hence a fully descriptive notation for v should be v(B, T, b). Since T and b are thought of as fixed in the present context, the concise notation v (B) is adequate. Observing that W(x, B) =0 for xECTB by IV.2.10), we see that (2) may be written in the equivalent form

v(B) = f W(x, B) d L. TB

Lemma 2. If Tis B Vb in D, then 'V (B) is a finite measure on BOREL

sets in D.

Proof. Let B1 , .•. , B m , .•. be a (finite or infinite) sequence of pair-wise disjoint BOREL sets in D. Put B = UBm . Then (see 1.1.2)

W(x, B) = .l: W(x, Bm) .

Integration yields (see 111.1.1, lemma 27)

v (B) = .l: v (Bm).

Thus v is count ably additive on BOREL sets in D, and we already noted above that v is finite-valued and non-negative. Thus v is a finite measure on BOREL sets in D (see 111.1.2).

Definition 2. The measure v is termed the measure induced in D by the transformation T and the base-function b, where T is assumed to be BVb in D. The derivative of v is denoted by D(u, v).

Observe that D (u, v) exists a.e. in D and is L-summable in D by 111.2.4, lemma 1. From the point of view of applications, it is important to obtain information concerning points u E D where D (u, v) = O. If B is a BOREL set in D, then (by 111.2.4, lemma 2)

v(B) = 0 implies D(u, v) = 0 a.e. on B. (4)


Accordingly, we proceed to determine BOREL sets B (D on which v vanishes. It will be assumed throughout that T £s B V b in D.

Lemma 3. v(D-Dbl =0 (see IV.2.1, definition 2).

Proof. Since W(x,D-Db)=O for xET(D-Db) by IV.2.1, lemma 2, the assertion follows directly from (3) with B=D-Db'

Lemma 4. If B is a BOREL set in D such that LTB = 0, then v(B) = O.

This is a direct consequence of (3).

Lemma 5. V(Db-D~)=O (see IV.2.1, definition 2).

Proof. In view of IV .2.1, lemma 3, applied with S = Db - Dr" the set T(Db - Db) is contained in the set where W(x, D) = + 00.

This latter set is of L-measure zero since W(x, D) is L-summable in R" (see 111.1.1, lemma 31). Thus LT(Db- D~) =0, and hence V(Db- Db) = 0 by lemma 4.

Lemma 6. If B is a BOREL subset of Dr" then v (B) = 0 if and only if LTB =0.

Proof. ByIV.2.1, lemma 4, one has W(x,B)21 for xETB. Hence, by (3), v(B)2LTB. Thus v(B}=O implies that LTB=O. Conversely, if LTB=O, then v(B)=O by (3).

Lemma 7. v(D-D/'J =0.

This is a direct consequence of lemmas 3 and 5. Lemma 8. If B is any BOREL set in D, then v(B)=v(BnDJ,J.

Proof. B is the union of the disjoint BOREL sets Bl =Bn (D - Db) and B2=BnD'b. Hence

v(E) = v(BI ) + v(B2 ).

In view of lemma 7, we have

V(Bl) ::;;;; v(D - Db) = 0

and thus v (EI) = O. Hence v (B) = v (B2 ), and the lemma is proved.

The preceding lemmas will now be applied to obtain statements concerning the vanishing of the derivative D (u, v) (see definition 2), it being assumed throughout that T is B V b in D.

Lemma 9. D(u, v) =0 a.e. on D-D~.

Proof. Since v (D - Db) = 0 by lemma 7, the assertion follows from (4).

Lemma 10. If S is a set in R" such that L S = 0, then D (u, v) = 0 a.e. on T-I S.

Proof. \Ve can select (see 111.1.1, lemma 7) a BOREL set B(R" such that SeB and LB=O. Then T-IB is a BOREL set in D (see IV.1.1,


theorem 1), and TT-IB(B. Thus LTT-IB=O, and hence v (T-IB) =0 by lemma 4. In view of (4) it follows that D (u, v) = 0 a.e. on T-IB, and hence also on T-IS, since T-IS(T-IB.

Lemma 11. If F is any set in R" such that L (F n T Db) = 0, then

D(u,v)=O a.e. on T-IF.

Proof. F is the union of the disjoint sets

FI = F neT Db and F2 = F n T Db'

Since LF2 =0 by assumption, we have by lemma 10

D(u,v)=O a.e. on T-IF2. (5)

On the other hand, clearly T-IFI(D-Db. As D(u,v)=O a.e. on D - Db by lemma 9, it follows that

D(u, v) = 0 a.e. on T-IFI . (6)

Since T-I F is the union of T-I FI and T-I F2, (5) and (6) imply that

D(u, v) = 0 a.e. on T-IF.

Lemma 12. If G is any set in D such that LTG=O, then D(u, v) =0 a.e. on G.

Proof. On setting S=TG, and noting that G(T-ITG=T-IS, one sees that lemma 12 is an immediate consequence of lemma 10.

IV.2.3. Absolute continuity with respect to a base-function. Let there be given a bounded continuous transformation T:D--7R" and a base-function b (u) in D, as in IV.2.2.

Definition. T is said to be absolutely continuous in D with respect to the base-function b (briefly, A C b in D) if T is B V b in D and the measure v generated by T and b is absolutely continuous with respect to L-measure in D (see IV.2.2 and III. 1.2).

Explicitly, T is A C b in D if the following holds. (i) The summatory function W(x, D) corresponding to b (see IV.2.1) is L-summable in R". (ii) v (B) = 0 for every BOREL set B (D such that LB = O. Suppose now that these conditions are satisfied, and consider any domain D (D. Then 0 ~ W(x, D) ~ W(x, D), and hence W(x, D) is also Lsummable. Thus condition (il still holds if D is replaced by D, and the same is obvious for condition (ii). Hence the following statement.

Lemma 1. If T is A C bin D, then T is also A C b in every domain D(D.

Rado and Reichelderfer, Continuous Transformations, 15


Lemma 2. Suppose that T is B V b in D. Then T is A C binD if and only if T satisfies the condition (N) on the reduced base-set Dr, (see IV.2.t, definition 2 and IV.1.4).

Proof. Suppose first that T is A C b in D, and consider any set 5 (D~ such that L 5 = O. There exists then a BOREL set B such that 5(B and LB=O. Since Db itself is a BOREL set, the set B n Db is a BOREL set such that

5 ( B n Db ( Dr;, L(B n Db) = o.

As v is absolutely continuous, it follows that v(B n Db) = 0, and hence LT(B n Db) = 0 by IV.2.2, lemma 6. In view of the inclusion T 5 (T(B n Db) we conclude that LT 5 = O. Thus it is established that T satisfies the condition (N) on Dr,. Suppose, conversely, that Tis B Vb in D and satisfies the condition (N) on Db' Consider any BOREL set B ( D such that L B = O. Then B n Dr, is also a BOREL set and L (B n D~) = O. Since T satisfies the condition (N) on Db, it follows that

LT(BnDb)=O. (1 )

In view of IV.2.2, lemma 6, we infer from (1) that

v(B n Db) = o. (2)

By IV.2.2, lemma 8 we have

v(B) = v(B n D~). (2) and (3) imply that v(B) =0. Thus v is absolutely continuous, and the lemma is proved.

Lemma 3. If T is A C b in D, then T satisfies the condition (N) on the base-set Db'

Proof. Let 5 be any set in Db such that L 5 = O. As 5 is the union of the sets

51 = 5 n (Db - Db) and S2 = 5 n Db,

we have T 5 = T 51 U T 52' and hence the lemma will be proved if we can establish the relations

L T S1 = 0,

LTS2 =0.

(4)

(5)

Now since 5 2 (Db and L52 =0, (5) is a direct consequence of lemma 2. As 51 (Db -Db, by IV.2.t, lemma 3 the set T51 is a subset of the set where W(x, D) = + 00. This latter set is however of L-measure zero, since W(x, D) is L-summable, and thus (4) follows.


Lemma 4. If T is A C b in D and L is any L-measurable set in D, then W(x, L) is L-measurable and L-summable in Rn.

Proof. By lemma 3, T satisfies the condition (N) on Db' and thus W(x, L) is L-measurable by IV.1.4, lemma 2 [recall that the base function b(u) is BOREL measurable]. Since W(x, D) is L-summable and clearly O:S:: W(x, L) :s:: W(x, D), it follows that W(x, L) is L-summable also.

Lemma 5. Suppose that T is A C b in D, and let 5 be a set in D such that LS=O. Then W(x, 5) =0 a.e. in R".

Proof. There exists (see 111.1.1, lemma 7) a BOREL set B such that S(B(D and LB=O. Since Tis ACb in D, we have v(B)=O. As W(x, B) ~O, in view of IV.2.2 (2) it follows that W(x, B) = 0 a.e. in Rn. Since O:S;;W(x,S):S::W(x,B), we conclude that W(x,S)=O a.e. in Rn.

IV.2.4. Transformation formulas for definite integrals. Let there be given a bounded continuous transformation T: D -l>- Rn and a basefunction b(u), as in IV.2.1.

Theorem 1. Assume that Tis BVb in D, and let v be the measure induced by T and b (see IV.2.2, definition 2). If h(u) is a (finite-valued) non-negative, BOREL measurable function in D and B is a BOREL set in D, then

J h(u) d l' = J W(x, B, h b) dL, (1 ) B

as soon as one of the two integrals involved exists. More explicitly: if either of the two integrals exists, then so does the other, and (1) holds.

Proof. The function hb is BOREL measurable in D, and hence the summatory function W(x, B, hb) is L-measurable in R" by IV.I.3. Since h is BOREL measurable, the measurability conditions necessary for the consideration of the integrals appearing in (1) are fulfilled. Take a positive number e. For each non-negative integer j denote by Bj the set of those points u E B where

ej:S::h(u) <e(f+1). (2)

Evidently the sets Bj , j = 0, 1, 2, ... are pair-wise disjoint BOREL sets whose union is B. Integration of the inequalities (2) over B j with respect to the measure v yields

ejv(Bj):s::J h(u) dv :S::e(j + 1) v(Bj). Bj

Since the base function b (u) is non-negative, multiplic3;tion of the inequalities (2) by b(u) yields

ejb(u) :S::h(u) b(u) ;;;;'e(j + 1) b(u), uEBj . (4) 15*


For the summatory functions W(x, B j , b), W(x, B j , h b) we obtain from (4) the inequalities (see 1.1.2)

Bj W(x, B j , b) :s;; W(x, B j , h b) :s;; B(j + 1) W(x, B j , b). (5)

Since Tis BVb in D, W(x, B j , b) is L-summable (see IV.2.2). Thus (5) shows that W(x, Bj , h b) is also L-summable. Integration of (5) yields

Bj v (Bj ) :S;;J W(x, B j , hb) dL:S;; B(j + 1) v (Bj) . (6)

From (3) and (6) we infer that

11 h(u) dv - J W(x, Bf , h b) dLI:S;; B v (Bj ) .

Hence for every integer m ~ 0 we have the inequality

Ij~olh(U)dV-i~JW(X,BJ,hb)dLI :S;;Bi~/(Bi) :S;;8'V(B). (7)

From (7) it is evident that the two series of non-negative terms 00 00

L Jh(u)dv and L JW(x,Bi,hb)dL (8) i~O B; i~O

are either both convergent or both divergent. Since B is the union of the pair-wise disjoint BOREL sets Bi , and h(u) is non-negative, it follows (see 111.1.2) that h is v-summable on B if and only if the first series in (8) is convergent, and that

00

J h(u) dv = L J h(u) dy B i ~O B;

if h is v-summable on B. On the other hand, by 1.1.2 we have 00

W(x,B,hb) =LW(x,Bj,hb). i~O

(9)

So it follows (see 111.1.2) that W(x, B, hb) is L-summable if and only if the second series in (8) is convergent, and that

JW(x, B, hb) dL = L J W(x, B i , hb) dL i~O

(10)

if W(x, B, hb) is L-summable. We conclude from the preceding observations that the first integral in (1) exists if and only if the second one exists. Assume now the existence of these two integrals. From (7), (9), (10) we obtain the inequality

11 h(u) dv- JW(x, B, hb) dLI :S;;Bv(B).

Since B > 0 was arbitrary, the relation (1) follows.

§IV.2. Absolute continuity with respect to a base-function. 229

Lemma 1. Assume that T is B V b in D, and let v be the measure induced by T and b. If H (x) is a (finite-valued) non-negative, BOREL measurable function in R" and B is a BOREL set in D, then

f H(T u) dv = f H(x) W(x, B) d L, (11) B


Proof. Put h(u) =H(Tu), uED. ( 12)

By IV.1.1, theorem 2, it follows that h (u) is a finite-valued, nonnegative, BOREL measurable function in D, and clearly

W(x, B, h b) = H(x) W(x, B). (13)

In view of (12) and (13), the present lemma appears as a direct consequence of theorem 1.

Theorem 2. Assume that T is B V b in D, and let v be the measure induced by T and b. If H(x) is a (finite-valued) real-valued, BOREL measurable function in R", and B is a BOREL set in D, then

f H(T u) dv = f H(x) W(x, B) dL, (14) B


Proof. Consider the auxiliary functions

(15 )

Observe that HI (x), H 2 (x) are finite-valued, non-negative, BOREL measurable functions. Hence, by lemma 1,

f HI (T ttl dv = f HI (x) W(x, B) d L, ( 16) B

as soon as one of the two integrals involved exists, and similarly

f H2(T u) dv = f H 2 (x) W(x, B) dL, (17) B

as soon as one of the two integrals involved exists. Now suppose that the first integral in (14) exists. By (15) and 111.1.1, lemma 26 it follows that the first integral in (16) also exists. Since clearly O~H2~2IHI =

2 HI' it follows further that the first integral in (17) exists. Hence (16) and (17) both hold, and on subtracting (17) from (16) we obtain (14), in view of (15). Suppose next that the second integral in (14) exists. The same reasoning as before [applied to the second integrals in (16) and (17)J yields again (14).

We proceed to consider the case when T is AC b in D.


Lemma 2. Assume that Tis ACb in D, and let D(u, v) denote the derivative of the measure v induced by T and b. If H(x) is an L-measurable (finite-valued) function in Rn and B is any BOREL set in D, then

J H(T u) D (u, v) dL = J H(x) W(x, B) dL, ( 18) B


Proof. There exists (see 111.1.1, lemma 23) a (finite-valued) BOREL measurable function H*(x) in Rn such that H*(x) =H(x) a.e. in R". On denoting by e the exceptional set where H* (x) '* H (x), we have

H(x) =H*(x) if xE! e, Le = O. (19)

We assert that

H(Tu) D(u, v) = H*(Tu) D(u, v) a.e. in D. (20)

Indeed, if uET-ICe, then H(Tu) =H*(Tu) by (19). Since D(u, v) exists a.e. in D (see IV.2.2), it follows that

H(Tu)D(u, v) =H*(Tu)D(u, v) a.e. on T-lCe. (21)

On the other hand, since D(u, v)=O a.e. on T-1e by IV.2.2, lemma 10, we have

H(T u) D (u, v) = H*(T u) D (u, v) a.e. on T-1e. (22)

As D=T-lCeUT-1e, clearly (20) follows from (21) and (22). Now since H*(Tu) is BOREL measurable by IV.1.1, theorem 2, and D (u, v) is L-measurable by 111.2.3, theorem 3, it follows from (20) that H(Tu)D(u, v)· is L-measurable in D (see 111.1.1, lemma 21). Also, W(x, B) is L-measurable by IV .1.3. Thus the measurability conditions necessary for the consideration of the two integrals in (18) are fulfilled. We proceed to discuss the existence and equality of those two integrals. In view of (20) it follows that

JH(Tu)D(u,v)dL=JH*(Tu)D(tt,v)dL, (23) B B

as soon as one of the two integrals involved exists. By III.2.4, theorem4, we have

J H*(Tu) D(u, v) dL = J H*(Tu) dv, (24) B B

as soon as one of the two integrals involved exists. By theorem 2 we have

J H*(T u) dv = J H*(x) W(x, B) d L, (25) B


as soon as one of the two integrals involved exists. Finally, in view of (19) it follows that

J H*(x) W(x, B) dL = J H(x) W(x, B) dL, (26)

as soon as one of the two integrals involved exists. Clearly, it follows that if one of the two integrals in (18) exists, then all the integrals appearing in (23) to (26) exist and have the same value. Thus the lemma is proved.

Theorem 3. Assume that T is A C b in D, and let D (u, 11) denote the derivative of the measure 11 induced by T and b. If H(x) is an Lmeasurable (finite-valued) function in R" and L is any L-measurable set in D, then

J H(T u) D(u, 11) dL = J H(x) W(x, L) dL, (27) L


Proof. Note first that W(x, L) is L-measurable in Rn by IV.2.3, lemma 4, and H(Tu)D(u, 11) is L-measurable in D, as it has been shown in the course of the proof of lemma 2. Note next that there exists a BOREL set B such that

L(B(D, L(B-L)=O. (28)

Clearly, therefore,

J H(T u) D (u, 11) d L = J H(T u) D (u, 11) d L, (29) L B

as soon as one of the two integrals involved exists. By lemma 2 we have

J H(Tu) D (u, 11) dL = J H(x) W(x, B) dL, (30) B

as soon as one of the two integrals involved exists. We assert that

W(x, B) = W(x, L) a.e. in Rn.

Indeed, since the sets Land B - L are disjoint, we have

W(x, B) = W(x, L) + W(x, B - L).

By IV.2.3, lemma 5, we have in view of (28)

W(x, B - L) = 0 a.e. in R".

Clearly, (31) follows from (32) and (33). From (31) we infer that

J H(xj W(x, B) d L = J H(x) W(x, L) d L,

(31)

(32)

(33)

(34)


as soon as one of the two integrals involved exists. It is now clear that if one of the two integrals appearing in (27) exists, then all the integrals appearing in (29), (30). (34) exist and have the same value, and thus the theorem is proved.

§ IV.3. Bounded variation and absolute continuity with respect to a multiplicity function.

IV.3.t. Admissible multiplicity functions. Let T: D ---+ R" be a bounded continuous transformation, where D is a bounded domain in Rn. The term multiplicity function has been already used in various contexts. In § 11.3, we considered the multiplicity functions K(x, T, D), K+ (x, T, D), K- (x, T, D), k(x, T, D), and the crude multiplicity function N(x, T, S) entered the picture on numerous occasions. Our objective in the present § IV.3 is the development of a general theory applicable to all those multiplicity functions that are relevant for our purposes. The transformation T is assumed given as above.

Definition 1. Let M(x, T, D) be a function defined for every point xERn and every domain D(D (and depending also 'upon T), such that each value of M(x, T, D) is either a non-negative integer or + 00.

Then M(x, T, D) is termed a non-negative multiplicity function for the given transformation T.

The examples mentioned above are non-negative multiplicity functions in this sense. A useful theory of such multiplicity functions will naturally depend upon additional properties that are available in relevant special cases. Before stating such properties, let us introduce a function b (u) associated with a given non-negative multiplicity function M(x, T,D) in the following manner. Let u be a point in D, and consider all those domains D which satisfy the conditions u ED (D. If D is such a domain, then M(Tu, T, D) is defined and is either a non-negative integer or + 00. We define b (u) as the greatest lower bound of all these values M(Tu, T, D). In symbols

b(u) =gr.l.b.M(Tu, T,D), uED(D. (1 )

Clearly, each value of b (u) is either a non-negative integer or + 00.

Since b(u) depends upon M, a more descriptive notation like b(u,M) may be used if M is not clearly identified by the context.

Definition 2. A non-negative multiplicity function M(x, T, D) is termed admissible (for the given transformation T) if the following conditions are satisfied.

(i) If xEf T D, then M(x, T, D) = o.

§ IV.3. Absolut\l continuity with respect to a multiplicity function. 233

(ii) For fixed x, M is sub-additive with respect to D. Explicitly: if D1 , ... , D m' D are domains in D such that D k (D, k = 1, ... , m, and Din Dk = 0 if i=t=k, then

m

L M(x, T, Dk ) ~ M(x, T, D). (2) k=l

(iii) If D is a domain in D and {Dm} is an infinite sequence of domains filling up D from the interior (see 11.3.2, remark 10), then

M(x, T, D) = lim M(x, T, Dm) .. m ..... oo

(iv) For fixed D, M(x, T, D) is L-measurable as a function of x in R", and the function b (u), associated with M by means of (1), is BOREL measurable in D.

We shall verify later on that all the multiplicity functions relevant for our purposes are admissible in the sense of the preceding definition.

Definition 3. The function b (u) defined by (1) is termed the basefunction generated by the admissible multiplicity function M.

As noted above, each value of b (u) is clearly a non-negative integer or + 00. Also, b (u) is BOREL measurable by (iv). Thus the term base-function is used in the preceding definition in conformity with IV.2.l, definition 1. The base-function b (u) determines a summatory function [see IV.2.l (2)J, given by the formula

W(x, T, 5) =Lb(u),· uE 5nT-1 x, (3 )

where 5 is any set in D. This summatory function depends upon M, and hence a more descriptive notation like W(x, T, 5, M) may be used if the multiplicity function M is not clearly identified by the context.

Definition 4. The function W, defined by (3), is termed the summatory function generated by the multiplicity function M.

Throughout this section, the transformation T and the admissible non-negative multiplicity function M are thought of as fixed. The symbols b, Ware used in the sense of (1) and (3) respectively.

Lemma 1. If D is any domain in D and x is any point in Rn , then

o ~ W(x, T, D) ~ M(x, T, D). (4)

Proof. The assertion is obvious if W(x, T, D) = O. SO assume that W(x, T, D) > 0, and take any real number ex such that

W(x, T, D) > ex::2: O. (5)


In view of the definition of W, there follows that existence of a finite sequence U1 , .. " Um of distinct points in D n T-l x such that

(6)

Select pair-wise disjoint domains D1 , ..• , D", such that ukEDk(D, k=1, ... , m. By (ii), (1), (6) it follows that

m m

M(x, T, D) ;;;;; 1: M(x, T, Dk) ;;;;; 1: b(uk ) > IX. (7) k=l k=l

Since IX was any number satisfying (5), clearly (7) implies (4).

Lemma 2. Given a point uED, there exists a domain Du such that

uEDu(D,

b (u) = WiT u, T, Du) =M(T u, T, Du)'

(8)

(9)

Proof. Since each value of M is either a non-negative integer or + 00, it is clear from the defining formula (1) that there exists a domain Du such that uEDu(D and

b(u) =M(Tu, T, D,J. (10)

On the other hand, by the definition of Wand by lemma 1 we have

b (u) ~ W(T u, T, Dul ~ M(T u, T, Dul,

and (9) follows in view of (10).

Lemma 3. Let x be a point in R n and D a domain in D such that

M(x, T, D) < 00. (11 )

Assign a compact set F in D. Then there exists a domain Do such that

F(Do, 15o(D, M(x, T,Do)=M(x, T,D). (12)

Proof. According to 1.2.3, exercise 9, we can select a sequence of domains {Dm} filling up D from the interior. By the definition of such a sequence we have then

and by (iii)

Dm(D, m=1,2, ... ,

F (Dm for m large,

lim M(x, T, Dm) = M(x, T, D). ",->-00

(13 )

(14)

( 15)

§ IV.3. Absolute continuity with respect to a multiplicity function. 235

Since the finite values of M are integers, (15) and (11) imply that

M(x, T, Dm) =M(x, T, D) for m large. (16)

On choosing m sufficiently large and setting Do=Dm, (12) will hold in view of (13), (14), (16).

Lemma 4. Let x be a point in Rn and D* a domain in D such that

(17)

Then for every domain D(D* we have

W(x, T, D) = M(x, T, D). (18)

Proof. (18) is obvious, in view of lemma 1, if M(x, T, D) =0. So assume that M(x, T, D»O. By (ii) we have then

M(x, T, D*) ~M(x, T, D) > o. (19)


0< W(x, T, D*) < 00. (20)

Let F* be the set of those points uED*n T-1x where b(u»O. Then [see (3)J

(21)

If D' is any domain in D*, then clearly [see (3)J

W(x, T, D') = L b(u), uE D'nF*. (22)

Since b (u) is integral-valued, we conclude from (20) and (21) that F* is a finite, non-empty subset of D*. Returning to the domain D, we have by (22) (applied with D' = D)

W(x, T, D) = L b (u), u ED n F*. (23)

Since D n F* is a finite set, it is compact (perhaps empty). By lemma 3 we have therefore a domain Do such that

D n F* (Do, 150 (D, M(x, T, Do) = M(x, T, D). (24)

In view of (22) (applied with D' = Do). the first two relations in (24) imply that

W(x, T, Do) = L b(u}, uE Do nF* = D nF*,

and hence by (23) W(x, T, Do) = W(x, T, D). (25)


Consider now the open set D*-Do' In view of (24), this set is nonempty, and it contains the set F* - D n F*. This latter set is finite (perhaps empty). Accordingly, we can select a finite number of components D1 , .•• , Dm of D* -Do such that

(26)

Note that each Dk is a domain, since D* -Do is open. The domains Do, D1 ,··· ,D", are pair-wise disjoint sub-domains of D*, and hence by (ii)

m

L M(x, T, Dk) ;;;;;M(x, T, D*) . (27) k~O

Furthermore, F* is contained in the union ofthe domains Do, D1 , •.. , Dm by (24) and (26). Hence [applying (22) with D'=Do,D1 , •.• ,Dm,D*]

m

L W(x, T, Dk) = W(x, T, D*). (28) k=O

Let us put

Yk =M(x, T, Dk ) - W(x, T, D kJ , k = 0,1, ... , m. (29)

Subtraction of (28) from (27) yields, in view of (17),

Since Yk:;;;;;O by lemma 1, it follows that Yk = 0, k = 0, 1, ... , m. In view of (29), (25), (24), the relation Yo=O is equivalent to (18), and the lemma is proved.

Lemma 5. Let x be a point in R" such that

M(x, T, D) < 00. (30)

Then there exists a positive number C = C (x) such that the following holds: if D is any domain such that (see 1.1.4, definition 4)

then D(D, bD <C(x),

W(x, T, D) =M(x, T, D).

(31) .

(32)

Proof. According to lemma 3, we can select domains D', D" such that

D'(D", D"(D,

M(x, T, D') = M(x, T, D).

(33)

(34)

If T-1 x=0, then it follows from (i) and lemma 1 that M(x, T, D) = W(x, T, D) = 0 for every domain D (D. Thus C (x) may be chosen


as any positive number in this case. So suppose that T-l x =1= 0. For each point uE T-l x there exists, by lemma 2, a domain Du such that uEDu(D and

W(x, T, Du) = M(x, T, Du). (35)

Note that since D is bounded, the set l5" n T-l x is bounded and closed, and hence compact. Clearly the domains D u , corresponding to the points uE T-l x, constitute an open covering of the compact setl5" n T-l x. Hence there exists (see 1.1.4, exercise 6) a positive number 1] such that the following holds: if S is any set satisfying the conditions

then S(D" for some point uEl5"nT- l x. Observe next that (33) yields e(D', CD"»O (see 1.1.4, definition 7). Choose now C(x) as the smaller one of the two positive numbers 17 and e (D', CD"). We proceed to show that this choice of C (x) is adequate for the purpose of the lemma. Take any domain D satisfying (31). It is convenient to consider three cases.

Case 1. Assume that D nD' = 0. Then D and D' are disjoint domains in D, and hence by (ii)

° S;M(x, T,D) +M(x, T,D') S;M(x, T,D). (36)

From (36) and (34) we conclude that M(x, T, D) =0, and thus (32) follows by lemma 1.

Case2. Assume that D nT-Ix = 0. By (i) we have then M(x,T,D) = 0, and (32) follows by lemma 1.

Case 3. There remains to discuss the situation when

D n D' =1= 0 and D n T-l x =1= 0. (37)

Since oD<C(x) S;e(D', CD"), the first relation in (37) implies that

D ( D" (l5". (38)

From (38) and the second relation in (37) one concludes that

D n l5" n T-l x =1= 0. (39)

Since oD < C (x) S; 1]. we infer from (39) that there exists a point u such that

(40)

In view of (35), (30), and lemma 4, applied with D* =D,,, (32) follows from the second relation in (40), and the proof is complete.


Our next lemma is concerned with the reduced base-set Db (see IV.2.1, definition 2) corresponding to the base-function b (u) given by (1). Let us recall that

Db={uluED, O<b(u) <oo}. (41)

Lemma 6. A point u E D belongs to the reduced base-set Db if and only if for every s>O there exists a domain D". (which depends upon u and s) such that the following holds.

(i) u ED".(D. (ii) bDu• < s. (iii) 1;;;;'M(Tu,T,Du.) <00.

(iv) If D is any domain such that u ED(Du., then M(Tu, T, D) =

M(Tu, T, Due). Proof. Assume first that a domain Due with the properties (i) to (iv)

exists for every s> o. From (1), (i) and (iii) we infer that

btu) < 00. (42)

Let us take now any domain D such that u t= D (D. Since D is open, from (i) and (ii) it follows that we shall have the inclusions u E Due (D if s is sufficiently small, and hence (if s is sufficiently small)

M(T u, T, Due) ;;;;. M(T u, T, D).

In view of (iii) it follows that

M(T u, T, D) ~ 1 if u E D (D,

and thus [see (1)J b (u) ~ 1. (43)

From (42), (43), (41) we see that u E Db. Suppose, conversely, that u E Db. Then

1;;;;'b(u) < 00. (44)

By lemma 2, there exists a domain D" such that

uE Du (D, btu) = M(Tu, T, Du). (45)

Now assign s>O arbitrarily. Since Du is open and uED", we can select a domain D"e such that

uED"e(Du, bDu.<s. (46)

We assert that Due satisfies the conditions (i), (ii), (iii), (iv). For (i) and (ii) this is clear in view of (46). Also, since u EDu.(Du, we have

btu) ;;;;'M(Tu, T, DUE) ;;;;'M(Tu, T, Du),


and hence (iii) is also satisfied in view of (44) and (45). To verify (iv), consider any domain D such that u ED ( D" e' Then [see (45) and (46)]

b (u) -;;. M(T u, T, D) -;;'M(T u, T, Du.) -;;. M(T u, T, Du) = b (u),

and hence the sign of equality must hold throughout. In particular

M(Tu, T, D) =M(Tu, T, Due)'

Thus (iv) is verified, and the lemma is proved.

IV.3.2. Bounded variation. Throughout this section, T denotes a transformation given as in IV.3.1, and we assume that there is assigned an admissible (non-negative) multiplicity function M in the sense of IV.3.1, definition 2. The symbols band W will be used in the sense of the definitions 3 and 4 in IV.3.1. The transformation T and the admissible (non-negative) multiplicity function M are thought of as fixed, and hence the corresponding base-function b and the summatory function Ware also considered as fixed.

Definition 1. T is said to be of bounded variation in D with respect to the admissible (non-negative) multiplicity function M(x, T, D) (briefly, B VM in D) if M(x, T, D) is L-summable in R".

Lemma 1. If Tis B VM in D, then it is also B VM in every domain D(D.

Proof. If D is a domain in D, then

0;;;;: M(x, T, D) ;;;;: M(x, T, D)

by the condition (ii) in IV.3.1, definition 2. Thus the L-summability of M(x, T, D) implies the L-summability of M(x, T, D), and hence T is BVM in D.

Assume that T is B VM in D. If D is any domain in D, then M(x, T, D) is L-summable, and hence we can define a function [J((D) of domains D (D by the formula

[J((D) = J M(x, T, D) dL, D (D. (1 ) Clearly

o ;;;;: [J((D) < 00 , D ( D. (2)

Since [J((D) depends upon T and M, the concise notation [J((D) is adequate only if T and M are thought of as fixed, which is the case at present.

Definition 2. The function of domains [J((D) , given by (1), is said to be generated by the admissible multiplicity function M.

Lemma 2. Assume that T is B VM in D, and let [J((D) be the function of domains generated by M. Then the following holds.

240 Part IV. Bounded variation and absolute continuity in RH.

(i) [Il(D) is sub-additive on the family of domains in D (see 111.2.4. definition 2).

(ii) The derivative D(u. [Il) of [Il [see 111.2.4 (2)J exists a.e. in D and is L-summable in D.

(iii) If D is any domain in D. then

J D(u. [Il) dL :;;;'[Il(D). (3) D

Proof. Take domains D1 ••..• Dm. D in D. such that Dk(D. k=1 • ...• m. and DjnDk= 0 for i=Fk. By condition (ii) in IV.3.1. definition 2. we have then

'" L M(x. T, D k ) :;;;, M(x, T, D). k~l

Integration yields the inequality m

L [Il(Dk) :;;;, [Il(D) • k~l

and (i) is proved. In view of (2) and (i), [Il is a non-negative, finitevalued, sub-additive function of domains D (D. Thus (ii) and (iii) follow directly from 111.2.4, theorem 1.

Lemma 3. Assume that Tis BVM in D, and let b denote the basefunction generated by M (see IV.3.1, definition 3). Then T is BVb in D (in the sense of IV.2.2, definition 1).

Proof. By IV.3.1, lemma 1. we have

o ~ W(x, T, D) ~ M(x, T, D) .

Since M(x, T, D) is now L-summable (see definition 1), it follows that W(x, T, D) is also L-summable, and thus T is B V b in D.

Once it is known that T is B Vb in D, we have at our disposal the finite measure [see IV.2.2 (2)J

v(B) = J W(x. T, B) d L, (4)

defined for all BOREL sets B (D. as well as the derivative D (u, v) of v (see IV.2.2, definition 2).

Summing up: if Tis BV},! in D, then M generates a non-negative, finite-valued. sub-additive function 9Jl(D) of domains D (D which possesses a derivative D (u. [Il) a.e. in D. On the other hand, ;n.1 generates a base-function b (u) and a corresponding summa tory function W, in the sense of IV.3.1, definitions 3 and 4. In turn, W yields the finite measure v, given by (4). which possesses a derivative D(u, v) a.e. in D. The relationships amongst these various functions will now be discussed.


Lemma 4. Assume that Tis BVM in D. Then the following statements hold.

(i) If D is any domain in D, then

y (D) ;;;;: ID1(D) .

(ii) The sign of equality holds in (5) if and only if

W(x, T, D) =M(x, T, D) a.e. in Rn.

(5)

(6)

(iii) If y (D) = ID1(D). then y (D) = ID1(D) for every domain D ( D.

Proof. If D is a domain in D, then

o ;;;;: W(x, T, D) ;;;;: M(x, T, D) (7)

by IV.3.1, lemma 1. In view of (1) and (4), integration of (7) yields (5). By 111.1.1, lemma 35, (7) implies that the sign of equality holds in (5) if and only if (6) holds. Thus (i) and (ii) are proved. Suppose now that 11 (D) = ID1(D). By (ii) it follows that

W(x, T, D) = M(x, T, D) a.e. in Rn. (8)

Since M(x, T, D) is now L-summable, we have (see 111.1.1, lemma 31)

M(x, T, D) < 00 a.e. in Rn. (9)

Take now any domain D( D. By IV.3.1, lemma 4 we conclude from (8) and (9) that

W(x, T, D) =M(x, T, D) a.e. in R n,

and the relation y (D) = ID1(D) follows by (ii).

Lemma 5. Assume that T is BVM in D. Then

D (u, ID1) = D (u, v) a.e. in D. (10)

Proof. As noted above, D (u, ID1) and D (u, v) exist a.e. in D. To prove (10) we consider an auxiliary function <p(int I) of open intervals in D, defined by the formula

<p (int I) = ID1(int I) - v (int I) , int I ( D. (11)

Note that <p is non-negative by (5). Since D (u, ID1) and D (u, y) exist a.e. in D, it follows from (11) that D (u, <p) also exists a.e. in D and

D (u, <p) = D (u, ID1) - D (u, v) a.e. in D.

Thus (10) will be established if we prove that

D(u, <p) = 0 a.e. in D. Rado and Reichelderfer, Continuous Transformations.

(12) 16


The proof of (12) will be based on the lemma in 111.2.3, which will be applied with 0 = D. Accordingly, Qj will now be a generic notation for a cube whose interior lies in D and which belongs to the subdivision Dpf' and Qj denotes the class of all such cubes Qj (for a fixed positive integer j). According to the lemma in 111.2.3, (12) will be established if we show that

(13 )

To prove (13), let us introduce for each positive integer j the function

gj(X) = L:'[M(x, T, int Qj) - W(x, T, int Qj)] , xE R", (14) QI

where the symbol L:' is used to indicate that the summation is extended only over those terms which are not of the indeterminate form 00 - 00.

By lemma 1 in IV.3.1 and by condition (ii) in IV.3.1, definition 2 we have then

( 15)

Since Tis BVM in D, M(x, T, D) is L-summable. Hence, on denoting by XX) the set of those points xER" where M(x, T, D) = 00, we have

LXoo = o. (16)

Consider now a point x such that

xE exoo • (17)

Then M(x, T, D) < 00, and hence by IV.3.1, lemma 5 there exists a positive number C (x) such that if D is any domain in D satisfying the condition oD < "x), then

M(x, T, D) = W(x, T, D).

On the other hand, clearly 0 Qj < C (x) for all the cubes Qj if j is sufficiently large, say j>io(x). Hence

A1(x, T, int Qj) - W(x, T, int Qj) = 0 if i> jo(x). (18)

From (14), (17) and (18) we conclude that

(19)

Since M(x, T, D) IS L-summable, (15) and (19) imply (see 111.1.1, lemma 38) that

lim f gj(x) d L = O. (20) 1--+ 00

In view of (14), (1), (4) and (11), the relation (20) is equivalent to (13), and the proof is complete.


Lemma 6. Assume that T is BVM in D. Then

D(u, m) = 0 a.e. on D - Db' (21)

where Db is the reduced base-set (see IV.2.l, definition 2) corresponding to the base-function b(u) generated by the multiplicity function M.

Proof· We have D(u, v) =0 a.e. on D-Db by IV.2.2, lemma 9, and D (u, v) = D (u, m) a.e. in D by the preceding lemma 5, and thus (21) follows.

Lemma 7. Assume that Tis BVM in D, and let D be any domain in D. Then

J D (u, m) d L = J D (u, v) d L :;;;;; v (D) :;;;;; m (D). (22) D D

Proof. Note that Tis B V bin D by lemma 3, and hence the measure v is available. The relation (22) follows then by direct application of lemma 5, part (i) of lemma 4, and III.2.4, lemma 1.

Lemma 8. Assume that T is B V M in D, and let G be any set in D such that LTG=O. Then

D(u, m) = 0 a.e. on G. (23)

Proof. Note that Tis B Vb in D by lemma 3 and hence the measure v [given by (4)J is available. We have D(u,v)=O a.e. on G by IV.2.2, lemma 12, and D(u, m) =D(u, v) a.e. in D by lemma 5, and thus (23) follows.

IV.3.3. Absolute continuity. In this section, we assume that a transformation T and an admissible (non-negative) multiplicity function M are given as in IV.3.2. The symbols b, W, m, v, D(u, m), D(u, v) are used in the same sense as in IV .3.2.

Definition. T is said to be absolutely continuous in D with respect to the admissible (non-negative) multiplicity function M (briefly, ACM in D) if Tis BVM in D and

J D(u, m) dL = m(D). (1 ) D

To justify and motivate this definition, let us recall that if T is BVM in D, then

J D (u, m) d L :;;;;; m (D) (2) D

by IV.3.2, lemma 2. Thus the property A C M in D requires that the sign of equality should hold in (2), or equivalently, that m (D) should be expressible as the L-integral over D of its derivative D (u, m).

16*


Lemma 1. Suppose that T is A C M in D. Then the following holds.

(i) T is A C b in D. (ii) v (D) = ~ (D).

Proof. Since T is A C M in D, it is also B V M in D by definition. By IV.3.2, lemma 3, T is therefore BVb in D. To prove (i), we have to show yet that the measure v (B) is absolutely continuous on BOREL sets in D (see IV.2.3). Now since T is BVM in D, by lemma 7 in IV.3.2 we have

JD(u,~)dL=JD(u,v)dL;;;;;v(D) ;;;;;~(D). (3) D D

In view of (1), the sign of equality must hold throughout in (3). This yields (ii), as well as the relation

J D(u, v) dL = v (D) . D

By 111.2.4, theorem 3 it follows that v is absolutely continuous on BOREL sets in D, and the proof is complete.

Lemma 2. Assume that T is B VM in D. Then T is A CM in D if and only if the following two conditions hold simultaneously.

(i) T is A C b in D. (ii) v (D) = ~(D).

Proof. If T is A CM in D, then (i) and (ii) hold by lemma 1. Suppose -conversely that T is B VM in D and (i) and (ii) hold. By IV.2.3, (i) implies that v is absolutely continuous on BOREL sets in D, and hence

J D(u, v) dL = v(D), (4) D

by 111.2.4, theorem 3. Also, since Tis B VM in D, we have D (u, v) =

D (u, ~) a.e. in D by IV.3.2, lemma 5. Hence

J D(u, v) dL = J D(u,~) dL. (5 ) D D

Now (ii), (4) and (5) imply (1), and thus T is A CM in D.

Lemma 3. Suppose that T is A CM in D. Let D be any domain in D. Then Tis ACM in D.

Proal. Note that T is B VM in D, and hence also B VM in D, by IV .3.2, lemma 1. Also, T is A C b in D by lemma 1, and hence T is ACb in D by IV.2.3, lemma 1. Since Tis ACM in D, we have 'jI (D) = ~(D) by lemma 1, and by part (iii) of lemma 4 in IV .3.2 we -conclude that v(D) = ~(D). On applying lemma 2 (with D replaced by D) it follows that T is A CM in D.


Lemma 4. Assume that T is B VM in D. Then T is A CM in D if and only if

J D(u, Wl) dL = Wl(D) (6) D

for every domain D ( D.

Proal. Suppose first that (6) holds for every domain D (D. Then (6) holds, in particular, for D = D, and thus T is A CM in D, by definition. Suppose, conversely, that T is A CM in D. Then T is A CM in every domain D(D by lemma ), and hence (6) holds by the definition of the A CM property.

Theorem. Assume that T is A CM in D. Let H(x) be a finite-valued, real-valued, L-measurable function of x E Rn and let D be any domain in D. Then

J H(Tu) D (u, Wl) dL = J H(x) M(x, T, D) dL, (7) D


Proal. By lemma 1, T is A Cb in D. Hence, by IV.2.4, theorem 3,

J H(Tu) D (u, v) dL = J H(x) W(x, T, D) dL, (8) D

as soon as one of the two integrals involved exists. By lemma 2, v (D) = Wl(D) , and hence v(D)=Wl(D) by part (iii) of lemma 4 in IV.3.2. From part (ii) of that same lemma it follows that

W(x, T, D) = M(x, T, D) a.e. in Rn,

and hence also

H(x) W(x, T, D) = H(x) M(x, T, D) a.e. in Rn.

In VIew of 111.1.1, lemma 25 we have therefore

J H(x) W(x, T, D) dL = J H(x)M(x, T, D) dL, (9)

as soon as one the two integrals involved exists. By IV.3.2, lemma 5, D (u, v) = D (u, Wl) a.e. in D and hence a.e. in D, and therefore also

H(T u) D (u, v) = H(T u) D (u, Wl) a.e. in D.

Consequently

J H(Tu) D(u, v) dL = J H(Tu) D(u, Wl) dL, (10) D D

as soon as one of the two integrals involved exists. Combining these facts one sees that if one of the two integrals in (7) exists, then all the integrals appearing in (8), (9), (10) exist and have the same value, and (7) follows.


IV.3.4. Some relevant examples of admissible multiplicity functions. Let there be given a bounded continuous transformation T:D-+R" as in IV.3.1, and a non-negative multiplicity function M(x, T, D) (see IV.3.1, definition 1).

Theorem 1. Assume that M(x, T, D) satisfies the conditions (i), (ii) , (iii) stated in IV .3.1, definition 2, and the additional condition:

(iv)* For each domain D( D, M(x, T, D) is a lower semi-continuous function of xERH (see 1.1.3, definition 28).

Then M(x, T, D) is admissible, in the sense of IV.3.1, definition 2.

Proal. For fixed D (D, M(x, T, D) is lower semi-continuous with respect to xE R" by assumption, and hence it is L-measurable in R" (see 111.1.1, lemma 20). Inspection of definition 2 in IV.3.1 reveals that there remains only to show that the base-function b (u) generated by M is BOREL measurable in D. To verify this point, take a real number oc: and a positive integer j, and denote by 5<1.j the set of those points uED which satisfy the following condition: There exists a domain D such that

uE D (D, fJD < 1Jj, M(Tu, T, D) > oc:. (1 )

We show presently that 5<1.j is an open set. This is obvious if 5<1.j is empty. So assume that 5<1.j is non-empty. Take a point u E 5<1.i and a domain D for which (1) holds. From the lower semi-continuity of M(x, T, D) it follows [in view of (1)J that there is an open set 0* (R" such that

TuEO*,

x E 0* implies M(x, T, D) > oc:.

(2)

(3)

Since T is continuous, it follows from (1) and (2) that there is an open set 0 such that

uEO(D,

u' E 0 implies T u' E 0*.

Consider any point u'EQ. From (5) and (3) we infer that

M(Tu', T, D) >oc:.

Combining (1), (4) and (6), we see that

u ' E D (D, fJD < iJj, M(Tu', T, D) > oc:.

(4)

(5)

(6)

Thus u' E 5<1.j' Since u' was an arbitrary point of 0, it follows that 0(5rxj • Summing up: if uE5<1.i' then there exists an open set 0 such


that uEO(S"i. Thus Sa; is open. Now let Sa. be the set of those points u ED where the base-function b (u) generated by M satisfies the inequality b(u) > IX. We assert that

00

sa.=n Sa.i. 1=1

Indeed, if u E Sa., then [see IV.3.t (1)J

IX < btu) :::;;'M(T ~t, T, D)

(7)

(8)

for every domain D such that u ED (D. Given any positive integer j, we can select a domain D such that

uED(D, r5D<1jj.

By (8) it follows then that u E Sai. Since u was an arbitrary point of Sa., we conclude that

for every positive integer j, and hence

(9)

Consider next a point

(10)

Then for every positive integer j there is a domain Di such that

Consider any domain D such that u ED ( D. From (11) it follows that

(12)

for j sufficiently large. If j is so chosen that (12) holds, then in view of (12) and condition (ii) in IV.3.t, definition 2 we have

M(T u, T, D) ~ M(T u, T, Di ) > IX.

Thus u ED (D implies M(T u, T, D) > IX. ( 13)

Since the finite values of M are integers, (13) implies [in view of IV .3.t( 1) ] that b (u) > IX. Hence u E Sa.. Since u was an arbitrary point satisfying (10), it follows that 00

.n S(1.i ( Sa.. (14) 1=1

Now (7) follows from (9) and (14). Since Sai is an open set, (7) shows that 5(1. is a BOREL set, and hence b (u) is BOREL measurable.


Remark 1. The preceding theorem will be applied below to the multiplicity functions K(x, T, D), K+(x, T, D), K-(x, T, D), k (x, T, D) discussed in § 11.3. In general, the statements in § II.3 are formulated for the case when D = D. However, since D was any bounded domain in which T is continuous and bounded, the concepts and results discussed in § 11.3 apply for any domain D (D. As regards situations involving local properties, section 11.3.5 contains the facts that should be kept in mind when one operates with subdomains of D.

Theorem 2. The multiplicity functions K(x, T, D), K+(x, T, D), K-(x, T, D) are admissible (see IV.3.t, definition 2).

Proof. From the definitions given in § 11.3 it is evident that each value of K, K+, K- is either a non-negative integer or + 00. The properties (i), (ii), (iii), (iv)* required in theorem 1 have been established for K, K+, K- in remark 2, theorem 2, remark 10, and theorem 3 of section 11.3.2, and thus the admissibility of K, K+, K- follows.

Remark 2. While theorem 2 covers the cases actually needed in the sequel, it is of interest to observe that there are various other multiplicity functions whose admissibility follows from theorem 1. For example, the multiplicity function k (x, T, D) (see 11.3.3) is readily seen to satisfy the conditions (i), (ii), (iii) required in theorem 1 as an immediate consequence of its definition, and it also satisfies condition (iv)* by 11.3.3, lemma 7. Thus k(x, T, D) is admissible. A further example is obtained as follows. Take a point xE Rn and a domain D( D. Denote by k+(x, T, D) the number (possibly infinite) of those essential maximal model continua C for (x, T, D) which satisfy the following condition: if 0 is any open set such that C (0 (D, then there exists a positive indicator domain D* for (x, T, D) (see 11.3.2) such that C(D*(O. One sees readily that k+(x, T, D) satisfies the condi-. tions required in theorem 1, and hence it is admissible. The same holds for the multiplicity function k-(x, T, D) obtained by replacing the term positive indicator domain by the term negative indicator domain in the definition of k+(x, T, D). Inspection of the literature reveals further examples of multiplicity functions that are admissible by virtue of theorem 1 .

. Theorem 3. The crude multiplicity function N(x, T, D) is admissible.

Proof. Note that N(x, T, D) is not lower semi-continuous with respect to x, and hence theorem 1 does not apply. However, N(x, T, D} obviously satisfies conditions (i), (ii), (iii) in IV.3.t, definition 2, and it is L-measurable as a function of x by IV.1.2. There remains to show that the base-function b(u) generated by N(x, T, D) is BOREL

measurable. Consider the set I(T, D) in 11.3.7, definition 2. If uE D - I (T, D), then every neighborhood of u contains some point

§ IVA. Essential bounded variation and absolute continuity. 249

u' =l= u such that T u' = T u. It obviously.follows that if D is any domain such that uED(D, then N(Tu, T, D) = 00. Hence

b(u) = 00 if uED-I(T,D). (15)

On the other hand, if u E I ( T, D), then there exists an open set 0 such that uEO( D and Tu' =l= Tu for u'EO, u' =l= u. Thus N(Tu, T, 0) = 1. Take any domain D such that uED(O. It follows that

N(T u, T, D) ;;;;: N(T u, T,O) = 1,

and thus b(u);;;;:1. On the other hand, if D is any domain such that uED(D, then N(Tu, T, D) ~1 and hence b(u):2:1. Thus we see that

b(u)=1 if uEI(T,D). (16)

Since I(T, D) is a BOREL set by 1I.3.7, lemma 1, it follows from (15) and (16) that b (u) is BOREL measurable.

§ IV.4. Essential bounded variation and absolute continuity.

IV.4.1. Essential bounded variation. Let there be given a bounded continuous transformation T: D-+Rn, where D is a bounded domain in Rn. Throughout the present § IV.4, constant use will be made of the results obtained in § IL3 about the essential multiplicity functions K(x, T, D), K+(x, T, D), K-(x, T, D), k(x, T, D) (see 1I.3.2, 1I.3.3) and the essential sets E(T, D), Ef(T, D) (see 1I.3.6). The reader should tum to the sections just referred to when in doubt about the meaning of terms or symbols.

Definition 1. T is said to be essentially of bounded variation in D (briefly, eB V in D) if the essential multiplicity function K(x, T, D) is L-summable in R".

Observe that K(x, T, D) is a lower semi-continuous function of x (see 1I.3.2, theorem 3). Thus K(x, T, D) is BOREL measurable and hence also L-measurable in R" (see IIL1.t, lemma 20). As noted in IV.3.4, K(x, T, D) is defined for every domain D (D, and is in fact an admissible multiplicity function in the sense of IV.3.t, according to IV.3.4, theorem 2. Comparison of definition 1 above with definition 1 in IV.3.2 reveals that the property eB V is equivalent to bounded variation with respect to the essential multiplicity function K(x, T, D). Hence the theory developed in § IV.3 applies to the present situation. To simplify formulas, we introduce the following notations. The function of domains generated by K, in the sense of IV.3.2, definition 2, will be denoted by sr (D). Thus explicitly

sr(D) = J K(x, T, D) dL, D (D. (1)


The base-function generated by K, in the sense of IV.3.1, definition 3, will be denoted by be (u). Thus

be(u) = gr.l.b. K(Tu, T, D), uE D C D. (2)

The corresponding summatory function [see IV.3.1 (3)J will be denoted by ff~. Thus, if S is a set in D, then

T1~(x, T, S) = L be(u) , uE S n T-l x.

The corresponding measure [see IV.3.2 (4)J will be denoted by Ve

(assuming that Tis eB V in D). Thus, if B is a BOREL set in D, then

ve(B) = J w,,(x, T, B) dL. (4)

The derivative of the function of domains Sf (D) will be denoted by De(u, T). Thus

De(u, T) = D(u, Sf), (5)

where the symbol D (u, st') is used in the sense of 11I.2.4 (2). The reduced base-set (see IV.2.1, definition 2) corresponding to the basefunction be will be denoted by 5S~. Thus

5B~ = {u iuE D, 0 < be(u) < co}. (6)

Since T and D are thought of as fixed in the present section, the preceding notations are adequate.

Lemma 1. 5B~=Ef(T, D). Proof. This is a direct consequence of 11.3.6, lemma 14 and IV.3.1,

lemma 6. Lemma 2. If T is eB V in D, then T is also eB V in every domain

DCD.

Proof. This is a direct consequence of IV.3.2, lemma 1.

Theorem 1. Assume that T is eB V in D. Then the following holds.

(i) The derivative De (u, T) exists a.e. in D and is L-summable there.

(ii) De (u, T) = 0 a.e. on D - Ef (T, D). (iii) If D is any domain in D, then [see (1)J

J De(u, T) dL s;;. st'(D). D

(iv) If G is any set in D such that LTG=o, then De(u, T) =0 a.e. on G.

Proof. (i) and (iii) follow directly from IV.3.2, lemma 2, and (ii) follows from lemma 1 and IV.3.2, lemma 6. Finally (iv) follows directly from IV.3.2, lemma 8.


Definition 2. Let D be any domain in D. Then the essential total variation v,;(T, D) is given by the formula v,;(T, D) = sr(D) [see (1)] if Tis eBV in D, while v,;(T, D) = 00 if T is not eBV in D.

Lemma 3. v,;(T, D) is finite if and only if T is eBV in D.

This is a formal consequence of the definitions involved.

Theorem 2. Given T as above, let T;.:Dr-~Rn, j=1, 2, ... , be a sequence of bounded continuous transformations which converge uniformly to T on compact subsets of D (see 11.3.2, remark 9). Then the following holds.

(i) v,;(T, D) :;:;:lim infv,;(Tj, D i ).

(ii) If lim inf v.: (Ti' Di ) < 00, then T is eB V in D. 1-->00

Proof. If the lim inf in (i) is infinite, the assertion is obvious. So we can assume that the lim inf is finite. Passing on to a subsequence if necessary, we can further assume that v.: (1j, Di ) < 00 for every j. By definition 2 we have then

v.:(1j, Di ) = J K(x, Ti , D i ) dL.

By 11.3.2, theorem 4 we have

K(x, T, D) :;:;: lim inf K(x, 1j, Di ). 1-->00

By the lemma of FATou (see 111.1.1, lemma 30) we conclude that K(x, T, D) is L-summable in Wand

J K(x, T, D) dL:;:;: lim inf J K(x, 1j, DI ) dL. 1-->00

(7)

In view of definition 2, the relation (7) is equivalent to (i). Finally, (ii) is a direct consequence of (i) and lemma 3.

IVA.2. Essential absolute continuity. The terminology of IV.4.1 will be used in the present section. The concept of essential absolute continuity, to be introduced presently, means absolute continuity with respect to the admissible multiplicity function K(x, T, D), in the sense of IV.3.3. The explicit wording is as follows [see IV.4.1 (1), (5)].

Definition. T is termed essentially absolutely continuous in D (briefly, eA C in D) if it is eB V in D and

J De(u, T) dL = J K(x, T, D) dL. (1 ) D

Note that if T is eBV in D, then both integrals appearing in (1) exist by IV.4.1, definition 1 and theorem 1, and thus the preceding definition is meaningful. As regards motivation, note that if T is eB V in D, then

J De(u, T) dL:;:;:J K(x, T, D) dL D


by IV.4.1, theorem 1, and thus the eA C property requires that the sign of equality should hold in the preceding relation.

Theorem 1. If T is eA C in D, then it is also eA C in every domain D (D, and hence

J De(u, T) dL = J K(x, T, D) dL, D (D. D

This is a direct consequence of IV.3.3, lemma 3.


K(x, T, D) < 00,

xE T[E(T, D) - Ef(T, D)]. Then

w. (x, T, D) < K(x, T, D).

(2)

(3 )

(4)

(5 )

Proof. Since xE T E(T, D) by (4), there exists at least one e.m.m.c. for (x,T,D), and thus k(x,T,D)?;,1. By 1I.3.4, theorem 3 it follows that K(x, T, D)?;,1. Hence, in view of (3),

1 ::;;;,K(x, T, D) <00. Note that

0::;;;' We (x, T, D) ::;;;, K(x, T, D)

(6)

(7)

by IV.3.1, lemma 1. Now if vv" (x, T, D) = 0, then (5) is obvious in view of (6). Hence we can assume [see (6) and (7)] that

0< vv,,(x, T, D) < 00.

By the definition of vv" it follows that there exists a finite (non-zero) number of points uET-lx where be(u)=l=O. Let u1 , ... , u'" be these points. Note that if D is any domain in D, then

(8)

Let D1 , ... , Dm be pair-wise disjoint domains in D such that

By (8) we have then m

vv,,(x, T, D) = L We (x, T, D j ). (9) j~l

By 11.3.6, lemma 15 the assumptions (3) and (4) imply the existence of a number 'Y) > 0 such that if the domains D j above satisfy the condition

(10)

§ IV.4:. Essential bounded variation and absolute continuity. 253

then ... L K(x, T, Di) < K(x, T, D). j=1

(11)

Now since the number of the points u1 , ... , u ... is finite, we can clearly comply with (10) in selecting the domains D1 , .•. , D.... Since

w" (x, T, Di) ;s;: K(x, T, Di )

by IV.3.l, lemma 1, the relations (9) and (11) yield

... ... w,,(x, T, D) = L w,,(x, T, D i) ~L K(x, T, Di) < K(x, T, D),

;=1 ;=1 and (5) is proved.

Lemma 2. Let x be a point in RH such that

K(x, T, D) < 00,

xEf T[E(T,D) - Ef(T, D)]. Then

We(x, T, D) = K(x, T, D).

Proof. (14) is obvious if K(x, T, D) =0, since

o ~ w,,(x, T, D) ~K(x, T, D)

by IV.3.l, lemma 1. So we can assume, in view of (12), that

0< K(x, T, D) < 00.

By 11.3.4, theorem 3 it follows that

O<k(x, T,D) < 00.

(12)

(13)

(14)

In other words, the number of essential maximal model continua for (x, T, D) is finite and non-zero. Furthermore, each of these continua reduces to a single point by (13). Let the points u1 , ••• , u'" be the essential maximal model continua for (x, T, D). Let us select pairwise disjoint domains D 1 , ••• , D", in D such that uiEDi' i = 1, ... , m. By 11.3.4, theorem 4 and 11.3.5, remark 3 we have then

and hence

m

K(x, T, D) = L lie (Uj' T)I, j=1

K(x, T, Di ) = lie (U;, T)I, i = 1, ... , m,

m

K(x, T, D) = L K(x, T, D;). ;=1

(15)


By IV.3.t, lemma 5, the domains D; can be so chosen that

W.(x,T,D;) =K(x, T,D,), j=1, ... ,m. (16)

Since the domains D1 , ••• , D", are pair-wise disjoint, clearly

... L w,,(x, T, D,) :;;;w,,(x, T, D). ;=1

Also, by IV.3.t, lemma 1,

W.(x, T, D) :;;;K(x, T, D).

Combining (15), (16), (17), (18), we obtain

K(x, T, D) :;;; W. (x, T, D) ;:;;;; K(x, T, D),

and (14) follows.

(17)

(18)

Lemma 3. Let x be a point in R n such that K(x, T, D) < 00. Then the relation

ly'(x, T, D) = K(x, T, D)

holds if and only if

x Ef T [E(T, D) - Ef (T, D)).

This is a formal consequence of lemmas 1 and 2.

Theorem 2. Assume that T is eA C in D. Then

LT[E(T, D) - Ef(T, D)] = o. (19)

Proof. Let Xoo be the set of those points in Rn where K(x, T, D) = 00.

Since T is also eB V in D, K(x, T, D) is L-summable in Rn, and hence

LXoo = O. (20)

Since T is eA C in D, by IV.3.3, lemma 1 we have 'lie (D) = St(D). By IV.3.2, lemma 4 it follows that

W.(x, T, D) = K(x, T, D) a.e. in RH.

Hence, on setting

X = {xJw.(x, T, D) =f= K(x, T, D)},

we have LX = 0, and hence also [see (20)]

L(XUXoo) =0.

Consider now a point x E Rn such that

(21)

(22)

(23)


Then K(x, T, D) < 00, K(x, T, D) = We (x, T, D),

and hence by lemma 3

xE[ T[E(T, D) - Ef(T, D)]. (24)

Thus (23) implies (24), and consequently

T[E(T, D) - Ef(T, D)] (X U Xoo. (25)

The relation (19) follows now from (25) and (22).

Lemma 4. Assume that Tis eB V in D and satisfies the condition (N) on the essential set E(T, D). Then

LT[E(T, D) - Ef(T, D)] = o. (26)

Proof. Since T satisfies the condition (N) on E(T, D), it also satisfies the condition (N) on the set E(T, D) -EP(T, D). But this latter set is either empty or else it is a union of non-degenerate maximal model continua for T. By IV.1.4, lemma 3 it follows that

LT[E(T, D) - P(T, D)] = o. (27)

Now consider any point

uE EP(T, D) - Ef(T, D). (28)

Thenu itself is an e.m.m.c. for (Tu, T, D) which is not essentially isolated. Hence K(Tu, T, D) = 00. Thus Tu E Xoo , where Xoo denotes again the set of those points x ERn where K(x, T, D) = 00. Since u was any point satisfying (28), it follows that

T[EP(T, D) - Ef(T, D)] (Xoo . (29)

But LXoo = 0 since K(x, T, D) is L-summable. Hence (29) implies that

LT[EP(T, D) - Ef(T, D)] = 0,


Theorem 3. Assume that T is eB V in D. Then T is eA C in D if and only if it satisfies the condition (N) on the essential set E(T, D).

Proof. Assume first that T is eA C in D. In view of theorem 2, T obviously satisfies the condition (N) on the set E( T, D) - Ef (Ti D). Also, T is A C be in D by IV .3.3, lemma 1, and hence by IV .2.3, lemma 2 it satisfies the condition (N) on the reduced base-set \B~ which corresponds to the base-function be (see IV.4.1). Since \B~=Ef(T, D) by IV.4.1, lemma 1, it follows that T satisfies the condition (N) on the


set Ef (T, D) also. Combining these facts, we see that T satisfies the condition (N) on E(T, D). Suppose, conversely, that T satisfies the condition (N) on E(T, D). Make the following observations.

(i) T is eB V in D by assumption, and hence it is also BV be in D by IV.3.2, lemma 3.

(ii) Since Ef(T, D) (E(T, D), T satisfies the condition (N) on Ef(T, D).

(iii) Since Ef (T, D) coincides, by lemma 1 in IVA.1, with the reduced base-set corresponding to be, (i) and (ii) imply (by IV.2.3, lemma 2) that T is A C be in D.

(iv) Since Tis eBVin D and satisfies the condition (N) on E(T, D), we have

LT[E(T, D) - Ef(T, D)] = 0

by lemma 4. (v) On setting again

Xoo = {xIK(x, T, D) = oo},

we have LXoo = 0 since T is eB V in D and hence K(x, T, D) is Lsummable in Rn.

Note now that (iv) and (v) yield

K(x, T, D) < 00, xEf T[E(T, D) - Ef(T, D)] a.e. in Rn.

By lemma 3 it follows that

TV;, (x, T, D) = K(x, T, D) a.e. in RH.

Integration yields [see IVA.1, (1), (4)]

Ye(D) = tr(D). (30)

From (iii) and (30) we conclude by IV.3.3, lemma 2 that T is absolutely continuous in D with respect to the multiplicity function K. By definition, this means that Tis eA C in D, and the theorem is proved.

IV.4.3. The essential generalized Jacobian J e (u, T). The terminology of IVA.1, IV.4.2 will be used in the present section. Let us recall the relations (see II.3.2, theorem 1 and II.3A, as well as remark 1 in IV.3A).

o ;;;;: K+(x, T, D) ;;;;: K(x, T, D),

0;;;;: K-(x, T, D) ;;;;:K(x, T, D),

K(x, T, D) = K+(x, T, D) + K-(x, T, D),

(1 )

(2)

(3)

fle(x, T, D) = K+(x, T, D) - K-(x, T, D) if K(x, T, D) < 00, (4)


where D is any domain in D. Let us also recall that K+(x, T, D) and K-(x, T, D) are admissible multiplicity functions (see IV.3.4, theorem 2). Accordingly, it is legitimate to consider the properties BVK+, BVK-, ACK+, ACK- in the sense of IV.3.2 and IV.3.3.

Lemma 1. Tis eBV in D if and only if it is both BVK+ and BVKin D.

Proof. If T is eB V in D, then K(x, T, D) is L-summable in Rn. In view of (1) and (2) it follows that K+(x, T, D) and K-(x, T, D) are also L-summable in ~, and hence T is both BVK+ and BVK- in D by definition. Conversely, if T is both BVK+ and BVK- in D, then K+(x, T, D) and K-(x, T, D) are L-summable in R". By (3) it follows that K(x, T, D) is also L-summable in Rn, and hence T is eB V in D by definition.

Assuming that Tis eBV in D, we introduce the following notations. Since Tis BVK+ in D by lemma 1, there arises a corresponding function of domains (see IV.3.2, definition 2) which will be denoted by st+ (D). Thus explicitly

st+(D) = J K+(x, T, D) dL, D (D. (5)

The base-function generated by K+ (see IV.3.t, definition 3) will be denoted by b; (u). Thus explicitly

(6)

By IV.3.2, lemma 2, the derivative D(u, st+) exists a.e. in D. We put

(7)

The reduced base-set (see IV.2.t, definition 2) corresponding to b; (u) will be denoted by B~+. Thus

(8)

Similarly, since T is also BVK- in D by lemma 1, we obtain analogous definitions relative to K-, as follows.

st-(D) = J K-(x, T, D) dL, D (D, (9)

b;(u) = gr.l.b. K-(Tu, T, D), uE D (D, (10)

D;(u, T) =D(u, st-), (11)

58;- = {uJu ED, 0 < b; (u) < oo}. (12) Rado and Reichelderfer, Continuous Transformations. 17


For convenient reference, let us recall the analogous definitions relative to K itself (see IV.4.1).

~(D) = f K(x, T, D) dl, D (D,

be(u) = gr.l.b. K(Tu, T,D), uED(D,

De(u, T) =D(u, ~),

IB~ = {ulu ED, 0 < be(u) < oo}.

(13 )

(14)

( 15)

(16)

The set of those points x where K(x, T, D) = 00 will be denoted by Xoo. Thus

Xoo = {xIK(x, T, D) = oo}. (17)

Let us introduce a new function of domains 2fe (D) by the formula

( 18)

In view of the geometrical interpretation obtained for ~+ and ~from (5) and (9), 2fe(D) may be thought of as the essential algebraic area of the image of D under T. Observe that the derivatives D; (u, T), V; (u, T) exist a.e. in D by IV.3.2, lemma 2, and hence in view of (18) the derivative D (u, 2fel of 2fe also exists a.e. in D. We put

J. (u, T) = D (u, 2fel. (19)

The following definition is motivated by the geometrical interpretation of 2fe (V).

Definition. J. (u, T) is termed the essential generalized Jacobian for T (where T is assumed to be eB V in D).

Theorem 1. If Tis eB V in D, then the essential generalized Jacobian J. (u, T) exists a.e. in D and is l-summable in D. Furthermore,

J.(u, T) =D;(u, T) -D;(u, T) a.e. in D. (20)

Proof. Since T is both BVK+ and BVK- in D, by IV.3.2, lemma 2 the derivatives D; (u, T) and D; (u, T) exist a.e. in D and are l-summable there. The existence of J. (u, T) a.e. in D as well as the formula (20) are thus direct consequences of (18) and (19). Finally, (20) yields the l-summability of J. (u, T) in D.

In preparation for the following discussion, some additional comments about the essential set Ef (T, D) will now be made (see I1.3.6). If u is a point of Ef (T, D), then u itself is an e.m.m.c., and it is essentially isolated, by definition. Accordingly, the essential local index ie (u, T)


is defined and its value is an integer different from zero. Let us put

Ef+(T, D) = {uluE Et(T, D), i.(u, T) > o}, (21)

Et-(T, D) = {uluE Ef(T, D), i.(u, T) < a}. (22)

Clearly Ef+(T, D) U Et(T, D) = Ef (T, D),

Ef+(T, D) n Ef-(T, D) = 0.

(23)

(24)

Lemma 2. The following inclusions hold [see (8), (12), (21), (22)].

Et(T, D) ( C \B~+'

Ef+(T, D) ( C \B~-.

Proof. Consider a point

uE Et(T, D).

(25)

(26)

(27)

Then u by itself is an essentially isolated e.m.m.c. for (Tu, T, D). Let 0 be a characteristic neighborhood of u (see 11.3.3, definition 2), and denote by D the component of 0 containing u. Then u is the only e.m.m.c. for (Tu, T, D) in the domain D, and ie (u, T) <0 by (27). By II.3.4(25), applied to the domainD, it follows that K+(Tu, T, D) =0. Since [see (6)]

0;:;;;; b; (u) ;:;;;; K+(T u, T, D),

we conclude that b; (u) = 0, and hence u Et 18:+. As u was an arbitrary point satisfying (27), the relation (25) follows. The proof of (26) is entirely similar.

Lemma 3. Assume that T is eB V in D. Then

D; (u, T) = 0 a.e. on Ef-(T, D),

D;(u, T) = 0 a.e. on Ef+(T, D).

(28)

(29)

Proof. Since the proof is entirely similar in either case, we discuss only (28) in detail. As Tis BVK+ in D by lemma 1, in view of IV.3.2, lemma 6 we have the relation

D;(u, T) = 0 a.e. on D - \B~+,

and (28) follows by (25).

Theorem 2. If T is eB V in D, then

lle(u, T)I = D.(u, T) a.e. in D.

(30)

(31) 17*


Proof. Let D be any domain in D. From (1), (2), (3), (13), (5), (9) it follows that

and hence clearly

o ~D;(u, T) ~De(u, T) a.e. in D,

o ~D-;(u, T) ~De(u, T) a.e. in D,

De (u, T) = D; (1£, T) + D; (1£, T) a.e. in D.

Note that De(u, T) = 0 a.e. on D - Ef(T, D)

by IV.4.1, theorem 1, and

(32)

(33)

(34)

(35)

le(u, T) = D;(u, T) - D;(u, T) a.e. in D (36)

by (20). From (35), (32), (33), (36), (34) we see that

ile(u, T)i =O=De(l£, T) a.e. on D-Ef(T,D). (37)

From (34), (36), (28) we obtain

ile(u, T)i = D;(u, T) = De(u, T) a.e. on Ef-(T, D). (38)

Similarly, from (34), (36), (29) we conclude that

ile(u, T)i = D;(u, T) = De(u, T) a.e. on Et(T, D). (39)

Since by (23)

D = [D - Ef(T, D)] U Ef+(T, D) U Ef-(T, D),

the relations (37), (38), (39) imply (31).

Theorem 3. Assume that T is eB V in D. Then

Jile(u, T)i dL ~J K(x, T, D) dL, (40) D

and the sign of equality holds if and only if T is eA C in D.

Proof. Since ile(u,T)i=De(u,T) a.e. in D by theorem 2, the present theorem is a direct consequence of IV.4.1, theorem 1 and of the definition of essential absolute continuity (see IV.4.2).

Lemma 4. Assume that T is defined and continuous in D and eB V in D. Let D be a domain such that

DCD, LTfrD=O. (41 )

§ IV.4. Essential bounded variation and absolute continuity. 261

Then Pe (x, T, D) = f.1. (x, T, D) a.e. in Rn,

~e(D) = J f.1.(x, T, D) dL.

Proof. By 11.3.4, theorem 2 we have

f.1.e(x, T, D) =f.1.(x, T, D)

at those points xE R" which satisfy the conditions

x EE T fr D, K(x, T, D) < 00.

(42)

(43)

(44)

Now since T is eBV in D, K(x, T, D) is L-summable in R" for every domain D (D, and hence K(x, T, D) < 00 a.e. in R". Thus in view of (41) the conditions (44) hold a.e. in R", and (42) follows. Furthermore, since K(x, T, D) < 00 a.e. in Rn , we have by (4)

f.1..(x, T, D) = K+(x, T, D) - K-(x, T, D) a.e. in R",

and (43) follows by integration in view of (42), (5), (9), (18).

Lemma 5. Assume that T is eB V in D. Let G be any set in D such that LTG=O. Then

Je(u, T) = 0 a.e. on G. (45)

Proof. We have 1J.,(u,T)I=De(u,T) a.e. in D by theorem 2, and De (u, T) = 0 a.e. on G by IV.4.1, theorem 1. Thus (45) follows.

IV.4.4. Transformation formulas for definite integrals. The terminology of IV.4.1, IV.4.2, IV.4.3 will be used in the present section.

Theorem 1. Assume that T is eA C in D. Let H(x) be a finite-valued L-measurable function in R". Then .

J H(Tu) IJe(u, T)I dL = J H(x) K(x, T, D) dL, D


Proof. Since the property eA C is equivalent to the property A CK (see IV.4.2), by the theorem in IV.3.3 we know that

J H(Tu) D(u, ~') dL = J H(x) K(x, T, D) dL, D

as soon as one of the two integrals involved exists. Now D (u, St') =

De(u, T) as a matter of notation [see IV.4.1 (S)]. Since T is also eBV in D by definition, we have De(tt,T) = 1J.,(u,T)1 a.e. in D by IV.4.3, theorem 2, and the present theorem follows.

Our next objective is the derivation of a transformation formula which involves J.,(u, T) instead of IJe(u, T)I. In preparation, we establish the following


Lemma. Tis eA C in D if and only if it is both A CK+ and A CKin D.

Proof. Assume that T is both A CK+ and A CK- in D. Then T is also both BVK+ and BVK- in D, and hence it is also e B V in D by IV.4.3, lemma 1. On setting (see IV.4.3)

I. = ~ (D) - J D (u, ~) d L, D

1.+ = ~+(D) - J D (u, ~+) dL, D

I. - = ~-(D) - J D (u, ~-) dL, D

we have 1.=1.++1.

by IV.4.3 (3), (7), (11), (15), (34), and

I. ;;;;;; 0, I. + ~ 0, 1.-;;;::: °

(1 )

(2)

(3 )

(4)

(5)

by IV.3.2, lemma 2. Now since T is both A CK+ and A CK- byassumption, we have ..1+=0,1.-=0 (see IV.3.3), and hence also 1.=0 by (4). Accordingly, T is eA C in D (see IV.4.2). Suppose, conversely, that T is eAC in D. Then Tis eBV in D, and hence also BVK+ and BVKin D by IV.4.3, lemma 1. Thus we can introduce again 1.,1.+,..1- by means of the formulas (1), (2), (3), and we have again at our disposal the relations (4) and (5). Since T is now eA C in D by assumption, we have 1.=0, and hence also 1.+=0, ..1-=0 in view of (4) and (5). Thus T is both A CK+ and A CK- in D (see IV.3.3).

Theorem 2. Assume that Tis eA C in D. Let H(x) be a finite-valued L-measurable function in Rn. Then (see 11.3.4)

J H(T u) ].(u, T) dL = J H(x) fle(x, T, D) dL, (6) D

as soon as the integral on the left exists. More explicitly: if the product

H(T u) ]. (u, T) (7)

IS L-summable in D, then the product

H(x) fle (x, T, D) (8)

is L-summable in Rn, and (6) holds.

Proof. Assume that the product (7) is L-summable in D. Then the product

H(Tu) /Je(u, T)/


is also L-summable in D (see 111.1.1, lemma 26). By theorem 1 it follows that the product

H(x) K(x, T, D)

is L-summable in RH. Since

o ;:;;; K+(x, T, D) ;:;;; K(x, T, D), 0;:;;; K-(x, T, D) ;:;;; K(x, T, D)

by IV.4.3 (1), (2), it follows further that the products

H(x) K+(x, T, D), H(x) K-(x, T, D)

are also L-summable in R" (note that H, K+, K- are known to be Lmeasurable in Rn). As T is both A CK+ and A CK- in D by the preceding lemma, we conclude from the theorem in IV.3.3 that the products

H(T u) D (u, ~+), H(T u) D (u, ~-)

are L-summable in D, and

J H(Tu) D(u, ~+) dL = J H(x) K+(x, T, D) dL, (9) D

J H(T u) D(u, ~-) dL = J H(x) K-(x, T, D) dL. (10) D

Since T is eB V in D, K(x, T, D) is L-summable in Rn, and hence K(x, T, D) < 00 a.e. in Rn. Accordingly, (see 11.3.4, definition 2) the algebraic multiplicity function fle (x, T, D) is defined a.e. in Rn, and

fle(x, T, D) = K+(x, T, D) - K-(x, T, D) a.e. in Rn, (11)

by 11.3.4, theorem 4. Also

Ie (u, T) = D (u, sr+) - D (u, ~-) a.e. in D, (12)

by IV.4.3 (20), (7), (11). Thus on subtracting (10) from (9) one obtains (6) in view of (11) and (12), and the theorem is proved.

Theorem 3. Assume that T is defined and continuous on ]j and eA C in D. Assume further that

L T Ir D = o. (13)

Let H(x) bea finite-valued L-measurable function in R". Then (see 11.3.4, theorem 2)

J H(T u) Ie(u, T) dL = J H(x) fl (x, T, D) dL, (14) D

as soon as the integral on the left exists.


Proof. By IV.4.3, lemma 4 (applied with D = D) we have

Pe(x, T, D) = p(x, T, D) a.e. in Rff.

Thus the present theorem appears as a direct consequence of theorem 2.

The special case when H(x) == 1 yields the following statement as a direct application of the preceding theorems.

Theorem 4. Assume that T is eA C in D. Then

JIJJu, T)ldL=JK(x, T,D)dL, D

J Je(u, T) dL = J Pe(x, T, D) dL. D

( 1S}

(16}

Furthermore, if T is defined and continuous on Jj and LT/rD=O, then

J fe(u, T) dL = J p(x, T, D) dL. ( 17} D

Note that Je(u, T) is L-summable in D by IV.4.3, theorem 1, and hence theorems 1, 2, 3 are applicable. The formulas (15), (16), (17) may be interpreted as yielding expressions, in terms of the essential generalized Jacobian, for the essential absolute area and the essential algebraic area of the image of D under T.

Remark. If T is eA C in D, then it is also eA C in every domain DeD, by IV.4.2, theorem 1. Accordingly, the preceding theorems 1, 2, 3, 4 remain valid if D is replaced by any domain De D.

IV.4.5. Closure theorems. Let there be given bounded continuous transformations

T:D-+R",

1j:Dj -+R", j=1,2, ... ,

(1 )

(2)

where D, D j are bounded domains in RI!. In the present context, by a closure theorem we mean a statement to the effect that if each one of the transformations Ti possesses the eA C property and the transformations Ti converge to T in an appropriate manner, then T also possesses the e A C property. In other words, a closure theorem states that the class of e A C transformations is closed under certain passages to the lirr.it.

Lemma 1. Given T and 1j as in (1) and (2), assume that the following conditions are satisfied.

(i) The sequence {If} converges to T uniformly on compact subsets of D (see 11.3.2, remark 9).


(ii) Tj is eA e in D j , j = 1, 2, .... (iii) lim inf f lIe (u, 1j) I d L < 00.

1-+00 D;

Then T is eB V in D, and

f IIe(u, T)I dL ~f K(x, T, D) dL ~lim inf f IIe(u, 1j) I dL. (3) D 1-+00 D;

Proof. Condition (ii) and IV.4.3, theorem 3 yield

f K(x, 1j, D j ) dL = f IIe(u, 1j)1 dL. D;

In view of (iii) and IV.4.1, theorem 2 it follows that Tis eB V in D and

f K(x, T, D) dL ~ lim inf f K(x, 1j, D j ) dL = lim}nf f IIe(u, 1j)1 dL. 1---700 1-)-00 Dj

Since (see IV.4.3, theorem 3)

f IIe(u, T)I dL ~f K(x, T, D) dL, D

the relations (3) follow.

Theorem 1. Given T and r;. as in (1) and (2), assume that the following conditions are satisfied.

(i) The sequence {1j} converges to T uniformly on compact subsets of D (see 11.3.2, remark 9).

(ii) Tj is eAe in D j , j=1, 2, .... (iii) T is eB V in D.

(iv) lim inf file (u, 1j) I d L ~ file (u, T) I d L. 1->00 D; D

Then T is eA e in D.

Proof. Since Tis eBV in D, Ie(u, T) is L-summable in D by IV.4.3, theorem 1. Thus, in view of (i), (ii), (iv), the assumptions of lemma 1 are satisfied and hence (3) holds. Combining (3) and (iv) we obtain the inequalities

f IIe(u, T)I dL ~f K(x, T, D) dL ~f II.(u, T)I dL. D D

It follows that

f IIe(u, T)I dL = f K(x, T, D) dL, 1)

and hence T is eA e in D by IV.4.3, theorem 3.

Theorem 2. Given T as in (1), assume that there exists a sequence of domains D j which fill up D from the interior (see 11.3.2, remark 10), such that T is eA e in D j , j = 1, 2, .. .. Assume further that the


sequence of integrals

J 11.(u, T)I dL, j = 1, 2, ... , (4) Dj

is bounded. Then T is eA C in D.

Proof. Since T is eA C in Df , 1. (u, T) exists a.e. in Di and is Lsummable there (see IV.4.3, theorem 1). On setting, for clarity, If= TID" clearly the sequence {Tj } converges to T uniformly on compact subsets of D, in the sense of 11.3.2, remark 9. Thus, in view of the boundedness of the sequence (4), the assumptions of lemma 1 are satisfied, and hence Tis eB V in D. Accordingly, 1.(u, T) is L-summable in D (see IV.4.3, theorem 1), and clearly Je(u, ~.) =1.(u, T) a.e. in Df . Hence

J 11.(u, If) I dL = J 11.(u, T)I dL;;;:J IJe(u, T)I dL. Dj Df D

Thus the assumptions of theorem 1 are obviously satisfied, and hence T is eAC in D.

Lemma 2. Given T as in (1), assume that the following conditions are satisfied.

(i) Tis eBV in D. (ii) T is eA C in every domain D such that

I5 ( D, L fr D = o. (5) Then TiseACinD.

Proof. There exists (see 1.2.3, exercise 9) a sequence of domains Di which fill up D from the interior, such that each D j satisfies the conditions (5). In view of (i),1. (u, T) is L-summable in D by IV.4.3, theorem 1. Hence

J IJe(u, T)I dL;;:;J 11.(u, T)j dL < 00.

D; D

Thus, in view of (ii), the assumptions of theorem 2 are satisfied, and hence T is eA C in D.

Theorem 3. Given T and If as in (1) and (2), assume that the following conditions are satisfied.

(i) The sequence {If} converges to T uniformly on compact subsets of D (see 11.3.2, remark 9).

(ii) T j is eAC in Df , i=1, 2, ....

(iii) T is eB V in D. (iv) For every oriented n-cube Q (D we have

.lim J 1J.(Zt, If)j dL = J j1.(u, T)j dL. I--'>CO Q Q

Then Tis eAC in D.


Proof. In view of lemma 2 it is sufficient to prove that T is e A C in every domain D such that

D(D, LfrD=O. (6)

In view of (iii), J. (u, T) exists a.e. in D and is L-summable there (see IV.4.3, theorem 1). In view of (6) we have

f lfe(u, T)I dL = f 1J.(u, T)j dL. (7) i) D

Given a positive number e, it follows from (6) and (7) that there exists (see 111.1.1, lemma 34) an open set O=O(e) such that

D(O(D,

f IJe(u, T)I dL < f 1J.(u, T)j dL + e. a D

(8)

(9)

In view of (8), we can select a finite system of oriented n-cubes Q1, ... , Qk such that

int Qi n int Qi = 0 for i 9= j, k

1) (.U Qj(O. .=1

From (10), (11) and (9) we find that k

L f IJe(u, T)I dL < f IJe(u, T)I dL + e. .=1 Ql D

Using (11), (8), and condition (iv), we obtain k k

(10)

(11)

(12)

.lim L f 1J.(u, 11)1 dL = L f IJe(u, T)I dL. (13) ' ..... 00.=1 Qi <=1 Qi

In view of condition (i) we have k U Q. (D· for j large .

• =1' 1

Thus (11) implies that k

f 1J.(u, 11)1 dL s: L f 1J.(u, 11)1 dL for j large. (14) D i=1 Qi


~inff IJ.(u, r,·)1 dL < f 1J.(u, T)I dL + e. ' ..... 00 D D

Since e> 0 was arbitrary, it follows that

~inff IJ.(u, T;)I dL ~f 1J.(u, T)j dL. ' ..... 00 D D

(15)


In view of (8) and condition (i) we have DC D j for j large, say j> jo. Set

Dt=D, 1f* = 1fID for j> jo· (16)

Clearly condition (i) implies that the sequence {1f*} converges to TID uniformly on compact subsets of D. From condition (ii) and IV.4.2, theorem 1 it follows that 1j* is eA C in D. From condition (iii) and IV.4.1, lemma 2 it follows that Tis eB V in D. Hence, in view of (15) and (16), the transformations TID and If* satisfy the assumptions of theorem 1, with D, D j replaced by D. Accordingly, Tis eA C in D, and the proof is complete.

Theorem 4. Given T and T j as in (1) and (2), assume that the following conditions are satisfied.

(i) The sequence {Tj } converges to T uniformly on compact sub-sets of D.

(ii) 1f. is eAC in D j , j=1, 2, ....

(iii) T is eB V in D. (iv) For every oriented n-cube QeD we have

lim f IJ~(u, T) - Je(u, If) I dL = o. 1---,.00 Q

Then T is e A C in D.

Proof. By 111.1.1, lemma 41, the present condition (iv) implies condition (iv) in theorem 3. In view of this fact, theorem 4 is an immediate consequence of theorem 3.

IV.4.6. Topological transformations. Let there be given a bounded continuous transformation

(1 )

which is one-to-one in the bounded domain DCR". Thus we assume now that if U 1 , U z are any two distinct points in D, then T u1 =F T Uz. By 11.2.4, theorem 5, the image T D of D is then again a domain. Let us put

D* = TD. (2)

Since T is bounded, D* is a bounded domain. Furthermore (see 11.2.4), T is a homeomorphism from D onto D*, and its inverse

T-l:D*-+Rn

is a homeomorphism from D* onto D. Under the circumstances just described, we shall term T a bounded homeomorphism.

Consider now a domain D such that l5 C D. Then T possesses an index t(T, D) relative to D (see 11.2.4). This index t(l', D) has the value + 1 or - 1, and is independent of the choice of D in the following


sense: if D1 , D2 are any two domains such that 151 (D, D2(D, then l(T, D1) = l(T, D2) (see II.2.4, theorem 5). This fact justifies the following definition.

Delinition 1. The bounded homeomorphism T, given as in (1), is termed even or odd in D according as l (T, D) = + 1 or l (T, D) = -1 for every domain D such that l5 (D.

Lemma 1. Given a bounded homeomorphism T and its inverse T-l as in (1) and 0), T and T-l are both even or both odd (in D and in D* respecti vel y) .

Proal. Take a domain D such that 15 (D. Put LJ = T D. By II.2.4 (6) we have then TD = TD =21, and thus (see II.2.4) LJ is a domain such that ,1(D*. Hence l(T-l, LJ) is defined. Also [see II.2.4 (7)J, TlrD= jr LJ. Thus TID is a homeomorphism from the pair (15, Ir D) onto the pair (,1, Ir LI), and T-11,1 is a homeomorphism from the pair (LT, Ir LJ) onto the pair (15, Ir D). By II.2.4, theorem 4 we conclude that

l(T-l T, D) = t{T, D) l(T-l,LJ).

But T-l T is the identity mapping in D. Hence (see II.2.4, theorem 2)

l (T-l T, D) = 1.

It follows that t{T, D) t{T-l,LJ) = 1. (4)

Now if T is even in D, then l(T,D)=1, and (4) yields t{T-l, LJ)=1. Thus T-l is even in D* (see definition 1). Similarly, if T is odd in D, then l(T,D)=-1, and (4) yields l(T-l,LJ)=-1. Thus T-! is odd in D*.

Lemma 2. Given a bounded homeomorphism T as in (1), let x be a point in Rn and D a domain such 15 CD and xE T D. Then p (x, T, D) = + 1 or p(x, T, D) = -1 according as T is even or odd in D.

Proof. By II.2.4, theorem 1 we have

/-l(x, T, D) = l(T, D) if xE T D.

Thus the present lemma is obvious in view of definition 1.

Lemma 3. Given a bounded homeomorphism T as in (1), the following holds (see II.3.2, II.3.4 for terminology).

(i) If xEfTD, then

K(x, T, D) = 0, K+(x, T, D) = 0, K-(x, T, D) = 0, (5)

/-l" (x, T, D) = 0. (6)


(ii) If X E T D and T is even in D, then

K(x, T, D) = K+(x, T, D) = 1,

K-(x, T, D) = 0,

fte(x, T, D) = 1.

(iii) If xET D and T is odd in D, then

K(x, T, D) = K-(x, T, D) = 1,

K+(x, T, D) = 0, fte(x, T, D) = -1.

(7)

(8)

(9)

(iv) If U is any point in D, then U is an e.m.m.c. for (Tu, T, D) (see 11.3.3, definition 1).

(v) E(T, D) = Ei(T, D) = Ef(T, D) = D (see 11.3.6).

(vi) ie U, T) = + 1 or ie (u, T) = -1 for uE D, according as T is even or odd in D.

Proal. (5) follows directly from 11.3.2, remark 2, and (6) is a direct consequence of (5) and 11.3.4, theorem 4. Thus (i) is verified. The proofs for (ii) and (iii) are entirely similar, and so we discuss only (ii) explicitly. Assume that T is even in D. Consider a point xoE T D. Since T is a homeomorphism, there exists a unique point uoE D such that Tuo=xo. Let D be a domain such that 15(D. We assert that D is an indicator domain for (xo, T, D) if and only if uoE D. Indeed, if uoED, then Xo = TUoET D, and hence ft(xo, T, D) = + 1 by lemma 2. Thus D is an indicator domain for (xo, T, D). Conversely, if D is an indicator domain for (xo' T, D), then ft(xo, T, D) =1=0 and hence xoET D by 11.2.3, theorem 2. Since T is a homeomorphism, it follows that uoE D. From the preceding remarks we draw the following conclusions (it is assumed that T is even in D).

(a) There exist no negative indicator domains for (xo, T, D) and hence (see 11.3.2, remark 2)

(10)

(b) If D is a domain such that 15 (D and uoE D, then D is an indicator domain for (xo, T, D), and ft (xo, T, D) = 1.

(c) If 6(xo, T, D) is any non-empty indicator system for (xo, T, D), then this system consists of a single indicator domain. Indeed, if D1 , D2 were two different indicator domains of the system, then we should have uoED1' uoED2 by the remarks above, in contradiction with the requirement that the domains in an indicator system must be disjoint.


Let D be the unique indicator domain of the system 6 (xo, T, D). Then ,u (xo, T, D) = 1, as noted above, and hence [see n.3.2 (1) J

M[6(xo, T, D)J = 1. (11)

Since (11) holds for every non-empty indicator system 6 (xo, T, D), it follows [see n.3.2 (10)J that

K(xo, T, D) = 1. (12)

Note that (see n.3.2, theorem 1)

K(xo, T, D) = K+(xo, T, D) + K-(xo, T, D). (13)

In view (10), (12), (13), the relations (7) and (8) follow. By (12) and n.3.4, theorem 4 we have

and (9) follows in view of (7) and (8). Thus (ii) is proved. Consider now any point uE D. Since T is a homeomorphism, it is clear that u is an m.m.c. for (Tu, T, D), and in fact u is the only m.m.c. for (Tu, T, D). Thus (iv) and (v) will be proved if we show that u is an e.m.m.c. for (Tu, T, D). Assign any open set 0 such that uEO(D, and select any domain D such that uED, D(O. By lemma 2 we have then

,u(Tu, T, D) = + 1 if T is even in D,

,u(Tu, T, D) = - 1 if T is odd in D.

(14)

(15)

Thus D is an indicator domain for (Tu,T, D) such that uED,D(O. Since 0 was any open set such that uEO(D, it follows that u is an e.m.m.c. for (Tu, T, D). Finally, (14) and (15) show that ie(u, T) = + 1 or ie (u, T) = -1 according as T is even or odd in D (see n.3.4, definition 1), and the proof is complete.

Theorem 1. Given a bounded homeomorphism T as in (1), the following holds.

(i) Tis eBV in D. (ii) The essential generalized Jacobian Je (u, T) exists a.e. in D

and is L-summable in D. (iii) Ie (u, T) ~ 0 a.e. in D if T is even in D, and Ie (u, T) ~ 0 a.e.

in D if T is odd in D. (iv) T is eA C in D if and only if it satisfies the condition (N) in D

(see IV.1.4).

Proof. By lemma 3, K(x, T, D) coincides with the characteristic function of th'e set T D. Since T is bounded, clearly LTD< 00, and


thus K(x, T, D) is L-summable in Rn. Hence Tis eB V in D. (ii) follows now directly from IV.4.3, theorem 1. By that same theorem, we have

Ie(u, T) = D;(u, T) - D;(u, T) a.e. in D. ( 16)

Assume that T is even in D. If D is any domain in D, then

o ~ K-(x, T, D) ~ J(-(x, T, D) = 0

in view of lemma 3. Thus ]C(x, T, D) = 0 for every domain D (D and for every point xER", and hence obviously [see IV.4.3 (9), (11)J

D;(u,T)=O a.e.inD. (17)

Since D;(u, T) is non-negative, it follows from (16) and ('17) that Ie (u, T) :;;;; 0 a.e. in D if T is even in D. An entirely similar argument shows that Ie(u, T) ~O a.e. in D if T is odd in D. Finally, (iv) is a direct consequence of (i), part (v) of lemma 3, and IV.4.2, theorem 3.

Definition 2. A bounded homeomorphism T, given as in (1), is termed measurable in D if for every L-measurable set 5 (D the image set T 5 is L-measurable in R".

Lemma 4. A bounded homeomorphism T, given as in (1), is measurable in D if and only if it satisfies the condition (N) in D (see IV.lo4).

Proof. Suppose first that T satisfies the condition (N) in D. By IV .1.4, corollary 2, for every L-measurable set L (D the image set T L is L-measurable, and thus T is measurable in D. Suppose, conversely, that T is measurable in D. Consider a set 5 (D such that L 5 = O. Deny that LT 5 = O. Observe that T 5 is L-measurable (since T is measurable). By III.lol, lemma 1 there follows the existence of a set 5* (T 5 such that 5* is not L-measurable. The set T-1S* is then a subset of 5, and hence LT-l 5* = O. Thus T-1S* is L-measurable, and yet its image T T-l 5* = 5* is not L-measurable. This contradicts the assumption that T is measurable in D, and the proof is complete.

Remark 1. Definition 2 remains meaningful and lemma 4 remains valid for general continuous transformations in Rn. However, this fact is not needed for our purposes.

Theorem 2. Suppose that the bounded homeomorphism T, given as in (1), is measurable in D. Let H(x) be a finite-valued L-measurable function in R n, and let D be any domain in D. Then

f H(Tu) IJe(u, T)I dL = J H(x) dL, D TD

as soon as one of the two integrals involved exists, and

JIJe(u, T)ldL = LTD. ( 18) D


Proof. Observe that T is eA C in D by lemma 4 and part (iv) of theorem 1. Accordingly, by theorem 1 in IV.4.4, we have (in view of the remark at the end of IV.4.4)

J H(Tu) lIe (u, T)I dL = J H(x) K(x, T, D) dL, D

as soon as one of the two integrals involved exists. Since the transformation TID is one-to-one, we can apply lemma 3 with D replaced by D. Hence, by that lemma, K(x, T, D) = 0 if x~ T D and K(x, T, D) = 1 if xET D. Thus

J H(x) K(x, T, D) dL = J H(x) dL, TD

as soon as one of the two integrals involved exists, and the first part of the theorem follows. On choosing H(x) == 1, one obtains the formula (18).

Definition 3. A bounded homeomorphism T, given as in (1), is termed bi-measumble in D if it is measurable in D and its inverse [given as in (3)J is measurable in the image domain T D.

Theorem 3. Assume that the bounded homeomorphism T, given as in (1), is bi-measurable in D. Let h(x) be a finite-valued L-measurable function defined a.e. in the image domain T D, and let D be any domain in D. Then the following holds.

(i) h(Tu) is defined a.e. in D. (ii) Jh(Tu)IIe(u,T)ldL=Jh(x)dL, as soon as one of the two

D TD

integrals involved exists.

Proof. Let 5* be the set of those points xE T D where h (x) is not defined. Then L 5* = 0 by assumption, and hence LT-l 5* = 0, since T-l is measurable in TD (see lemma 4). As h(Tu) is defined for uED-T-lS*, (i) follows. Introduce now an auxiliary function H(x) in R" as follows: H(x) = 0 if x~TD - 5*, and H(x) = h(x) if xET D - 5*. Then H(x) is clearly finite-valued and L-measurable in R", and hence, by theorem 2,

J H(Tu) IIe(u, T)I dL = J H(x) dL, D TD

as soon as one of the two integrals involved exists. Since obviously h(x)=H(x) a.e. in TD and h(Tu)=H(Tu) a.e. in D, the theorem follows.

Theorem 4. Given a bounded homeomorphism T as in (1), let

( 19) Rado and Reichelderfer 1 Continuous Transformations. 18


be a bounded continuous transformation from the image domain T D into R". Assume that T is bi-measurable in D and T* is e A C in T D. Then the following holds.

(i) T*T is eA C in D. (ii) Ie (u, T* T) = Ie (u, T) Ie (T u, T*) a.e. in D.

Proal 01 (i). Clearly, the transformations T*T and T* are topologically similar in the sense of II.3.8. Accordingly, by 11.3.8, lemma 4 we have the relations

l\(x, T*T, D) = K(x, T*, T D), x ERn,

T E(T*T, D) = E(T*, T D).

(20)

(21)

Now since T* is eAC in TD and hence K(x,T*,TD) is L-summable in R n, (20) shows that K(x, T*T, D) is L-summable in Rn. Thus T*T is eBV in D, and hence (in view of IV.4.2, theorem 3) (i) will be proved if we show that T*T satisfies the condition (N) on the essential set E(T*T, D). So take any set 5 such that

S(E(T*T,D), LS=o. (22)

In view of (21) we have then

T 5 ( E(T*, T D). (23)

Since T is measurable in D, the second relation in (22) implies that

LT 5 = O. (24)

By assumption, T* is eAC in TD, and hence (see IV.4.2, theorem 3) T* satisfies the condition (N) on the essential set E(T*, T D). Accordingly, (23) and (24) imply that

LT* T 5 = O. (25)

Thus (22) implies (25). In other words, T*T satisfies the condition (N) on the essential set E(T*T, D), and (i) is proved.

Proof 01 (ii). There arise two cases, according as T is even or odd in D. Since the argument is the same in either case, we discuss only the case when T is odd in D. By theorem 1 we have then

Ie(u, T);;:;;O a.e. in D. (26)

Furthermore, if D is any domain such that 15 (D, then (see definition 1)

l(T, D) = -1. (27)


Let x be any point in R". We proceed to verify the relations

K+(x, T*T, D) = K-(x, T*, T D),

K-(x, T*T, D) = K+(x, T*, T D).

Let D be an indicator domain for (x, T*T, D). Then

(28)

(29)

15 (D, x (J. T*T fr D, flex, T*T, D) =t= o. (29a)

By lemma 1 and lemma 2 in 11.3.8 it follows that [see (27)J

TD(TD, x(J.T*frTD, fl(x,T*,TD)=-fl(x,T*T,D). (30)

From (29a) and (30) we conclude that if D is a positive indicator domain for (x, T* T, D), then T D is a negative indicator domain for (x,T*,TD). Accordingly, if D1 , •.. ,D", is a positive indicator system for (x, T*T, D), then T D1 , .•• , T D", is a negative indicator system for (x, T*, T D) (see 11.3.2), and

m m

L flex, T*T, Dj ) = - L fl(x, T*, T Dj ).

i-I i-I

By the definition of K-(x, T*, T D) (see 11.3.2), the summation on the right does not exceed K-(x, T*, T D). Hence

'" Lfl(X,T*T,Dj ) s;;.K-(x, T*, T D). i-I

Since D1 , ••• , D", was an arbitrary positive indicator system for (x, T*T, D), there follows the inequality

K+(x, T*T, D) s;;.K-(x, T*, T D). (31 )

Starting with negative indicator systems for (x, T*T, D), a similar argument yields the inequality

K-(x, T*T, D) s;;. K+(x, T*, T D). (32)

Note that in deriving (31) and (32) we used only the fact that T*T and T* are topologically similar. Since this relationship is symmetric, an entirely similar argument shows that

K+(x, T*, T D) s;;.K-(x, T*T, D),

K-(x, T*, T D) s;;. K+(x, T*T, D),

and (28) and (29) follow in view of (31) and (32). Observe now that if D is any domain in D, then TID is a homeomorphism from D onto

18*


T D and T*I T D is defined and continuous. Accordingly, in the argument leading to (28) and (29) we can replace D by D, and hence

K+(x, T*T, D) = K-(x, T*, T D),

K-(x, T*T, D) = K+(x, T*, T D),

(33)

(34)

for every domain DCD. Using the remark at the end of IV.4.4, let us apply now IV.4.4, theorem 4 to T*T in D. In view of IV.4.4 (11), applied to T*T in D, there follows the formula

J Ie(u, T*T) d L = J K+(x, T*T, D) dL - J K-(x, T*T, D) dL. (35) D

Applying the same references to T* in T D, we obtain

J fe(x, T*) dL = J K+(x, T*, T D) dL - J K-(x, T*, T D) dL. (36) TD

The relations (33) to (36) yield

J fe(u, T*T) dL = - J Ie (x, T*) dL. (37) D 1'D

Application of theorem 3, with h(x) = fe(x,T*), yields

J Ie(Tu, T*) IIe(u, T)I dL = J fe(x, T*) dL. D TD

In view of (26), this last formula is equivalent to

J fe(Tu, T*) fe(u, T) dL = - J fe(x, T*) dL. (38) D TD

Combining (37) and (38) we obtain

J Je(u, T*T) dL = J fe(T1t, T*) fe(u, T) dL. (39) D D

Since D was any domain in D, (39) implies (ii) by 111.1.1, lemma 44, and the theorem is proved.

Theorem 5. Given a bounded homeomorphism T as in (1), let T* : T D -7- Rn be a bounded homeomorphism defined in the image domain T D. Assume that T is bi-measurable in D and T* is bimeasurable in T D. Then the following holds.

(i) T* T is a bounded bi-measurable homeomorphism. (ii) fe(u, T*T) =fe(u, T) Ie (Tu, T*) a.e. in D.

Prool. (i) is a direct consequence of the definition of the terms involved. Noting that T* is eA C in TD by theorem 1 and lemma 4, (ii) appears as a direct consequence of theorem 4.

§ IV.S. Bounded variation and absolute continuity in the BANACH sense. 277

Remark 2. Assume that T, given as in (1), is the identity mapping in.D (that is, T u = u for every point uE D). Clearly, T is then a bounded bi-measurable homeomorphism in D. Take any domain Dsuch that 15 (D. Then TID is the identity mapping in 15, and hence (see 11.2.4, theorem 2) t(T, D) = 1. Thus T is even in D (see definition 1). Consider now any domain D(D. Then TID is the identity mapping in D. We can now apply lemma 3 with D replaced by D. Noting that TID is even in D and T D = D, we obtain from lemma 3 the following information. K+(x,T,D)=1 or 0 according as xED or xEfD, and K-(x, T, D) = o. Accordingly [see IV.4.3 (5), (9)J ~+(D) =·LD and ~-(D) =0, and hence 2r.(D) = ~+(D) - ~-(D) = LD [see IV.4.3 (18)]. Since this holds for every domain D(D, it follows that J. (u, T) = D(u,2re) exists and J.(u,T) =1 at every point uED [see IV.4.3 (19)].

Theorem 6. Given a bounded homeomorphism T and its inverse as in (1), (2), (3), assume that Tis bi-measurable in D. Then

J. (u, T) J. (T u, T-I) = 1 a.e. in D. (40)

Proof. In view of definition 3, T-I is clearly bi-measurable in the image domain T D. Thus we can apply theorem 5 with T* = T-l, obtaining

J. (u, T) J. (T u, T-I) = J. (u, T-I T) a.e. in D.

Since T-I T is the identity mapping, J.(u, T-I T) = 1 in D by remark 2, and (40) follows.

The theorems 5 and 6 correspond to familiar theorems in classical Analysis, where of course one assumes that the transformations involved are continuously differentiable and one uses the ordinary Jacobian. An important special case of theorem 4 arises if the homeomorphism T stands for a change of the coordinate system. Theorem 4 suggests the question whether a similar product theorem holds if one drops the assumption that T is a homeomorphism. While several remarkable special results are available in the literature, ultimate clarification of this point seems to be remote as yet.

§ IV.5. Bounded variation and absolute continuity in the BANACH sense.

IV.S.l. Bounded variation. Let there be given a bounded continuous transformation T: D-+Rn, where D is a bounded domain in RH. If I is an n-interval such that int I (D, then the set T(int I) is bounded and L-measurable (see IV.1.l, theorem 4). Accordingly, we can define a function G (int I) of open intervals in D by the formula

G (int I) = L T(int I) , int I ( D. (1)


Since G depends upon T, a more descriptive notation like G (int I, T) may be used if T is not clearly identified by the context. An equivalent formula for G is obtained as follows. If F is a set in Rn, then let us denote again by c (x, F) the characteristic function of F. That is,

{ 1 if xE F, }

c(x,F) = 0 if xE CF.

(2)

Then (1) may be re-written in the form

G(int I) = J c(x, T intI) dL. (3 )

Since T is bounded by assumption, we have LT D< 00. Thus clearly

G (intI) ~ LTD < 00, int I ( D. (4)

From (4) we see that G is bounded in D.

Definition 1. T is said to be of bounded variation in D in the BANACH sense (briefly, BVB in D) if G(int I) is of bounded variation in D (see IIL2.I, definition 5).

We shall show presently that this concept of bounded variation is equivalent to that of bounded variation with respect to the special base-function b (u) == 1. In anticipation of this result, we denote the base-function b (u) == 1 by bB (u). Thus

bB (1£) == 1, 1£ ED. (5)

The corresponding summatory function W [see IV.2.t (2)J will be denoted by WB . Clearly, if 5 is any set in D, then

W B (x, T, 5) = N(x, T, 5), (6)

where N is the crude multiplicity function (see IV.lo2). By IV.2.2, definition 1, we have therefore the following statement.

Lemma 1. T is BV bB in D if and only if the crude multiplicity function N(x, T, D) is L-summable in R".

Assuming that T is BV bB in D, we denote by VB the measure V

induced by T and bB (see IV.2.2, definition 2). If B is any BOREL set in D, in view of (6) we have then

VB (B) = J N(x, T, B) dL. (7)

To simplify notations, we put (see IV.2.2, definition 2)

DB(u, T) = D(u, VB). (8)

§ IV.s. Bounded variation and absolute continuity in the BANACH sense. 279

If ° is any open set in D, we put [see (1) and 111.3.1, definition 4]

Vs (T, 0) = V(G, 0). (9)

Definition 2. VB (T, 0) is termed the total variation of T, in the BANACH sense, on the open set ° (D [observe that VB (T, 0) may be equal to + 00]'

Lemma 2. Let ° be an open set in D. Then VB (T, 0) is finite if and only if the crude multiplicity function N(x, T, 0) is L-summable in R". Furthermore

Vs(T,O)=JN(x,T,O)dL if Vs(T,O) <00. (10)

Proof. Take an oriented n-cube Q such that D (Q. Using the subdivision Dp; of Q introduced in 1.2.3, definition ), for each positive integer j let 0: denote the class of those cubes Q of Dpj for which int Q(O. Then 0; is clearly a system 0(0) in the sense of 111.2.1, definition 1. In terms of the terminology in 111.2.1 we have then [in view of (9)J

L [G, OJ] :s; VB (T, 0). (11 )

Assume first that VB (T, 0) < 00. (12)

Introduce, for each positive integer j, the auxiliary function [see (2)]

fj(X)=Lc(x,TintQ), QEOi', xER". (13)

By IV.1.2 (10), (11) we have then the relations

o :S;fj(x) :S;N(x, T,O),

lim fj(x) = N(x, T, 0). 1-+ 00

Integration of (13) yields, in view of (3), (11), and (12),

J fj(x) dL :s; VB (T, 0) < 00.

(14)

( 15)

(16)

By the lemma of FATOU (see 111.1.1, lemma 30) we conclude from (14), (15), (16) that N(x, T, 0) is L-summable and

J N(x, T, 0) dL ;;;;;;VB(T, 0). ( 17)

Suppose, conversely, that N(x, T, 0) is L-summable in R". Take any system 0(0) (see III.2.1, definition 1). Clearly

N(x, T, 0) ~ L c(x, T intI), IE 0(0).


Integration yields, in view of (3), the inequality

L [G, 0"(0)] ~J N(x, T, 0) dL < 00.

Since 0"(0) was arbitrary, it follows that

VB (T, 0) :;;;; J N(x, T, 0) dL < 00. (is)

Thus VB(T, 0) is finite. Finally, (10) follows by (17) and (is).

Theorem 1. T is BVB in D if and only if the crude multiplicity function N(x, T, D) is L-summable in Rn.

Proof. In view of 111.2.1, lemma 1, Tis BVB in D if and only if VB (T, D) < 00. Thus the theorem follows from lemma 2 on choosing there O=D.

Theorem 2. T is BVB in D if and only if it is BVbB in D.

This is a formal consequence of lemma 1 and theorem 1.

Theorem 3. Assume that T is BVB in D. Then the derivative DB (u, T) [see (8)] exists a.e. in D and is L-summable there. Furthermore, if 0 is any open set in D, then

J DB(u, T) dL ~ J N(x, T, 0) dL < 00. (19) o

Proof. Since T is BVbB in D by theorem 2, the present theorem is a direct consequence of 111.2.4, lemma 1, in view of (S) and (7).

Assume that T is BVB in D. A remarkable feature of the BANACH

theory is the fact that the derivative DB (u, T) can be replaced by the derivative D(u,G) of G [see (1)]. Let us note that G is generally not sub-additive, and hence theorem 5 in 111.2.3 cannot be applied directly to G. The following lemma will enable us to apply that existence theorem in an indirect manner.

Lemma 3. If 51' ... ' 5", are pair-wise disjoint subsets of a set SeD, then [see (2)J

m

L [N(x, T, Sk) - c (x, T Sk)] ~ N(x, T, 5) - c(x, T 5). (20) k~l

Proof. (20) is obvious if N (x, T, 5) = 00. So assume that N(x, T, 5) < co. It is convenient to consider two cases.

Case 1. C(x, TSk ) =0, k=1, ... ,m. Then xEfTSk , and hence N(x, TSk) =0, k=1, ... ,m. Thus the summation on the left in (20) has the value zero, while clearly

N(x, T, 5) - c(x, T 5) ::2:0.

Thus (20) holds in this case.

§ IV.5. Bounded variation and absolute continuity in the BANACH sense. 281

Case 2. c(x, T 5 k) = 1 for some k. Then

As 51' ... , 5m are pair-wise disjoint subsets of 5, clearly

m L N(x, T, 5 k ) -;;;;'N(x, T, 5).

k=1

As c(x, T 5) :;;;;:1, we have

N(x, T, 5) - '1-;;;;' N(x, T, 5) - c(x, T 5),

and (20) follows now by (21), (22), (23).

(22)

(23)

Theorem 4. Assume that T is BVB in D. Then the function G of open intervals [see (1)] possesses a derivative D(u, G) a.e. in D, and [see (8)]

D(u, G) = DB(u, T) a.e. in D. (24)

Proof. Note first that DB (u, T) exists a.e. in D by theorem 3. Introduce now an auxiliary function cp of open intervals in D by the formula

cp(int I) = 'liB (int I) - G(int I), int I CD. (25)

In view of (3) and (7) we have explicitly

cp (int I) = J [N(x, T, int I) - c (x, Tint I)] d L. (26)

Thus clearly cp is non-negative. Furthermore, from the representation (26) we infer immediately, by means of lemma 3, that cp is sub-additive in the sense of 111.2.3, definition 3. By 111.2.3, theorem 5 we conclude that the derivative D (u, cp) of cp exists a.e. in D. Since the derivative of 'liB also exists a.e. in D, (25) yields the existence of the derivative D(u, G) a.e. in D, as well as the relation [see (8)]

D(u,cp) = DB(u, T) - D(u, G) a.e. in D.

Thus the proof will be complete if we show that

D(u, cp) = 0 a.e. in D. (27)

For this purpose, we shall use the lemma in 111.2.3. Taking 0 = D in that lemma, it follows that (27) is established if we prove that

,lim L;cp(int Qi) = 0, ' ..... 00 Qf

(28)


where Qj is used in the same sense as in the lemma in 111.2.3. For each positive integer j we introduce an auxiliary function gj(x), xERn ,

by the formula

gj (x) = L [N(x, T, int Qj) - C (x, Tint Qj) J. (29) Qj

By IV .1.2 (8), (9), (10), (11) we have then

0;;;;; gj(x) ;;;;; N(x, T, D),

lim gj(x) = 0 if N(x, T, D) < 00. 1-,>00

(30)

(31 )

Now since T is BVB in D, by theorem 1 we know that N(x, T, D) is L-summable in Rn. Hence N(x, T, D) < 00 a.e. in R" (see III.1.t, lemma 31). Thus (31) holds a.e. in R", and in view of (30) we conclude from III. 1. t , lemma 38 that

limfgj(x)dL=O. (32) 1 ..... 00

By (26) and (29) the relation (32) is equivalent to (28), and the proof is complete.

In the sequel, it will be a matter of importance to have information about points u ED where DB (u, T) vanishes. Such information is contained in the following statements.

Theorem 5. Assume that Tis B VB in D, and let S be a set in D such that

LT S = o. Then

DB(u, T) = 0 a.e. on S.

(33 )

(34)

Proof. Note that T is B V bB in D by theorem 2. Accordingly, by IV.2.2, lemma 12 the relation (33) implies that D(u, l'B)=O a.e. on S, and thus (34) follows in view of (8).

Theorem 6. Assume that T is B VB in D, and consider the set I (T, D) of those points uED which possess a neighborhood clear of relatives (see II.3.7, definitions 1 and 2). Then

DB (u, T) = 0 a.e. on D - I (T, D). (35)

Proof. Denote by Xro the set of those points x ERn where N(x, T, D) = 00. Since N(x, T, D) is L-summable in R" by theorem 1, from III. lot, lemma 31 we obtain the relation

LXoo= o. (36)

If u ED - I (T, D), then by definition every open set 0 containing u must contain a relative of u (distinct from tt). There follows the existence


of an infinite sequence of distinct points ukED, k = 1,2, ... , such that uk=l=u and TUk=Tu. Thus N(Tu, T, D)= 00 and hence TuEXooSince u was an arbitrary point of D-I(T, D), we conclude that

T[D - I(T, D)J (Xco , (37)

and (35) follows from theorem 5 in view of (37) and (36).

IV.5.2. Absolute continuity. The terminology of IV.5.t will be used in this section.

Definition. T is said to be absolutely continuous in D in the BANACH

sense (briefly, A C B in D) if the function G (int I) is absolutely continuous in D [see IV.5.t (1) and lIL2.2).

Theorem 1. If T is ACB in D, then it is also BVB in D.

Proof. Noting that G is bounded [see IV.5.t ~4)], we see that the assertion follows directly from lIL2.2, lemma 3.

Theorem 2. Tis ACB in D if and only if it is ACbB in D.

Proof. Assume first that Tis ACB in D. By theorem 1 and IV.5.t, theorem 2, T is then also BVbB in D. There remains to show (see IV.2.3) that the measure VB induced by T and bB is absolutely continuous in D with respect to L-measure. So let us consider any BOREL set B (D such that

LB=O. (1 )

We have to verify that vB(B) = O. (2)

Assign s > O. By lIL2.2, lemma 2 there exists an 1') (s) > 0 such that if 0 is any open set satisfying the conditions

OeD, LO<11(S)

then [see IV.5.t (9)J

VB(T, 0) < s. (4)

On the other hand, (1) implies (see lIL1.t, lemma 3) the existence of an open set 0 0 such that

B(Oo(D, LOo<17(s). (5)

From (3), (4), (5) it follows that

VB (T, 0 0) < c. (6)

284 Part IV. Bounded variation and absolute continuity in RfJ.

In view of IV.S.l (10) the relation (6) implies that

f N(x, T,Oo) dL < e,

and by (5) and IV.S.l (7) we conclude that

PB(B) ;:;;;;:PB(OO) = f N{x, T, 00) dL < e.

Since e was arbitrary, (2) follows. Assume, conversely, that T is A CbB in D. Then, by definition (see IV.2.3), the measure PB is available and absolutely continuous in D with respect to L-measure. Hence, if e>O is assigned, then there exists an 1](e}>O such that

'VB (B) < e if L B < 1] (e) , (7)

where B is a BOREL set in D. Take now (see 111.2.1 for notations) any system a (D) such that

L[a(D)] <1]{e). (8) On setting

BO= Uintl, lEa(D}, (9)

clearly (8) implies that LBo<1](e}. From (7) it follows then that 'VB (Bo) < e. Hence [see (9) and IV.S.l (3), (7)]

L [G, a (D)] = L f c(x, Tint l) dL :;;;, f N(x, T, Bo) dL = PB(Bo} < e. [Ea(D)

In summary, (8) implies that

L [G, a(D)] < e.

Thus G(int l} is absoltitely continuous in D (see 111.2.2), and hence T is A CB in D by definition.

From the preceding theorem and from theorem 2 in IV.S.l it appears that the BANACH theory of bounded variation and absolute continuity corresponds to the special case of the general theory in § IV.2 when one selects the base-function bB(u) =1. This relationship will be exploited in proving the following statements.

Theorem 3. Assume that Tis BVB in D. Then T is ACB in D if and only if it satisfies the condition (N) in D (see IV.l.4).

Proof. In view of theorem 2 and IV.S.l, theorem 2, the terms BVB and ACB can be replaced by the terms BVbB and ACbB respectively. Assuming that T is BV bB in D, T is A C bB in D if and only if it satisfies the condition (N) on the reduced base-set where 0 < bB (u) < 00

(see IV.2.3, lemma 2). Since bB (u) = 1 in D, clearly the reduced baseset coincides with D, and the theorem follows.

§ IV.5. Bounded variation and absolute continuity in the BANACH sense. 285

Theorem 4. Assume that T is A CB in D. If H(x) is an L-measurable, finite-valued function in R" and L is any L-measurable set in D, then

f H(Tu) Ds(u, T) dL = f H(x) N(x, T, L) dL, L


Proof. Note that T is A Cbs in D by theorem 2, and that Ds (u, T) and N(x, T, L) agree with the derivative D (u, 'VB) and the summa tory function W corresponding to bB respectively [see IV.S.1 (8), (6)]. Thus the present theorem is a direct consequence of theorem 3 in IV.2.4.

Theorem 5. Assume that T is A CB in D, and let L be any L-measurable subset of D. Then N(x, T, L) is L-summable in R", and

f N(x, T, L) dL = f Ds(H, T) dL. L

Proof. Note that Tis BVB in D by theorem 1, and hence N(x, T, D) IS L-summable in Rn by IV.S.1, theorem 1. Since

N(x, T, L) :s;: N(x, T, D),

it follows that N(x, T, L) is also L-summable in R". The present theorem appears thus a consequence of theorem 4, corresponding to the special case H(x) == 1.

Theorem 6. Assume that T is BVB in D. Then

f DB(u, T) dL ~f N(x, T, D) dL < 00,

D

and the sign of equality holds if and only if T is A CB in D.

( 10)

Proof. Note that (10) holds by IV.S.1, theorem 3. Note also that by IV.S.1 (7), (8) the inequality (10) is equivalent to the inequality

f D(u, 'VB) dL ~'VB(D). (11) D

By III.2.4, theorem 3 the sign of equality holds in (11) if and only if 'VB is absolutely continuous in D with respect to L-measure, and hence (see the definition in IV.2.3) if and only if T is A C bB in D. Since (by theorem 2) this latter condition is equivalent to T being A CB in D, the proof is complete.

IV.S.3. The generalized Jacobian J s (u, T). Comparison of the multiplicity functions K(x, T, D) and N(x, T, D). The terminology of IV.S.1 and IV.S.2 will be used in this section. Assume that Tis BVB in D. Then the derivative Ds (u, T) exists a.e. in D (see IV.S.1, theorem 3). Making use of the local index iB (u, T) discussed in II.3.7, a generalized


I acobian IB (u, T) can be introduced in the BANACH theory by means of the formula

IB (u, T) = iB (u, T) DB (u, T). (1 )

In view of 11.3.7, lemma 2 and 111.2.3, theorem 3 it is clear that this generalized Jacobian is L-measurable in D. The question of the Lsummability of IB (u, T) gives rise to the following comments. While the property eBV implies the L-summability of the generalized Jacobian Ie(u, T) (see IV.4.3, theorem 1), the property BVB generally does not imply the L-summability of the generalized Jacobian IB (u, T). If T is BVB in D, then DB(u, T) is L-summable in D (see IV.5.l, theorem 3). But since the local index iB (u, T) may assume arbitrarily large integral values, the L-summability of DB (u, T) does not necessarily imply the L-summability of IB (u, T). It is interesting to observe that the situation in this respect depends essentially upon the dimension n of the Euclidean space R" in which one operates. We shall see later on that in the cases n = 1, 2 the discrepancy referred to above does not arise. On the other hand, in the general case the fruitfulness of the BANACH theory seems to be limited to those particular situations where the generalized Jacobian IB (u, T) may be shown to be L-summable by virtue of further assumptions. These observations may serve to motivate the following definitions.

Definition 1. T is termed sBVB (strongly of bounded variation in the BANACH sense) in D if it is BVB in D and IB (u, T) is L-summable in D.

Definition 2. T is termed sA CB (strongly absolutely continuous in the BANACH sense) in D if it is A CB in D and IB (1£, T) is L-summable in D.

Our main objective in this section is to show that the sA CB transformations are included in the class of eA C transformations. In preparation, various preliminary facts will be first established. Essential use will be made of the set I(T, D) of those points 1£ED which have some neighborhood clear of relatives (see 11.3.7). For each integer s, we denote by Is (T, D) the set of those points 1£E I (T, D) where iB (u, T) = s. To simplify formulas, we introduce the notations

I=I(T,D),Is=Is(T,D),

which are adequate as long as T and D are kept fixed. Clearly

I = U IS' s = 0, ± 1, ± 2, ... ,

(2)

(3 )

(4)


Let us introduce further auxiliary sets by the formulas

1_ = U L s ' s = 1, 2, ... ,

1* = U Is, lsi =F 1.

(5 )

(6)

(7)

The set I is a BOREL set by II.3.7, lemma 1. As we noted in the proof of lemma 2 in 11.3.7, the sets Is are also BOREL sets. It follows that the sets 1+, L, 1* are also BOREL sets.

Lemma 1. Let x be a point in R" such that

N(x, T, D) < 00. (8)

Then the following relations hold (see 11.3.2).

00

L s N(x, T, Is) = K+(x, T, D), (9) s=1

00

L s N(x, T, I_s) = K-(x, T, D). ( 10) s=1

Proof. Assume first that N(x, T, D) = O. Then x Et T D, and thus the summations in (9) and (10) have the value zero, and K+(x, T, D) =0, K-(x, T, D) =0 by 11.3.2, remark 2. Thus (9) and (10) trivially hold in this case. So we can assume that

0< N(x, T, D) < =.

Then the set T-1 x is non-empty and finite. Let u1 , ... , Um be the points of T-1 x. Each of these points is then clearly in the set I [see (2) J. It is convenient to consider now two cases.

Case 1. u j E 10 , j = 1, ... , In [see (2)]. Then clearly the summations in (9) and (10) have the value zero. Inspection of the definition of the local index iB (u, T) reveals (see 11.3.7) that there exist now no essential maximal model continua for (x, T, D), and hence (see 11.3.4, theorem 3) we have K(x,T,D) =0 and thus a fortiori K+(x,T,D) =0, K-(x,T,D) =0. Thus (9) and (10) trivially hold in this case also.

Case 2. Uj E 1+ U L for some j. By the definition of iB (u, T) and Is we have clearly

00

L s N(x, T, I.) = LiB (Uj, T), Uj E 1+ , (11) s=1

00

L s N(x, T, I_s) = - L iB (Uj, T), Uj E L. ( 12) s=1


Now since uiEI+ U L if and only if Uj by itself is an e.m.m.c. for (x,T,D), we have [see the definition of iB (u, T) in 1I.3.7J

( 13)

As those points ui which lie in 1+ U L are now precisely the essential maximal model continua for (x, T, D), from 11.3.4 (25), (26) we obtain the relations

K+(x, T,D) =2:ie(Uj, T), uiEI+,

K-(x, T, D) = - 2: ie(ui , T), uiE 1_,

and (9) and (10) follow in view of (11), (12) and (13).

Lemma 2. Assume that T is BVB in D. Then

IB (u, T) = 0 a.e. on (D - I) U 10 , (14)

Proof. On the set (D - I) U 10, iB (Zt, T) = 0 by definition [see (2) and 1I.3.7J. Since DB (u, T) exists (and is finite) a.e. in D, the relati~n (14) follows in view of (1).

Lemma 3. Assume that T is A C B in D. If s is any integer, then Is (u, T) is L-summable on the set Is, and

JIB (u, T) dL = s J N(x, T, Is) dL. ( 15) I,

Proof. By IV.S.2, theorem 5 (applied with L=Is) we have

J Ds(u, T) dL = J N(x, T, Is! dL. ( 16) I.

On multiplying (16) by the constant s, we obtain (15), since is (u, T) = s on Is.

Theorem 1. Assume that T is sA C BinD (see definition 2). Then the following holds.

(a) T is eAC in D. (b) IB (u, T) = 1. (u, T) a.e. in D.

Proof. By assumption, Is (u, T) is L-summable in D and hence also on every L-measurable set in D. For the set 1+ [see (5)J we obtain in view of lemma 3 the relation (see 111.1.1, lemma 39)

co

J ls(u, T) dL = 2: J sN(x, T, Is) dL. ( 17)

Since sN(x, T, Is) ~ 0 in (17), by 111.1.1, lemma 40 we conclude from (17) that co

2: s N(x, T, I.) (18) s=1

§ IV.s. Bounded variation and absolute continuity in the BANACH sense. 289

is an L-summable function of x in Rn. Observe now that T is B VB in D by IV.5.2, theorem 1, and hence N(x, T, D) is L-summable in Rn by IV.5.1, theorem 1. Accordingly (see 111.1.1, lemma 31)

N(x, T, D) < 00 a.e. in Rn.

In view of lemma 1 it follows that 00

L s N(x, T, Is) = K+(x, T, D) a.e. in Rn. (19)

Since the function (18) is L-summable in Rn , it follows from (19) that K+(x, T, D) is L-summable in R". Since all the terms in (19) are nonnegative, integration of (19) yields (see 111.1.1, lemma 27)

00

L J s N(x, T, Isl dL = J K+(x, T, D) dL. (20)

From (17) and (20) we obtain

J Js(u, T) dL = J K+(x, T, D) dL. (21)

An entirely similar reasoning, applied to the set L [see (6)J yields the conclusion that K-(x, T, D) is L-summable in Rn and

J Is (14, T) dL = - J K-(x, T, D) dL. L

Now since K(x, T, D) = K+(x, T, D) + K-(x, T, D)

by 11.3.2, theorem 1, the L-summability in Rn of K(x, T, D) follows. Thus T is eBV in D (see IV.4.1). Observe now that T satisfies the condition (N) in D by IV.5.2, theorem 3. Hence T also satisfies the condition (N) on every subset of D, and in particular on the essential set E(T, D). Thus Tis eA C in D by IV.4.2, theorem 3, and part (a) of the theorem is proved. Turning now to part (b) of the theorem, note that since Is (14, T) = 0 a.e. on the set (D - I) U 10 by lemma 2, we have

J Is (u, T) dL = J Is(u, T) dL + J Is(u, T) dL. D 1+ I

In view of (21) and (22) there follows the relation

J Is(u, T) dL = J K+(x, T, D) dL - J K-(x, T, D) dL. (23) D

Observe now that since clearly T is also sAC B in every domain De D, the preceding conclusions apply if D is replaced by any domain DeD. In particular, application of this remark to (23) yields the relation

JIs(u,T)dL=JK+(x,T,D)dL-JK-(x,T,D)dL, DeD. (24) D



Since we already know that Tis e A C in D, we have by IV .4.4, theorem 4,

J Je(u, T) dL = J K+(x, T, D) dL - J K-(x, T, D) dL, DCD. (25) D

(24) and (25) yield

J JB(U, T) dL = J J.(u, T) dL (26) D D

for every domain D CD, and part (b) of the theorem follows by 111.1.1, lemma 44.

The preceding theorem shows that the theory of sA C B transformations is a special case of the theory of eA C transformations studied in § IVA. There naturally arises the question of the precise relationship between the crude multiplicity function N and the essential multiplicity function K. The following discussion throws some light upon this issue: More detailed information will be obtained, for the case of differentiable transformations, in Part v.


N(x, T, D) < 00. (27) Then [see (2)]

00 00

K(x, T, D) = L s N(x, T, Is) + L s N(x, T, C s), (28) s=l s=l

00 00

N(x, T, D) = N(x, T, 10) + L N(x, T, Is) + L N(x, T, C s) . (29) s=l s=l

Proal. Since

K(x, T, D) = K+(x, T, D) + K-(x, T, D)

by 11.3.2, theorem 1, the relation (28) is a direct consequence of lemma 1. As regards (29), note that D is the union of the pair-wise disjoint sets

and thus (29) will be proved if we show that

N(x, T, D - I) = O. (30)

In view of the definition ofthe set! = I(T, D) (see 11.3.7), the relation (30) is an immediate consequence of (27), and the proof is complete.

Lemma 5. Denote by Xoo the set of those points x E R n where N(x, T, D) = 00, and let 1* be the set defined by (7). Let x be a point such that

xEf Xoo U TI*. (31 )


Then N(x, T, D) = K(x, T, D). (32)

Proof. Since xEfXeo , the formulas (28) and (29) hold by lemma 4. Furthermore, since xEfTI*, it follows that xEfTIs if lsi =1=1, and hence N(x, T, Is) = 0 if lsi =1= 1. Thus (28) and (29) yield now (32).

Theorem 2. Assume that T satisfies the following conditions.

(i) T is BVB in D. (ii) LTI*=O [see (7)].


(a) N(x, T, D) =K(x, T, D) a.e. in Rn. (b) T is eBV in D. (c) T is sBVB in D.

Proof. Let Xeo denote again the set of those points xE Rn where N(x, T, D) = 00. Since N(x, T, D) is L-summable in Rn by IV.5.1, theorem 1, the set Xoo is of L-measure zero. In view of the assumption (ii) we have therefore

L(Xeo U TI*) = 0,

and (a) follows by lemma 5. AsN(x, T, D) is L-summablein Rn, (a) implies that K(x, T, D) is also L-summable in R", and (b) is established. To prove (c), there remains to show that fs (u, T) is L-summable in D. Now

fs (u, T) = 0 a.e. on D - I (33

by IV.5.3, lemma 2. Furthermore, the assumption (ii) implies (by IV.5.1, theorem S) that Ds(u,T)=O a.e. on 1*, and in view of (1) it follows that

fs(u, T) = 0 a.e. on 1*.

By (1) and (2) we have

Ifs (u, T) I = Ds (u, T) a.e. on II U L 1 .

N ow since [see (3) and (7) J

D = (D - I) U 1* U II U L 1 ,

we conclude from (33), (34), (35) that

Ifs(u, T)I ;:;;;;,Ds(u, T) a.e. in D,

(34)

(35)

and as Ds (u, T) is L-summable in D by IV .5.1, theorem 3, the L-summability of fs (u, T) in D follows, and the theorem is proved.

19*

292 Part V. Differentiable transformations in R".

Lemma 6. Denote by i the set of those points uED where liB (u, T) I > 1. Then

K(x, T, D) :s:; N(x, T, D) for x Ef T I. (36)

Proof. Note that

Hence, if x is a point in Rn such that xEfTI, then xEfTIs if lsi ;;;;;2, and thus

N(x, T, Is) = 0 for lsi ~ 2.

Observe now that (36) is obvious if N(x, T, D) = 00. Hence we can assume that

N(x, T, D) < 00. (38)

Lemma 4 yields, in view of (38) and (37),

K(x, T, D) = N(x, T, IJ ) + N(x, T, L t ) :s:; N(x, T, D),

and (36) is thus proved.

Part V. Differentiable transformations in Rtf.

§ V.l. Continuous transformations in RI.

V.1.1. Preliminaries. A point of Euclidean 1-space Rl has just one coordinate x, and thus it should be denoted by (x), in accordance with the general terminology introduced in 1.2.1. However, as a matter of convenience, we shall speak of the point x instead of the point (x). If this terminology is adopted, then the points of Rl are merely the real numbers, and if x', x" are two real numbers then their (Euclidean) distance is equal to I x" - x'l, the absolute value of their difference (see 1.2.1). While, in general, there exists a significant conceptual difference between an object x and the set consisting of the single object x, it should be noted that in the present case we could operate directly with the space whose points are the real numbers, the distance of two real numbers x', x" being defined as Ix" - xl The space so obtained is the real number-line of Analysis. In other words, the notational agreement of writing x instead of (x) has the effect that Rl becomes formally identical with the real numher-line. Let now U be a non-empty subset of Rl, and consider a transformation T: U --+Rl. Then T associates with every real number uE U a definite real number x = T(u). Thus T may be thought of as a real-valued function x = T(u) of the real variable uE U. Accordingly, the general theory developed in Part IV may be interpreted in the case n = 1 as a geometrical approach

§ V.1. Continuous transformations in RI. 293

to the theory of real-valued ,functions of a real variable. Our main objective in the present § V.l is to show that the various concepts of bounded variation, absolute continuity, generalized Jacobian (introduced in Part IV for general Euclidean n-space Rn) reduce in the case n = 1 to the classical concepts of bounded variation, absolute continuity, derivative in the theory of real-valued functions of a real variable. In preparation, we shall review in this section a few elementary facts relating to real-valued functions of a real variable.

We have noted earlier (see 1.2.1, exercise 8) that in Rl a bounded domain is merely an open (bounded) interval. It will be useful, therefore, to introduce convenient notations for intervals in RI. If a, b are real numbers, then we put

(a, b) = {x I a < x < b} if a< b, (1 )

(a,b)={xlb<x<a} if b < a, (2)

(a, b) = 0 if a= b. (3 )

Thus if a =1= b, then (a, b) is the open interval with end points a, b. Next we put

[a, b] ={xla~x;;':;;b} if a;;,:;;b, (4)

[a, b] = {xlb;;':;; x:;;;; a} if b;;':;;a. (5)

Thus if a=l=b, then [a, b] is the closed interval with end points a, b. For emphasis, we shall speak of non-degenerate and degenerate open and closed intervals (a, b), [a, b] according as a=l=b or a=b. The following relations are obvious.

(a, b) = (b, a), [a, b] = [b, a], (a, a) = 0, [a, a] = a.

int [a, b] = (a, b), (a, b) = [a, b] if a =1= b.

Definition 1. Let 5 =1= 0 be a subset of Rl which does not reduce to a single point, and f(x) a real-valued finite function defined for xE S. The total variation V(f, 5) of f on 5 is defined as the least upper bound of all the sums

L If(bj ) - f(a;)l, i

where [aI' bl], [a 2 , b2J .... is any finite sequence of non-degenerate closed intervals whose interiors are pair-wise disjoint and whose end points belong to S. If V(f, 5) < 00, then f is said to be of bounded variation on 5 (briefly, BV on 5).

Definition 2. Given 5 and f as in definition 1, f is said to be absolutely continuous on 5 (briefly, A C on 5) if for every e>O there exists an

294 Part V. Differentiable transformations in Rn.

1]=1](c) >0 such that the following holds: if [~, b1], [a2 , b2], ... is any finite sequence of non-degenerate closed intervals whose interiors are pair-wise disjoint and whose end points belong to 5, then the inequality L I bi - ail < 1] implies the inequality L II (bi ) - I (ai) 1< c.

If [a, a] is a degenerate closed interval such that aE 5, then I/(a) - l(a)1 =0, and thus clearly we would obtain equivalent definitions if we would allow some of the intervals [ai' b.J to be degenerate.

Delinition 3. Given 5 and I as in definition 1, the oscillation OJ (I, 5) of I on 5 is the least upper bound of all the numbers I/(b) - l(a)l, where a, b are any two numbers belonging to 5.

The following lemmas 1 through 12 are easy consequences of the definitions involved (the proofs of most of these lemmas may be found in the SAKS treatise listed in the Bibliography).

Lemma 1. Given 5 and I as in definition 1, let 5* be a subset of 5. Then the following holds.

(i) V(f, 5*) ;;;;, V(f, 5). (ii) If I is BV on 5, then I is also BV on 5*.

Lemma 2. Given 5 and I as in definition 1, assume that I is A C on 5. Then j is also A C on every subset of 5.

Lemma 3. Given 5 and I as in definition 1, assume that I is BV on 5. Then I is bounded on 5.

Lemma 4. Given 5 and I as in definition 1, the total variation V(f, 5) is lower semi-continuous with respect to I, in the following sense: if III (x), n = 1, 2, ... , is any sequence of real-valued, finite functions on 5 such that In(x)---+/(x) on 5, then V(f,5);;;;;liminfV(f,,, 5).

Lemma 5. Assume that the real-valued function I (x) is bounded, continuous, and A C in the open (bounded) interval (a, b). Then I is BV in (a, b).

Lemma 6. Let I (x) be a real-valued continuous function in the closed interval [a, b], where a=l=b. Then

V(t, [a, b]) = V(t, (a, b)).

Lemma 7. Let I (x) be a real-valued continuous function in the closed interval [a, b], where a=l= b. Then I is BV in [a, b] if and only if it is BV in (a, b).

Lemma 8. Let I (x) be a real-valued continuous function in the closed interval [a, b], where a =l= b. Then I is A C in [a, b] if and only if it is A C in (a, b).

Lemma 9. Assume that the real-valued function I (x) is A C in [a, b], where a=l=b. Then t is BV in [a, bJ.

§ V.i. Continuous transformations in Rl. 295

Lemma 10. Assume that the real-valued function I (x) is BV in the open interval (a, b), where a=l=b. Then I(x) approaches definite (finite) limits as x approaches the end points a, b.

Lemma 11. Let I(x) be a real-valued, bounded, continuous function defined in a non-degenerate (bounded) open interval (a, b). Its oscillation w (I, (a, b)) on open intervals (a, b) (a, b) is then a function of open intervals in (a, b), and (see 111.2.1, definition 4)

V(t, (a, b)) = V(w, (a, b)).

Lemma 12. Let !(x) be a real-valued, bounded, continuous function defined in a non-degenerate (bounded) open interval (a, b). Then I is BV in (a, b) if and only if its oscillation w(l, (a, b)), considered as a function of open intervals (a, b) (a, b), is of bounded variation in (a, b).

We shall now make some elementary comments about the derivative /,(x) of a (real-valued) function I (x), defined in an open interval (a, b). The statement that the derivative /,(x) exists and has the value /'(xo) at a point xoE (a, b) is equivalent to the following statement: for every e>O there exists a <5= <;(xo, e) >0 such that if

0< Ihl< <;, (6) then

xo+hE(a, b) and I~!l+hl-f(xo) -/'(Xo)I<8. (7)

Consider now the function LI(a, b) of open intervals (a, b) (a, b) defined as follows. The notations being so cho;;en that a < b, we set LI (a, b) = I(b)-/(a) if a, b both lie in (a, b), and LI(a, b)=O otherwise. Then the statement that the derivative D (x,LI) of LI exists and has the value D (xo,LI) at a point xoE (a, b) is equivalent (see 111.2.3, definition 2) to the following statement: for every e > 0 there exists a <;*= c5*(xo, e) > 0 such that if

a < Xo < band b - a < c5*, (8) then

The following lemma describes the relationship between the derivatives /,(x) and D(x,LI).

Lemma 13. Assume that I(x) is real-valued and continuous in the open interval (a, b), and let Xo be a point of (a, b). Then /'(x) exists at Xo if and only if D (x, LI) exists there, and

(10) if both exist.


Prool. Assume first that f'(x) exists at Xo' Assign 8> 0, and let 15 > ° correspond to this 8 in the sense of (6) and (7). Consider any two numbers a, b such that

a < a < Xo < b < b, b - a < 15. (11)

Then 0< b - Xo < 15, 0< Xo - a < 15, and hence (7) yields (for h = b - Xo

and h = a - Xo respectively)

f'(Xo)-8< f(b~=::xo} <f'(Xo) +8, (12)

(13 )

Consider now the identity

f(b}-f(a}_f(a}-f(xo} xo-a+f(b}-f(xo} b-xo (14) ~a- ---a-=-~b-a ~X;;--b-a-'

Noting that

Xo - a 0 IJ.. - Xo > ° . Xo - a + _b~_:::o = 1 b-a>' b-a ' b-a b-a '

we obtain from (12), (13), (14) the inequalities

f'(xo) - 8 < f(b)-::-f(a} < f'(xo) + 8. b-a

It follows that

II L0} __...1J!:l - f'(xo) I < 8 b-a

whenever (11) holds. Thus D(x,Ll) exists at Xo and is equal there to I'(XO)' Assume, conversely, th~t D(x,Ll) exists at xo' Assign 8>0,

and let 15* > ° correspond to this 8 in the sense of (8) and (9). Denote by ~ the smallest one of the numbers ~*, b-xo, xo-a, and consider any number h such that

o <[hi < 15.

Assume first that h > O. Then (15) implies that

Xo < Xo + h < b, 0 < h < ~*.

(15)

(16)

Let a be any number such that a < a < Xo' If a is sufficiently close to xo, then in view of (16) we shall have Xo + h - a < 15*. By (9), applied with b = Xo + h, we have then the inequality

I f(Xo+h}-f(a} -D(x ,Ll)I<8. Xo + 1z - a 0

Since I is continuous, for a -+ Xo there follows the inequality

I f(xo+ hl-f(Xo} -D(xo, Ll)I:s;;8. (17)


An entirely similar reasoning yields (17) in the case when h<O. Thus (17) holds whenever (15) holds. It follows that I'(x) exists at Xo and is equal there to D(xo,LI), and the lemma is proved.

Remark 1. Let t (x) be a real-valued, bounded, continuous function which is A C in the open interval (a, b). By lemma 5, t(x) is then BV in (a, b), and hence by lemma 10 the limits

IX = lim t(x), (3 = lim t(x), x-+-a x-+-b

exist. Define now a function t*(x), xE[a, b], as follows: f*(x) =t(x) if a< x< b, and f*(a) = IX, t*(b) =(3. Clearly, t*(x) is then continuous in the closed interval [a, b], and by lemma 8 is follows that t*(x) is AC in [a, b]. Briefly, t(x) admits of a continuous extension to the closed interval [a, b], and this extension is A C in [a, b].

Remark 2. Let t (x) be real-valued, continuous, and BV in the open interval (a, b). If (a, b) is any non-degenerate open interval in (a, b), then tis BV in (a, b) by lemma 1. Thus V(t, (a, b)) is a function of open intervals (a, b) ((a, b). It can be shown that there exists a non-negative, countably additive function W(j, B) of BOREL sets B((a, b) such that W(t, (a, b)) =V(t, (a, b)) for every open interval in (a, b). Briefly, the function V(t, (a, b)) of open intervals admits of a non-negative, countably additive extension W(j, B) to BOREL sets B( (a, b). On the other hand, if B is a BOREL set in (a, b), then there arises the total variation V (j, B) in the sense of definition 1. Simple examples show that generally V(j, B) =l= W(j, B). In the literature, frequently W(j, B) is termed the total variation of t on the BOREL set B. The quantity V(j, B), described in definition 1, is frequently referred to as the weak total variation of t on B.

V.l.2. Index functions and multiplicity functions in RI. Let there be given a bounded continuous transformation T:D--+RI, where D is a bounded domain in RI. As noted above, D is then merely a nondegenerate (bounded) open interval (a, b). Accordingly, we write

T:(a, b) --+RI, (1 )

where a =l= b. Then x = T(u) is a real-valued, bounded, continuous function of the variable uE (a, b). Throughout the present section a, b, T are considered as fixed. Let (a, b) be an open interval in (a, b) such that [a, bJ ((a, b), a =l= b. Consider a point xE RI. For the topological index fl (x, T, (a, b)) to be defined (see 11.2.2), it is then necessary and sufficient that x~Ttr(a, b). Now Ttr(a, b) consists of the points T (a), T (b) (which mayor may not be distinct). Accordingly, we have the following statement.

Part V. Differentiable transformations in R".

Lemma 1. P (x, T, (a. b)) is defined if and only if

a=l=b, [a.b] C (a, b), x=l=T(a),T(b). (2)

The next lemma yields the explicit value of p (x, T, (a, b)). whenever this quantity is defined.

Lemma 2. Assume that a<b and ,u(x, T, (a, b)) is defined. Then the followjng holds.

a<b, [a,b]C(u, b), x=!=T(a).T(b). (3)

p(x. T, (a, b)) = 0 if T(a) = T(b). (4)

,u(x. T, (a, b)) = 0 if xE C[T(a), T(b)J. (5)

p (x. T, (a, b)) = sgn (T(b) - T(a)) if xE (T(a), T(b)). (6)

Proof. In view of lemma 1, the relations (3) are direct consequences of the assumptions made. To verify (4), (5), and (6), consider the (unique) linear transformation

x = S3(u) = IXU+ f3

which satisfies the conditions

Clearly (see 11.2.5)

S3 (a) = T(a), S3 (b) = T(b).

det S3 = IX = T(b) - T(a) . b-a

(7)

(8)

(9)

Since the points a, b constitute the frontier of (a, b), the conditions (8) mean that S3=T on fr(a, b). By 11.2.3, remark 7 it follows that p(x, ~, (a. b)) is also defined and

p, (x. T, (a, b)) = P (x, S3, (a, b)) . (10)

Assume now first that T,a) =l= T(b). (11)

In view of (9), S3 is then non-singular. From (8) we see that

S3(a, b) = (T(a). T(b)) , S3[a, b] = [T(a), T(b)J.

By 11.2.5, theorem 4 it follows that

p, (x, £, (a, b)) = sgn det S3 if xE (T(a), T(b)) , (12)

p,(x,S3,(a,b))=O if xEC[T(a),T(b)]. (13)

§ V.i. Continuous transformations in RI. 299

Thus (5) and (6) follow from (9), (10), (12), (13), while (4) is of course now vacuous in view of (11). Assume next that

T(a) = T(b). (14)

Note that (6) is now vacuously true, since (T(a), T(b)) = 0 in view of (14). To verify (4) and (5), observe that (14) implies the relations

,\\ (a, b) = ,\\ (a) = T(a) = T(b).

Since x=f=T(a), T(b) by (2), it follows that we have now the relation xE C,\\ (a, b), and hence,u (x, '\\, (a, b)) = 0 by U.2.3! theorem 2. Thus (4) and (5) follow in view of (10), and the proof is complete.

Lemma 3. Let (a, b) be an open interval such that a < band [a,bJ((a,b). Then

J,u (x, T, (a, b)) dL = T(b) - T(a). (15)

Proof. Assume first that

T(a) =f= T(b). (16)

By lemma 1, ,u (x, T, (a, b)) is then defined for x=f=T(a), T(b) and by lemma 2 we have,u(x, T, (a, b)) = ° if xEf [T(a), T(b)] and,u (x, T, (a, b)) =

sgn (T(b) -T(a)) if xE (T(a), T(b)). Thus clearly

J ,u (x, T, (a, b)) d L = I T(b) - T(a) I sgn (T(b) - T(a)) = T(b) - T(a).

Thus (15) is proved under the assumption (16). Suppose next that

T(a) = T(b). (17)

By lemma 1 and lemma 2, ,u (x, T, (a, b)) is defined and has the value zero for x =f= T(a) = T(b). Thus now clearly

J ,u(x,T, (a, b)) dL = 0,

and (1 5) follows in view of (17).

Lemma 4. If the essential local index ie(C, T) is defined (see 11.3.4), then ie (C, T) = ± 1.

Proof. If ie (C, T) is defined, then C is an essentially isolated e.m.m.c. for some point xE RI, and there exists an open interval (a, b) ( (a, b) such that

ie(C, T) =!l(X, T, (a, b)). ( 18)

If the notation is so chosen that a < b, then lemma 2 yields the fact that

p(x, T, (a, b)) = ± l' or O. (19)


Since ie(C, T) is always a non-zero integer (see 11,3.4, remark 3), the relations (18) and (19) imply that ie (C,T)=±1, and the lemma is proved.

Lemma 5. If x is any point in RI, then (see 11.3.2, 11.3.3)

K (x, T, (a, b») = k (x, T, (a, b»). (20)

Proof. If k (x, T, (a, b») is equal to either zero or 00, then (20) holds by 11.3.4, theorem 3, and so we can assume that 0< k (x, T, (a, b») < 00.

By 11.3.4, theorem 4, remark 2, and remark 5 we have then the formula

K(x, T, (a, b») = L.:lie(C, T)I, CE @(x, T, (a, b»).

Since lie(C,T)1 =1 by lemma 4, it follows that K(x,T, (a,b») is equal to the number of the continua CE@(x,T, (a, b»). As this number is equal to k(x, T, (a, b») by definition (see 11.3.3, definition 1), the relation (20) follows.

Lemma 6. The local index £s (u, T) (see 11.3.7) takes on no value different from 0, 1, -1.

Proof. By its definition, is (u, T) is equal either to zero or else to ie (u, T), and in the latter case its value is either 1 or -1 by lemma 4.

Lemma 7. If (a, b) is an open interval in (a., b) and x is a point in RI, then (a, b) is an indicator domain (see 11.3.2) for (x, T, (a, b») if and only if

[a, b] ((a, b) and (T(a) - x) (T(b) - x) < 0. (21)

Proof. Assume that (a, b) is an indicator domain for (x, T, (a, b»). Then p.. (x, T, (a, b») must be defined and different from zero. By lemma 1 it follows that [a, b] ( (a, b), and by lemma 2 it follows further that x E (T(a), T(b»). This last inclusion is clearly equivalent to the second relation in (21). Assume, conversely, that (21) holds. The second relation in (21) is clearly equivalent to the inclusion x E (T(a), T(b»). Thus we have now at our disposal the relations

[a,b]((a, b), xE(T(a),T(b»). (22) It follows that

(23)

since otherwise we would have T(a) =T(b), and the second relation in (21) could not hold. Furthermore

x =F T(a), T(b) , (24)

since otherwise we would have (T(a) - x) (T(b) - x) = 0, in contradiction with (21). From (22), (23), (24) we conclude, by lemma 1 and lemma 2, that f1( x, T, (a, b») is defined and is equal to sgn (T(b) - T(a»)


or sgn(T(a)-T(b)), according as a<b or b<a [see (23)]. In either case, fl(X, T, (a, b)) = ± 1 [note that the second relation (21) implies that T(a) =l= T(b), a fact which we already used above]. Thus fl( x, T, (a, b)) is defined and different from zero, and hence (a, b) is an indicator domain for (x, T, (a, b)).

Our next objective is to compare the multiplicity functions N(x, T, (a, b)) and K(x, T, (a, b)) (see 1.1.2 and 11.3.2). In preparation, we consider two auxiliary sets Smax, Smin defined as follows. A point uE (a, b) belongs to Smax if and only if there exists a number h=h(u) >0 such that

[u - h, u + h] ((a, b) and

T(u') ::s;; T(u) if u' E [u - h, u + h].

Similarly, a point uE (a, b) belongs to Smin if and only if there exists a number h*=h*(u) >0 such that

and T(u') ~ T(u) if u'E [u - h*, u + h*].

Thus Smax is the set of those points uE (a, b) at which the function x = T(u) has a weak local maximum. Similarly, Smin is the set of those points uE (a, b) at which the function x=T(u) has a weak local minimum.

Lemma 8. The set T( Smax U Smin) is countable (perhaps empty). Proof. Clearly it is sufficient to show that the sets T Smax and

T Smin are countable. Since the proof is entirely similar in both cases, we consider explicitly only TSmax ' For each positive integer n, let S" denote the set of those points uE (a, b) for which it is true that

[u -~, u +~] ( (a, b) n n

(25)

and

T(u') :::;; T(u) if u'E [u -~, u +~]. n n

(26)

Clearly Smax=USn and hence TSmax=UTSn. Thus it is sufficient to prove that TS" is countable. In fact, we shall find that Tsn is a finite set. If Tsn= 0, this is obvious. So assume that T sn=l= 0, and consider a point xE T S". Then there exists at least one point uES" such that T(u)=x. For each xETS", select such a point u and denote it by u(x). Then

u(x) E sn, T(u(x)) = x. (27)

302 Part V. Differentiable transformations in RH.

Consider now any two points Xl' x2 such that

(28) We assert that

(29)

Assume, indeed, that lu(x2) -u(xl )/ s.1/n. We can then apply (26) with u=u(xl), u'=u(x2), obtaining

(30)

Also, we can apply (26) with u=;=u(x2),U'=U(xl), obtaining

(31 )

From (30), (31) it follows that Xl = x2 , in contradiction with (28), and thus (29) is verified. Denote by U the set of the points U (x) corresponding to the points xETSH. Then (29) shows that the distance between any two distinct points of U exceeds 1/n. Since n is now a fixed positive integer and U is a subset of the (bounded) open interval (a, b), it follows that U is a finite set. As TU =TSn by (27), it follows finally that sn is a finite set, and the lemma is proved.

Lemma 9. Let X be a point in Rl such that

X Ef T( Smax U Smin) . (3 2)

Then (see 1.1.2 and 11.3.2)

K(x, T, (a, b)) = N(x, T, (a, b)). (33)

Proof. In view of lemma 5, (33) is equivalent to

k (x, T, (a, b)) = 1'1(x, T, (a, b»). (34)

To prove (34) we first note that

k (x, T, (a, b»);;;;: N(x, T, (a, b)). (35)

Indeed, (35) is obvious if k(x, T, (a, b») =0. If k(x, T, (a, b))>O, then select any integer m such that

k(x, T, (a, b») ~m~ 1.

Then there exist, by the definition of k, distinct continua C1 , ... , Cm

in (a, b) such that TC j = x, j = 1, ... , m. Since each C j contains at least one point, it follows that N(x, T, (a, b») ~ tn. As m was any integer satistying (36), the inequality (35) follows [observe that (32) is

§ V.i. Continuous transformations in Ri. 303

not needed to establish (35)]. We show next that

N(x, T, (a, b));:;;;:k(x, T, (a, b)) (7)

if (32) holds. IfN(x, T, (a, b)) =0, then (37) is obvious. If N( x,T, (a, b)) > 0, then select any integer m such that

N(x, T, (a, b)) ~m ~ 1.

Then there exist distinct points U I • •.•• um in (a. b) such that

T(Ui)=X. j=1 ..... m,

(38)

(39)

(40)

Since ttl' ...• U m are distinct points in (a. b), we can select a number h > 0 such that on setting

Ii = [ui - h, ui + h] • j = 1, ... , m,

we have the relations

Ii((a.b). j=1 ....• m,

I, n Ii = 0 if i =f= j.

(41)

(42)

(43)

In view of (40) there follows, for each j = 1 •...• m, the existence of points ai' bi such that ai' biE Ii' T(ai) > T(ui) = x, T(bi) < T(uj) = x. Clearly. these relations imply that

[aj' bi] (Ii ((a, b). (T(aj) - x) (T(bi) - x) < O. (44)

By (42), (43). (44) and lemma 7 it follows that (ai • bI ). ... , (am. b",) are pair-wise disjoint indicator domains for (x. T, (a. b)). Since each of these indicator domains contains (by 11.3.3. lemma 4) at least one e.m.m.c. for (x. T. (a. b)). jt follows that there exist at least m distinct essential maxirpal model continua for (x. T, (a, b)). and hence k(x, T. (a, b)) ~ m. As m was any integer satisfying (38), the inequality (37) follows. Since (37) and (35) imply (34), the lemma is proved.

Observe that the set T(Smax U Smin) is countable by lemma 8. and hence this set is of L-measure zero. Thus lemma 9 implies that (33) holds a.e. in RI. Since we only assumed that T is continuous and bounded in (a. b). it is clear that (a. b) can be replaced by any nondegenerate open interval (a, b) ((a. b). Accordingly, we have the following statement.

Lemma 10. If (a. b) is any non-degenerate open interval in (a, b). then

K(x. T, (a, b)) = N(x, T, (a, b)) a.e. in RI.


Let us recall that T may be thought of as a real-valued function x = T(u), uE (a, b), and accordingly we can consider the derivative T(u), which mayor may not exist at a given point uoE (a, b).

Lemma 11. Assume a< b. Consider a point uoE (a, b), such that the derivative T'(UO) exists and is different from zero. Then (see 11.3.7)

Proof. The letters a, b will designate two numbers such that

a < a < uo < b < b. (45)

In view of V .1.1, lemma 13, the existence of T (uo) implies the relations

_~(bl - T(uo) -?- T(uo) b - U o

for b -?- UO' (46)

T(a) - T(uol -?- T(uo) for a -?- uo , (47) a - U o

T(b) - T(a) T() -?- Zlo

b-a for a -?-'lto, b -?- uo' (48)

It is convenient to consider the cases T(uo) > 0 and T(uo) < 0 separately. Since the argument is entirely analogous in both cases, we consider explicitly only the case when

T(uo) < o. (49)

We have to show then that

iB (uo, T) = - 1. (50)

l<rom (45) to (49) we infer the existence of a number 1]> 0 such that

T(b)-T(uo) <0 if uo<b<Uo+17, (51)

T(a) - T(uo) > 0 if U o -1] < a < uo, (52)

T(b)-T(a)<O if uo-1]<a<uo <b<uo+1]· (53)

From (51) and (52) it follows that T(u)=I=T(uo) if O<lu-uol<1]. Thus U o belongs to the set of those points I = I (T, (a, b)) which possess a neighborhood clear of relatives (see 11.3.7, definition 2). As explained in 11.3.7, U o by itself is therefore an m.m.c. for (T(uo), T, (a, b)). From lemma 1 it follows further that if a, b satisfy the inequalities

U o - 1] < a < U o < b < U o + 1] , (54)

then f1 (T(uo), T, (q;, b)) is defined. Since (51) and (52) imply that T(uo) E (T(a), T(b)), lemma 2 yields [see (53)]

f1(T(uo), T, (a, b)) = sgn(T(b) - T(a)) = -1. (55)


Thus in every neighborhood of U o there are indicator domains (a, b) for (T(uo), T, (a, b)) such that a< uo< b. Hence U o is an e.m.m.c. for (T(uo), T, (a, b)). Therefore (see II.3.7)

(56)

On the other hand, ie(uo,T) is equal to tt(T(uo),T, (a, b)) if (a, b) is an indicator domain for (Ttuo),T, (a, b)) such that a<uo<b and [a, bJ is contained in a characteristic neighborhood of U o (see II.3.4, definition 1). Clearly, the open interval (uo-1}, u o+1J) is a characteristic neighborhood of uo, since T(u) =l= T(uo) if u =l= uo' uE (uo -1}, u o+1}), as we noted above. Thus (54) and (55) yield ie (uo,T)=-1, and (50) follows in view of (56).

V.1.3. Bounded variation. The bounded continuous transformation T: (a, b) -+Rl, given as in V.1.2 (1), is considered fixed throughout this section.

Lemma 1. In Rl the properties BVB and sBVB (see IV.5.l, definition 1 and IV.5.3, definition 1) are equivalent. Explicitly: Tis BVB in (a, b) if and only if it is sBVB in (a, b).

Prool. If T is sBVB in (a, b), then it is also BVB in (a, b) by definition. Assume, conversely, that T is BVB in (a, b). Then (see IV.5.l, theorem 3) the derivative Ds (u, T) exists a.e. in (a, b) and is L-summable there. Now since

Is (u, T) = is (~t, T) Ds (u, T) a.e. in (a, b)

by definition, and lis (u, T) I ::s;: 1 by V.1.2, lemma 6, it follows that Is(u, T) is L-summable in (a, b). Thus T is sBVB in (a, b), and the lemma is proved.

Lemma 2. If (a, b) is any non-degenerate open interval in (a, b), then [see IV.5.1 (1) and V.l.l, definition 3J

G(a, b) =w(T, (a, b)). ( 1)

Proof. In (1), T is interpreted (in connection with w) as a realvalued function T(u), uE (a, b). Recall that

G (a, b) = L T(a, b) . (2) Let us put

A. = gr.l.b. T(u) , A = l.u.b. T(u) , u E (a, b). (3)

Since T is bounded, clearly A. and A are finite, and A. :;;;:A. We first verify that

G(a, b) :::;:A - A.. (4) Rado and Reichelderfer, Continuous Transformations. 20


Indeed, if uE(a,b), then J,.;;;;:T(u);;;;:A by (3). Thus T(a,b)([A,A], and (4) follows in view of (2). Next we verify that

w(T, (a, b));;;;:G(a, b). (5)

Assign any 13 > o. By the definition of w, there exist numbers IX, fJ such that

1X,j3 E (a, b), I T(f3) - T(IX)J > w(T, (a, b») - 13. (6)

It follows that T(oc), T(fJ) E T(a, b). Furthermore, the continuity of T implies that if xE[T(oc), T(f3)], then x=T(u) for some uE[oc,fJJ. Thus

[T(a), T(fJ)] (T[a, fJ] (T(a, b).

In view of (6) we conclude that

I T(fJ) - T(a) J ;;;;: G (a, b). (7)


w (T, (a, b)) < G(a, b) + 13.

As 13 > 0 was arbitrary, (5) follows. Finally we verify that

A-A;;;;:W(T, (a, b)). (8)

Assign again any 13 > o. In view of (3) there exist numbers uI , U 2 such that uI , u2 E (a, b), T(uI ) < A + 13, T(u 2»A-13, and hence

A - A - 213 < T(u 2) - T(u I ).

Since T(u 2)-T(uI );;;;:w(T,(a,bl), we conclude that A-J,.-213< w(T, (a, b)). As 13>0 was arbitrary, (8) follows. Clearly, (4), (5), (8) imply that

A - J,. = w (T, (a, b)) = G (a, b),


Theorem 1. V(T, (a, b)) = VB(T, (a, b)) = V.(T, (a, b)).

Proof. By V.1.t, lemma 11 we have

V(T, (a, b)) = V(w, (a, b)), (9)

where w stands now for the oscillation w (T, (a, b)) of T in (a, b) ((a, b). By definition (see IV.S.t, definition 2)

VB(T, (a, b») = V(G, (a, b»). (10)

§ V.1. Continuous transformations in RI.

Now since G(a, b) =w(T, (a, b)) for (a, b) ((a, b)

by lemma 2, (9) and (10) imply that

V(T, (a, b)) = Vs(T, (a, b)). Recall that

307

(11)

K(x, T, (a, b)) =N(x, T, (a, b)) a.e. in Rl (12)

by V.1.2, lemma 10. Thus K(x, T, (a, b)) is L-summable in RI if and only if N( x, T, (a, b)) is L-summable in RI. If both are L-summable in RI, then

VB(T, (a, b)) = J N(x, T, (a, b)) dL (13 )

by IV.S.l, lemma 2, and

v,;(T, (a, b)) = J K(x, T, (a, b)) dL (14)

by IV.4.l, lemma 3 and definitions 1 and 2. In view of (12), we see from (13) and (14) that

VB(T, (a, b)) = v,;(T, (a, b)) ( 15)

in this case. On the other hand, if K(x, T, (a, b)) and N(x, T, (a, b)) are not L-summable in Rl, then

VB(T, (a, b)) = (Xl = v,;(T, (a, b))

by the same references. Thus (15) holds always, and hence in view of (11) the theorem is proved.

Theorem 2. In Rl the properties BV, BVB, sBVB, eBV are equivalent. Explicitly, if T possesses in (a, b) one of these four properties, then it also possesses the other three.

Proof. BV is equivalent to the condition V(T, (a, b)) < 00 by V.1.1, definition 1; eBVis equivalent to the condition v,; (T, (a, b)) < 00

by IV.4.l, lemma 3; and BVB is equivalent to the condition VB (T, (a, b)) < (Xl by III.2.l, lemma 1 and IV.5.l, definition 1. In view of theorem 1 it follows that the properties BV, eBV, BVB are equivalent. Since BVB and sBVB are equivalent by lemma 1, the theorem follows.

V.1.4. Absolute continuity. Again, the bounded continuous transformation T: (a, b)--+Rl, given as in V.1.2 (1), is considered fixed.

Lemma 1. If T is A CB in (a" b), then T is also sA CB in (a, b) (see IV.S.2 and IV.S.3, definition 2).

20*


Proof. Assume that T is A CB in (a, b). By IV.S.2, theorem 1, T is then also BVB in (a, b). By V.1.3, lemma 1, it follows that the generalized Jacobian fa (u, T) is L-summable in (a, b), and hence T is sA CB in (a, b).

Lemma 2. Suppose that T [considered as a function T(u), u E (a, b)] is A C in (a, b), in the sense of V.1.t, definition 2. Then T is A CB in (a, b).

Proof. Consider any non-degenerate open interval (a, b) ((a, b). Since T is bounded, its oscillation w( T, (a, b)) is clearly finite. If a> 0 is arbitrarily assigned, then by the definition of w (T, (a, b)) there exist numbers ct., f3 such that

ct., f3 E (a, b), I T(f3) - T(ct.) I > w (T, (a, b)) - u.

Assign now s> O. Consider any finite system of non-degenerate open intervals (aI' bI)' ... , (am, bm) such that

(a i ,b,)((a,b), i=1, ... ,m,

(a i ,bi)n(aj,bi)=0 if i=t=j.

By the preceding remarks, there exist numbers ct.;, f3i such that

I T({Ji) - T(ct.;ll > w(T, (a" bi)) __ 8 , 2m

for i = 1, ... , m. Observing that

by V.1.3, lemma 2, we obtain the inequality

m m

L G(a;, bi) < ~ + L I T({J,) - T(ct.;)I· ;=1 ;=1

Now since T is A C in (a, b) by assumption, we shall have m

L I T({Ji) - T{ct.;ll < ~ , ,=1

provided only that

.f !{J; - ct.,! < rj = rj ( ~) , ,=1

(1 )

(2)

(4)

(5)

where rj corresponds to s!2 in the sense of the definition of absolute continuity. Since rJ.i' {Jj E (ai' bi ), (5) will certainly hold if

m

L!bi-ail<rj· (6) ;=1


Accordingly,. if (1), (2), (6) are satisfied, then from (3) and (4) it follows that m

L G (ai' bi ) < s. ;=1

Thus the function G (a, b) of open intervals (a, b) ((a, b) is absolutely continuous in (a, b). Hence, by definition, T is A CB in (a, b), and the lemma is proved.

Lemma 3. If T is eA C in (a, b), then it is also A C in (a, b).

Proof. Consider any non-degenerate closed interval [a, b ]((a, b), a< b. By IV.4.4 (17) and the remark at the end of IV.4.4, we have the formula

b

f Ie (u, T) d L = f fl ( x, T, (a, b)) d L. a

On the .other hand, V.l.2, lemma 3 yields

f fl (x, T, (a, b)) dL = T(b) - T(a) ,

and hence b

f Ie(u, T) dL = T(b) - T(a). (7) a

Note that since Tis eAC in (a, b), it is also eBV there. Thus Ie(u,T) is L-summable in (a, b) by IV.4.3, theorem 1. Hence (see IlL 1. 1 , lemma 33) for every e > ° there exists an 1'] = 1'] (e) > osuch that

f 1.Te(u, T)I dL < e 5

whenever 5 is an L-measurable set satisfying the conditions 5 ((a, b), L 5 < 17. Assign now e> 0, and consider any finite system of nondegenerate closed intervals [aI' b1], •.• , [am, bm] such that

a,<b" [a"biJ((a, b), i=1, ... ,m,

(ai,b,)n(aj,b j )=0 if i~j,

m

Llbi -ail<1']· ;=1

Put 5 = U (ai' bi), i = 1, ... , m. Then LS <1'], and hence

m bi

flJe(u,T)ldL=L fl.Te(u,T)dL<s. S i=l aj

In view of (7) it follows that m

L I T(b;) - T(ai) I < s ;=1

(8)

(9)

(10)


whenever (8), (9), (10) hold. Thus T is A C in (a, b), and the lemma is proved.

Theorem. In Rl the properties A C, A C B, sA C B, eA C are equivalent. Explicitly, if T possesses in (a, b) one of these four properties, then it also possesses the other three.

Fig. 43.

Proof. The diagram in Fig. 43, in which the arrows stand for logical implication, summarizes the available relevant information.

The arrows from ACB to sACB, from AC to A C B, and from eAC to AC represent the lemmas 1, 2, and 3 respectively. Finally, the arrow from sA C B to eA C represents IV.5.3, theorem 1, according to which the sA C B property implies the eAC property. It is clear

from the diagram that anyone of the properties A C, A C B, sA C B, eA C implies the other three, and theorem follows.

V.l.5. Derivatives. The bounded continuous transformation T: (a, b) .-+Rl, given as in V.1.2, is considered fixed throughout this section. As noted in V.1.1, T may be thought of as a real-valued function x=T(u), uE(a, b). Accordingly, we can speak of the derivative T'(u) , which mayor may not exist at a given point u E (a, b). Our objective in this section is to compare the derivative T'(u) with the generalized Jacobians JB(u,T) and Je(u,T) (see IV.5.3 and IV.4.3). We assume throughout that a< b.

Lemma. Assume that T is eBV in (a, b), and consider a point U o E (a, b). Then T'(uo) exists if and only if JJuo, T) exists, and

(1) if both exist.

Proof. Take any two numbers a, b such that

a < a < 1£0 < b < b. (2)

By IV.4.3, lemma 4 we have then

'lfe( (a, b)) = J fh (x, T, (a, b)) d L,

while V.l.2, lemma 3 yields

J fh (x, T, (a, b)) dL = T(b) - T(a).

Hence 'lfe( (a, b)) = T(b) - T(a). (3)

§ V.1. Continuous transformations in Rl. 311

By definition (see IV.4.3), the existence of Ie(uo, T) is equivalent to the relation

2{e((a, b)) ---* 1. (u T) for a ---* uo, b ---* uo. b - a e 0'

(4)

On the other hand (see V.1.t, lemma 13), the existence of T(uo) is equivalent to the relation

T(b) - T(a) ~ T'(uo) f b ~ or a ---* U o , ---* U o . b-a (5)

In either case, a and b are subjected to the conditions (2). In view of (3), the equivalence of (4) and (5) as well as the validity of (1) is obvious.

Theorem. Assume that T is eBV in (a, b). Then the derivative T(u) and the generalized Jacobians ls(u, T) and Ie(u, T) exist a.e. m (a, b), and

T(u) = Ie (u, T) = Is (u, T) a.e. in (a, b). (6)

Proof. fe(u, T) exists a.e. in (a, b) by IV.4.3, theorem 1. In view of the preceding lemma it follows that T(u) exists a.e. m (a, b) and

T(u) = Ie(u, T) a.e. in (a,b). (7)

Note .that T is BVB in (a, b) by V.1.3, theorem 2. Hence Is (u, T) exists a.e. in (a, b) by IV.S.3. So there remains to show that

Ie(u, T) = Is (1£, T) a.e. in (a, b). (8)

Now we have

Ife(u,T)I=De(u,T) a.e. m (a, b) (9)

by IV.4.3, theorem 2, and

Is(u, T) =is(u, T) Ds(u, T) a.e. m (a, b) (10)

by IV.S.3 (1). Since lis (1£, T)I;s1 byV.1.2, lemma 6, we infer from (10) that

I Is (1£, T) I;s Ds (1£, T) a.e. in (a, b). (11)

By definition [see IV.S.t (7), (8)], Ds(u, T) is the derivative of the interval function

~'s((a, b)) = J N(x, T, (a, b)) dl, (a, b) ((a, b),


and [see IV.4.3 (13), (15)J De(u, T) is the derivative of the interval function

sr((a, b)} = J K(x, T, (a, b)) dL, (a, b) ((a, b).

By V.1.2, lemma 10 we have

K(x,T,(a,b)}=N(x,T,(a,b)) a.e. in Rl,

and hence sr((a, b)) = Va ((a, b)), (a, b) ((a, b).

Thus clearly De (u, T) = Da (u, T) a.e. in (a, b). (12)

From (11), (12), (9) we obtain

Ifa(u,T)I~IL(u,T)1 a.e. in (a,b). (13)

Now denote by U the set of those points uE (a, b) where L(u, T) exists and is different from zero. From (13) we see that

la(u, T) = 0 = fe(u, T) a.e. on (a, b) - u. (14)

Consider now a point uoE U. Then fe(uo, T) exists and is different from zero. By the preceding lemma, it follows that T(uo) exists and

( 15)

From V.1.2, lemma 11 we conclude now, in view of (15), that

ia(uo, T) = sgn T(uo) = sgnfe(uo, T) if uoE U. (16)

By (10), (12), (9), (16) it follows that

fa (u, T) = Da (u, T) ia (u, T) = De (u, T) iB (u, T)

= Ife (u, T) I sgn fe (u, T) = fe (u, T)

a.e. on U. Thus (8) follows in view of (14), and the theorem is proved.

V.1.6. Applications to real-valued functions of a real variable. Let there be given a real-valued, bounded, continuous function T(u), uE (a, b), where a< b. On assigning to each point uE (a, b) the point x = T(u) of Rl as its image, we obtain a bounded continuous transformation T: (a, b) --">-Rl. The terms trans/ormation T and function T will be used interchangeably in this section, and T will be considered as fixed. The general theory of continuous transformations in R n ,

developed in Part IV, yields in the case n = 1 a series of fundamental classical theorems for real-valued functions of a real variable, as we


shall see presently. It should be noted that these theorems can be derived far more economically in a direct manner. Furthermore, the assumption of continuity (necessitated by the formulation of the general theory in Part IV) could be dispensed with in some cases. Yet the discussion of these applications to functions of a real variable serves a useful purpose, since additional light is thrown upon the basic concepts and the scope of the general theory. Furthermore, the equivalence theorems discussed in V.L2 to V.LS yield a variety of illuminating geometrical interpretations for the concepts of bounded variation, absolute continuity, and derivative.

Theorem 1. T is BV in (a, b) if and only if the crude multiplicity function N(x, T, (a, b)) is L-summable in RI.

Proof. By V.L3, theorem 2, T is BV in (a, b) if and only if it is BVB there. By IV.S.I, theorem 1, T is BVB in (a, b) if and only if N(x,T, (a, b)) is L-summable in RI, and the present theorem follows.

Theorem 2. If T is BV in (a, b), then

V(T, (a, b)) = J N(x, T, (a, b)) dL.

Proof. By assumption, V(T, (a, b)) < 00, and thus

VB(T, (a, b)) = V(T, (a, b)) < 00

in view of V.L3, theorem 1. Hence

VB (T, (a, b)) = J N(x, T, (a, b)) dL

by IV.S.I, lemma 2, and the theorem follows.

Theorem 3. Assume that T is BV in (a, b). Then the derivative T'(u) exists a.e. in (a, b) and is L-summable there. Furthermore

b

J I T'(u) I dL;;;;;Y(T, (a, b)), a

where the sign of equality holds if and only if T is A C in (a, b).

Proof. By theorem 2 in V.L3, Tis eBV in (a, b). Hence (see IV.4.3, theorem 3 and IV.4.I, definition 2), Je(u, T) exists a.e. in (a, b) and IS L-summable there, and

b

J I Je(u, T)I dL~V.(T, (a, b)), a

where the sign of equality holds if and only if T is eA C in (a, b). By the theorem in V.LS it follows that T'(u) exists and is equal to 1. (u, T)


a.e. m (a, b). Thus b b

J IJe(U, T)I dl = J I T'(U) I dl. a a

Furthermore V.(T, (a, b)) = V(T, (a, b))

by V.1.3, theorem 1. Finally, the property eA C is equivalent to the property A C by the theorem in V.1.4, and thus the present theorem follows.

Theorem 4. Assume that T is BV in (a, b). Then T is A C in (a, b) if and only if it satisfies there the condition (N) (see IV.1.4).

Proof. Tis BVB in (a, b) by V.1.3, theorem 2. By IV.5.2, theorem 3 it follows that T is A CB in (a, b) if and only if it satisfies there the condition (N). Since the properties A CB and A C are equivalent by the theorem in V.1.4, the present theorem follows.

The preceding theorems are concerned with a function T(u) defined in an open interval. In Analysis, it is more customary to consider functions given in a closed interval. Let us note that the passage to this latter situation is practically trivial. Consider, indeed, a realvalued function T(u) which is defined and continuous in a non-degenerate closed interval [a, b], where a< b. Make the following observations.

(i) By V.1.t, lemmas 7 and 8, T is BV (or A C) in [a, b] if and only if it is BV (or A C) in (a, b).

(ii) By V.1.t, lemma 6, we have V(T, [a, b]) = V(T, (a, b)). (iii) Since [a, b] -(a, b) consists of just the two points a and b,

it is clear that the terms a.e. in [a, b] and a.e. in (a, b) are equivalent. It is also clear that T satisfies the condition (N) in [a, b] if and only if it satisfies the condition (N) in (a, b). Finally,

N(x, T, [a, b]) = N(x, T, (a, b)) a.e. in RI,

since this relation clearly holds if x*T(a), T(b).

In view of these remarks, the following theorem appears as an immediate corollary of the preceding theorems [parts (a), (b), (c), (d) corresponding to the theorems 1, 2, 3, 4 respectively].

Theorem 5. Let T(u) be a real-valued continuous function in the closed interval [a, b], where a < b. Then the following holds.

(a) T is BV in [a, b] if and only if N (x, T, [a, b]) is l-summable in RI.

(b) If Tis BV in [a, b], then

V(T, [a, bJ) = J N (x, T, [a, bJ) dL.


(c) Assume that Tis BV in [a, bJ. Then the derivative T'(u) exists a.e. in [a, bJ and is L-summable there. Furthermore

b

J I T'(u) I dL:s;; V(T, [a, b]), a

where the sign of equality holds if and only if T is A C in [a, b J. (d) Assume that Tis BV in [a, bJ. Then T is A C in [a, bJ if and

only if it satisfies there the condition (N).

We now proceed to discuss transformation formttlas for definite integrals in RI. In this connection, the bounded continuous transformation T: (a, b) ~RI will be assumed to be A C in (a, b). By V.Lt, remark 1, T can then be extended continuously to the closed interval [a, b]. Accordingly, we can assume without loss of generality that T is defined, continuous, and A C in the closed interval [a, b] (see V.Lt, remark 1). To conform to general usage in the theory of functions of a single real variable, we shall write dx (or du) instead of dL in formulas involving definite integrals.

Theorem 6. Assume that the real-valued continuous function T(u) is defined and A C in the closed interval [a, b], where a < b. Let H(x) be a real-valued, finite-valued, L-measurable function in RI. Then

b

J H(T(u)) 1T'(u)ldu = J H(x)N(x, T, [a, b]) dx, a


Proof. T is eA C in (a, b) by V.Lt, lemma 2 and V.L4. Observe that

le(u, T) =T'(u) a.e. in (a,b)

by V.1.5, and

K(x, T, (a, b)) = N(x, T, (a, b)) a.e. in RI

by V.1.2, lemma 10. We noted above that

N(x, T, (a, b)) = N(x, T, [a, b]) a.e. in RI.

In view of these remarks, the present theorem appears as a direct consequence of IV .4.4, theorem 1, according to which

b

J H (T(u)) II,(u, T)I du = J H(x) K (x, T, (a, b)} dx, a


Theorem 7. Assume that the real-valued continuous function T(u) is defined and AC in the closed interval [a, b], where a<b. Let H(x)


be a real~valued, finite-valued, L-measurable function in Rl. Then

b T(b)

J H(T(u)) T'(u) au = J H(x) ax, .. T(o)


Proof. T is eA C in (a, b) and T'(u) =]. (u, T) a.e. in (a, b), as it has been noted in the proof of theorem 6. Thus if H(T(u)) T'(u) is L-summable in [a, b] and hence also in (a, b), then H(T(u))].(u,T) is also L-summable there, and

b b

J H(T(u)) T'(u) au = f H(T(u)) I.(u, T) au = f H(x) f-l (x, T, (a, b)) ax .. a

by IV.4.4, theorem 3. By V.lo2, lemma 1 and lemma 2 we have

f-l (x, T, (a, b)) = sgn (T(b) - T(a)) if x E (T(a), T(b)) ,

f-l(x, T, (a, b)) = 0 if xE C[T(a), T(b)J.

Thus obviously T(b)

f H(x) f-l(x, T, (a, b)) ax = f H(x) ax, T(o)

and the theorem follows.

If T is A C in [a, b], then it is also A C in every closed interval [a, b] in [a, bJ. Accordingly, the preceding two theorems remain valid if [a, b] is replaced by any closed interval [a, bJ ( [a, b J.

A noteworthy special case arises when H(x) == 1. Since T'(u) is L-summable in (a, b) by theorem 3, we obtain for H(x) == 1 the following statements [see also part (c) of theorem 5 and V.1A (7)].

Theorem 8. Assume that the real-valued continuous function T(u) is defined and A C in the closed interval [a, b], where a < b. Let a, b be two numbers such that a< a< b< b. Then

b b

f I T'(u) I au = V(T, [a, bJ), f T'(u) au = T(b) - T(a). a a

Observe that theorem 7 presents a lack of symmetry with respect to the two integrals involved (in contradistinction with theorem 6 which is symmetric from this point of view). Examples show that this discrepancy is unavoidable in the general case. On the other hand, a symmetric form of theorem 7 is valid in the special case when T(u) is monotone. Let us recall the following definitions. If T(u) is a realvalued function in [a, b], where a< b, then T(u) is termed strictly monotone in [a, bJ if either always T(a) < T(b) whenever a;;;;;;'a < b S;; b,

§ V.1. Continuous transformations in Rl. 317

or always T(a»T(b) whenever a;:;;;'a<b;:;;;.b. Furthermore, T(u) is said to be monotone in the large sense in [a, b] if either always T(a) ;:;;;. T(b) whenever a;:;;;'a< b;:;;;. b, or always T(a);;;;:; T(b) whenever a;:;;;'a< b;:;;;. b; in the first case, T is said to be non-decreasing in [a, b], while in the second case T is said to be non-increasing in [a, b J. Now if T is strictly monotone in [a, b], then (considered as a transformation) it is oneto-one there, and accordingly IV.4.6, theorem 2 yields a symmetric form of theorem 7 for this case. We shall show presently that a similar conclusion holds if T is monotone in the large sense.

Lemma. Assume that the real-valued function T(u) is continuous and monotone in the large sense in [a, b], where a < b. Then

N(x,T,[a,b])=O if xEf[T(a),T(b)],

N(x, T, [a, b]) = 1 a.e. in [T(a), T(b)].

(1 )

(2)

Proof. Suppose, for definiteness, that T is non-decreasing in [a, b] (the case when T is non-increasing can be settled in an entirely similar manner). If ztE [a, b], then a;:;;;'u;:;;;'b and hence T(a) ~T(u) ;:;;;'T(b). Thus clearly

T[a, b] ([T(a), T(b)],

and (1) follows. If xE [T(a), T(b)], then T(a) ;:;;;'x;:;;;'T(b). Since T is continuous, it follows that x=T(u) for some uE [a, bJ. Hence

N(x, T, [a, bJ) ~ 1 if xE [T(a), T(b)l

Consider now the auxiliary set

F = {xIN(x, T, [a, bJ) > 1}. We assert that

F (T Smax,

(4)

(5)

(6)

where Smax is the set of those points uE (a, b) where T(u) has a weak local maximum (see V.1.2, lemma 8). To verify (6), note that if xoEF, then there exist (at least) two points 1£' and u"> u' in [a, b] such that T(u') = T(u") = Xo' Since T is non-decreasing, it follows that T(u) = Xo

for u' < 1£ < 1£". Thus if uoE (1£',1£"), then clearly T has a weak local maximum (and indeed also a weak local minimum) at 1£0' Hence uoESmax and T(uo) = xo, and thus xoE T Smax. So (6) is verified. In view of (1), (4), (6) we have

N(x, T, [a, bJ) = 1 if xE [T(a), T(b)] - TSmax ' (7)

Now the set TSmax is countable (and hence of L-measure zero) by V.1.2, lemma 8. Thus (7) implies (2), and the lemma is proved.


Theorem 9. Assume that the real-valued function T(u) is continuous, A C, and monotone in the large sense in the closed interval [a, b J, where a< b. Let H(x) be a real-valued, finite-valued, L-measurable function in RI. Then

b T(b)

J H (T(u)) T(u) du = J H(x) dx, (8) a T(a)


Proof. The two cases when T is non-decreasing and non-increasing respectively can be treated in an entirely similar manner, and so we consider explicitly only the case when T is non-increasing in [a, b J. If the integral on the left in (8) exists, then (8) holds by theorem 7. There remains to verify (8) for the case when H(x) is known to be L-summable in [T(a), T(b)J. In view of the preceding lemma, we have

H(x) =H(x)N(x, T, [a, bJ) a.e. in [T(a), T(b)] (9)

and thus H(x) N(x, T, [a, bJ) is also L-summable in [T(a), T(b)J. By theorem 6 we have therefore

a

JH(T(u)) IT(u)ldu=JH(x)N(x,T, [a,bJ)dx. (10) b

Now since T is non-increasing, clearly

T'(u):;;;;: 0 a.e. in [a, b], and furthermore

T(b) :;;;;: T(a).

(11 )

(12)

Since N(x, T, [a, b]) = 0 for xE/: [T(a), T(b)] by the preceding lemma, we have Tlb)

J H(x) N(x, T, [a, b]) dx = - J H(x) dx T(a)

( 13)

in view of (9) and (12), and b b

J H(T(u)) I T(u)1 du = - J H(T(u)) T(u) du (14) a a

in view of (11). The formula (8) now follows from (10), (13), (14).

In the special case when T is strictly monotone, theorem 9 yields the classical transformation formula for single integrals. Theorem 7 may be now interpreted as revealing the fact that the classical transformation formula remains valid even if T is not monotone, provided that the integral obtained by the substitution x = T(u) exists. The assumption of absolute continuity on the part of T is essential, as simple examples show.

§ V.2. Local approximations in RH. 319

§ V.2. Local approximations in RD.

V .2.1. Additional comments on linear transformations inRD. Various basic facts relating to linear transformations in RH have been already discussed in 11.2.5. According to the terminology adopted there. a linear transformation 2 from ~ into RH is given by equations of the form n

2:xi = bi + L aijUi • i = 1 •...• n. j=l

(1 )

where bi • ai ,. are real cons tan ts and x = (Xl. .... xH ) is the image of U= (ul ..... ~11) under 2. The determinant of the matrix (aii) is termed the determinant of 2 and is denoted by det 2. Thus

det 2 = det (ai ;). (2)

Recall that 2 is termed non-singular or singular according as det 2 =F 0 or det 2 = O. If 2 is non-singular. then (see 11.2.5) 2 is a homeomorphism from Rn onto RH. and its inverse 2-1 is again a non-singular linear transformation. In the following discussion. it will be convenient to use'the vector notation explained in 1.2.1. Given 2 as in (1). put

where max laiil denotes the largest one of the numbers laii!' Lemma 1. Let 2 be given as in (1). If

are any two points in Rn. then

Proof. Put

Then (1) yields

and hence

n

IIx2 - xlll:s;; L Ix~ - x~l;;;;: IIu2 - ulll n2 max !aul i=l


(,)

(4)


The preceding lemma shows that 53 is Lipschitzian in Rn (see V.2.3, definition 1). The constant A (53) could be obviously replaced by a smaller one. However, the advantage of the choice (3) for .1(53) is that this constant is always positive.

If 53 is non-singular and n

53-1 ;Ui = bi+1:,tiii xi , i=1, ... ,n, 1~1

is the inverse of 53, then we put [in conformity with (3)]

A (B-1) = 1 + n2 max laill. Application of lemma 1 to B-1 yields the following statement.

(5)

Lemma 2. Assume that B is non-singular, and let u1 , U 2 be any two points in Rn. Then

(6)

Given 53 as in (1), select a point uo=(u~, ... , u~) and put

53 uo = x 0 = (x~, ... , x~) . (7)

Let u = (u1, •.. , un) be a variable point in Rn. Put

(8)

The point U o is thought of as fixed. From (1), (7), (8) we obtain for B the representation

(9)

which is convenient in studying B in the vicinity of an assigned point uo'

Lemma 3. Assume that 53, given as in (9), is singular. Then there exist (constant) vectors

bi = (bi, ... , bj), j = 1, ... , n, such that

(i) II bjll = 1, j = -1, .. _ , n, (ii) bi • bj = 0 for i =4= j, (iii) (53. u - xo) • bn = ° for every u ERn.

(10)

Proof. In (ii) and (iii), the indicated products are scalar products (see 1.2.1), and in (iii) the point Bu- Xo is interpreted as a vector. Consider the system of linear equations

n

Laij!Ji=O, j=1, ... ,n, (11) i=l

§ V.2. Local approximations in R". 321

where the coefficients aij are taken from (9), and {p, ... , fJ" are the unknowns. Since 2 is singular, the determinant of the system vanishes. Thus the system has a non-trivial solution, and hence it also has a solution {P, ... , /r such that the sum of the squares of pI, ... , P" is equal to 1. For the vector

b,. = (fJl, ... , P")

we have then Ilbn l\=1, and from (11), (9), (8), (7) it follows by direct calculation that (iii) holds. Since b" is a unit vector in R", we can select (see 1.2.1, exercise 11) vectors bl , ... , bn- l such that (i) and (ii) hold, and the lemma is proved.

Lemma 4. Assume that 2, given as in (9), is non-singular. Let aI' ... , an be the vectors given by the formulas

aj=(alj, ... ,anj), j=1, ... ,n, (12)

where the components aij are taken from (9). For each positive number fJ, denote by Q (fJ) the oriented n-cube with center at U o and side length 2fJ (see 1.2.2). Put

P(fJ) = 2 Q (fJ) . (13)


(i) P(fJ) is the n-parallelotope with center Xo = 2uo and edge vectors 2fJ aI' ... , 2fJ an (see 1.2.2).

(ii) L P(fJ) = 1 det 21 L Q (fJ)· (iii) The parameter of regularity n(P(fJ)) of P(fJ) (see 111.1.1,

definition 3) satisfies the inequality

( (fJ) Idet 21 :z P ) ~ nnA(2)n'

Proof. The points u = (~tl, ... , un) E Q ({J) are characterized by the fact (see 1.2.2, exercise 2) that there exist real numbers AI"'" A.,. such that

ui - ui = 2 Aj {J , j = 1, ... , n,

IA,I;:;;;;1/2, j=1, ... ,n.

In view of (9) and (13) it follows that the points

x = (Xl, ... , x") E P({J)

are characterized by the fact that there exist real numbers ~, ... , An such that

n

Xi = X~ + L, 2 Aj {J ai i ' i = 1, ... , n, i=l

Rado and Reichelderfer J Continuous Transformations. 21

(14)

(15)


In view of (12), the summation in (14) is equal to the i-th component of the vector

Thus the equations (14) are equivalent to the vector equation

" % = %0 + L, 2 Ad] aj. (16) j=1

By definition (see 1.2.2, exercise 4), the relations (16) and (15) yield precisely the points % of the n-parallelotope with center %0 and edge vectors 2{Ja1 , .•• , 2{Jan , and thus (i) is proved. From (i) it follows that the matrix of the components of the edge vectors of the parallelotope P({J) coincides with the matrix (2{Ja i j)' Hence (see 111.1.1, lemma 16)

L P({J) = Idet (aij) I 2n {l". (17) Now since

L Q ({J) = 2n (In,

and det(aij) =detB by (2), the relation (ii) follows from (17). Consider now any point

In view of (13) there exists a point

u = (ul, ... , un) E Q ((3)

such that %= Bu. From (9) and (3) it is obvious that

Ixi-x~l;;;;;:n{JA(B), i=1, ... ,n.

Thus P({J) is contained in the oriented n-cube with center Xo and side length 2n{JA (B)~ The volume of this cube being equal to

n H A (Bt 2n {3" = n" A (B)n L Q({J),

we conclude from (ii) and 111.1.1, definition 3 that

n(P{{3)) ~ nnA~B~n(~)Q(lf) = ,::~~ln' and the proof is complete.

V.2.2. Local linear approximations in Rn t. Let there be given a bounded continuous transformation

T:D-+R", (1 )

t The first systematic use of the topological index in conjunction with total differentials has been made by H. RADEMACHER, tJber partielle und totale Differenzierbarkeit von Funktionen mehrerer Variabeln [Math. Ann. 79, 340-359 (1919) and 81, 52-63 (1920)].


where D is a bounded domain in Rn. We can also represent T in the form (see 11.2.5)

(2)

where fI" .. , f,. are real-valued, bounded, continuous functions in D of the coordinates u I , ... , un of the point

and Xl, ... , xn are the coordinates of the image point

Select a point U o = (u~, ... , u~) ED,

and put Tuo = Xo = (x~, ... , x~). (4)

We shall discuss presently certain rather elementary facts which relate to the behavior of T in the vicinity of the assigned point uoE D. To avoid trivialities, we assume throughout this section that n> 1 (note that the case of continuous transformations in RI has been fully discussed in § V.l). Assume that the first partial derivatives of the coordinate functions Ii of T [see (2)J exist at UO' and put

_ Ofi(uo) "-aij --,,-.-, t,] -1, ... , n. uu1

Introduce the linear transformation [see V.2.l (9)] n

£:x' - x~ = L aij(u i - ui), i=I

The determinant of the matrix

(a .. ) = (0 t i ("!;ol) '7 ou7

i=1, ... ,n.

(5)

(6)

is termed the ordinary Jacobian J (uo, T) of Tat uo. Thus the ordinary Jacobian exists at a point uoED if and only if the first partial derivatives of the coordinate functions of T exist at that point. In case J(uo, T) exists, we have [see V.2.l (2)J

J(uo, T) = det £. (7)

As stated above, we assume now that J (uo, T) exists at the assigned point U o ED. Observe that the summation in (6) represents the differential of the coordinate function fi at uo, in the sense of 111.1.3. Accordingly, the vector whose i-th component is equal to the summation in (6) ma,y be considered as the differential of T at uo' Let us

21*


introduce an auxiliary vector-valued function fu of u by the formulas

f u = T u - ~ for u ED, u =1= uo, Ilu - uoll fuo = 0,

(8)

(9)

where the 0 in (9) stands for the zero vector (0, ... , OJ. From (8) and (9) there follows the formula

Tu = 2u + Ilu - uollfu, uE D. (10)

If ex is any positive number, then Q (ex) will denote the oriented n-cube with center at U o and side length 2ex (see V.2.l). Thus the relation

(11)

is equivalent to the relations

IUi-u~l~ex, i=1, ... ,n. (12)

Since uoE D and D is open, we can select a number exo> 0 such that

( 13)

In the sequel it will be assumed throughout that ex is so chosen that (13) holds. We define

y(ex} =maxllfull, uEfrQ(ex),

r(ex) =l.u.b.llfull, uEQ(ex).

(14)

( 15)

Since T is continuous in D, clearly Ilfull is continuous at every point u=l=uo in D. Thus the use of max Ilfull in (14) is legitimate. Clearly Y (IX) is finite and non-negative. On the other hand, Ilfu II is generally not continuous at uo, and r(ex} may be equal to +00. Obviously

(16)

Observe that y(ex}, for example, depends also upon T and uo' However, T and U o are thought of as fixed, and thus it is unnecessary to use more involved notations like y(IX, T, uo}.

Definition 1. T is said to possess a total differential at the point uoED if (a) the first partial derivatives of the coordinate functions of T exist at Uo and (b) r(IX) -+0 for ex-+O.

Definition 2. T is said to possess a 7PJeak total differential at the point uoED if (a) the first partial derivatives of the coordinate functions of T exist at U o and (b) there exists a sequence

exl > ... > IX", > ... > 0, IX", -+ 0 ,


Note that, as a direct consequence of (16), if T possesses a total differential at uo, tl;len T also possesses a weak total.differential at uo'

Our objective in this section is to study the kind and degree of approximation to T by S3 in the vicinity of U o [see (5), (6)]. The following situations are relevant for subsequent applications.

Situation 1. T possesses a total differential at uo, and J(uo, T) =0. Under these conditions, we assert that

lim L TQ(/X) = o. ", .... 0 L Q (/X)

(17)

To verify this relation, note that the assumption] (uo, T) = 0 implies, by (7), that S3 is singular. By V.2.1, lemma 3 we can therefore select (constant) vectors bI , ... , b" such that

II bi II = 1 , j = 1, ... , n,

b, • bi = 0 for i =f= j

(S3 u - xo) . bn = 0 for u E R".

(18)

(19)

(20)

Let b}, ... , bi be the components of bi , j = 1, ... , n. The relations (18) and (19) imply (see 1.2.1, exercise 10) that

(21)

Thus the vectors bI , ... , b" are linearly independent, and hence every vector' v in R" can be represented in the form

" v = L Aibi'

i=I (22)

where AI, ... , An are real numbers. From (19) and (22) it follows that Ai=v' bi' and hence (see 1.2.1, exercise 12)

Since II bill = 1, there follows the inequality

(23)

Consider now any point uE Q(oc). (24)

For the vector Tu - S3u we have a representation of the form

" Tu-S3u=LAjbi · (25) i=1


From (10) and (15) it follows [in view of (23)J that

IAil ~ II T u - 53 ull ;$; Ilu - uoll r(~). Since Ilu-uoll~n~ by (24), we conclude that

IAil ;$;nocF(~), j=1, ... ,n. (26)

Similarly, for the vector 53u - Xo we have a representation of the form


and hence

A: = (53 u - xo) . bn = 0,

n-l

53 u - Xo = L At bi · i~1

Since Xo= 53uo' lemma 1 in V.2.t yields

II 53u - xoll ~ A (53) IIu - uoll·

(27)

As Ilu-uoll~n~ by (24), we obtain in view of (23) the inequalities

IAjl~nA(53)~, j=1, ... ,n-1. (28)

From (25) and (27) we obtain, by addition, n-l

T u - Xo = L (Ai + Ai) bi + ;.: b". (29) i~1

By (26) and (28) we have

IAi+;';I~12oc(A(£)+r(~)), j=1, .... 12-1, (30)

(31 )

Assign 8> O. Denote by p.*(~) the parallelotope with center at Xo

and edge vectors

212~(A(£) +r(oc)) b1 , ... , 2noc(A(£) +r(oc)) b"-I' 2noc(8+r(~)) b".

From (29), (30), (31) we conclude that TUEP.*(oc). Since u was an arbitrary point of Q (oc), it follows that

and hence T Q (~) ( P.* (~) ,

L T Q (~) ~ LP'*(~). (32)

§ V.2. Local approximations in Rfl.

In view of (21) and m.1.t. lemma 16. we have

Since LQ(oc) =2f1 oc", and F(oc)-+o for oc-+O, (32) and (33) yield

limsup LTQ(ot) ':::;;:nnA(~t-le. "'-..0 L Q (ot)

As e> 0 was arbitrary. (17) follows.

327

Situation 2. T possesses a weak total differential at uoand J(uo,T) +0. Then there exists (see definition 2) a sequence {ex",} such that

Consider any sequence {oc",} satisfying (34). Choose a sequence {17",} such that

(35)

and put (36)

Note that the assumption J(uo, T) +0 implies, in .view of (7), that ~ is non-singular. and hence the positive constant A (~-l) is available [see V.2.t (5)]. In view of (34) and (35) the inequalities

(37)

will hold for m sufficiently large. Discarding, if necessary, a finite number of terms in the sequences (34) and (35), we can therefore assume that (37) holds for every integer m~1. Put

oc;" = (1 - J ..... ) iXrn'

oc~ = (1 + A.m ) oc'" .

(38)

(39)

Consider the oriented n-cubes Q (oc~), Q (oc",), Q (oc~) with center at U o and side length 2oc;,., 2oc"., 2oc;;, respectively. Since oc'" -+ 0 and ;.". -+ 0, these cubes will be contained in D for m sufficiently large. Discarding, if necessary. a finite number of these cubes, we can therefore assume that this is the case for every integer m ~ 1. We have then the inclusions (see 1.1.5. exercise 10)

Uo E int Q (oc;n) , Q (oc;,,) ( int Q (oc",). }

Q (oc",) ( int Q (oc;;,), Q (oc;;,) ( D. (40)


Let us put

The following diagram, which illustrates the case n = 2, may help the reader in understanding O the discussion. [ 1 In this diagram, the three concentric squares and the three concentric parallelograms corre-

Fig. 44. spond to Q (~), Q (0:",), Q (O::n) and p~, P"" P,:

respectively. The following discussion is concerned, of course, with the situation for the case of an arbitrary n. From V.2.1, lemma 4 we infer the following facts.

(i) P,~, Pm' P~ are parallelotopes with center xo. (ii) The volumes LP~, LPm , L P~ of these parallelotopes satisfy

the relations [see (7)]

L~~;;J = L~~:'n) = L ~-TJ;,.) = l](uo, T)l·

(iii) The parameters of regularity of P,;" P,~ satisfy the inequality [see (7)]

(P.' ) (P.") '- Jl (~o~ I_ n m' n m ,,;;;;. nn A ('\!)" .

Since ~ is non-singular as a consequence of the assumption ](uo, T) =j=O [see (7)], ~ is a homeomorphism from R n onto Rn, and hence (see 1.1.5, exercise 10) the inclusions (40) imply that

We assert that LP'

LQ(:m) --+IJ(uo, T)I,

for m--+ 00. Indeed, by (ii) we have

Since [see (38)]

L Q (o:~) = 2n 0:::. (1 - Amr = (1 - Am!" L Q (0:",),

(42)

(43)

and Am--+O, the first relation in (43) follows. The second relation in (43) is verified in a similar manner, using (39). Next we assert that

(44)

§ V.2. Local approximations in Rn. 329

for m~oo. Consider any two -points x~, x~EP;. In view of (41), there exist points u~, u~ such that

From V.2.1, lemma 1 we infer that

As x~, x; were arbitrary points of P~, it follows that

op'~ :s:: 2n IX: A (.2),

and (since IX~ ~O) the second relation in (44) is established. The first relation in (44) is verified in a similar manner. We proceed to discuss the topological index fl (x, T, int Q (IXm)) (see 11.2.2). We shall establish the fundamental relations

( T . Q( )) = {sgn](uo, T) = ± 1 if xEP~, } fl x, ,lnt IXm ." o If xE CPm .

(45)

The proof of (45) requires several steps. First we verify that

T Ir Q (tXm) ( P; - P~. (46)

Since 53 is a homeomorphism, the inclusion (46) is equivalent [in view of (41)J to the inclusion

(47)

In turn, the inclusion (47) is implied, in view of (38) and (39), by the following statement: if u is any point such that

u E fr Q (IXml. then

1153-1 Tu - ull < AmlXm ·

(48)

(49)

This latter statement is proved as follows. From V.2.1, lemma 2 we obtain the inequality

1153-1 Tu - ull ~A(53-1) IITu - 53ull· Since [in view of (10), (48), (14)J

II T u - 53 ull ~ Ilu - uolilif ull ~ nlXm y(ocm),

by (36) it follows that

1153-1 T u - ull ~ nA (53-1) y (tXm) OCm < Am OCm'


Thus (49) holds, and hence (46) is established. From (46) we conclude that

c P~ ( C T tr Q (1Xm) ,

P~ ( C T tr Q (1Xm) .

(50)

(51 )

These inclusions imply (see 11.2.2) that ft (x, T, int Q (IXm)) is actually defined on C P~ and on p,~. Observe now that the sets C Q (IX:) and Q (IX;") are connected. Hence, by 1.1.5, exercise 8, the sets

are also connected. Accordingly, by 11.2.3, remark 1, the inclusions (50) and (51) imply that

ft(x, T, int Q(am)) is constant on CP;:', (52)

ft (x, T, int Q (a",)) is constant on P~. (53)

Since the set CP~; is unbounded, from (52) and 11.2.3, remark 3 we conclude that

ft(x, T,intQ(a",)) =0 for xECP~.

To establish (45), there remains to show that

ft (x, T, int Q(am )) = sgnJ(uo, T) = ± 1 if xEP~. (54)

Note that since xoEP~, we have by (53) the relation

ft(x,T,intQ(am))=f.1(xo,T,intQ(a",)) for xEP~. (55)

Since 5:\ is a linear transformation and [see (41), (42)J

Xo E int P,~ = 5:\ int Q (a;,,) ( 5:\ int Q (am) ,

we conclude from (7) and 11.2.5, theorem 4 that

ft (xo, 5:\, int Q (am)) = sgn det 5:\ = sgn J(uo, T) = ± 1, (56)

where we used the fact that J(uo, T) =FO by assumption. We shall now verify the relation

f.1( xo, T, int Q (a",)) = ft (xo, 5:\, int Q (am)) . (57)

For u E tr Q (am) we have [see (10) and (14) J

II T u - 5:\ull s Ilu - uoll Y(IXm) ;;;. nam Y(O'-m)·


Hence (see 1.1.5, definition 5)

(58)

Consider now any point xE ~ Ir Q (rx",). Then there exists a point u such that

(59)

Lemma 2 in V.2.t yields

On the other hand, (61)

since uElrQ(rx",) and u.o is the center of Q{rx",). From (60) and (61) we conclude that

Since x was an arbitrary point of the set ~ Ir Q (a",), there follows the inequality (see 1.1.4, definition 7)

e (xo, B Ir Q (am));;;;' A ~-l) . (62)

Comparison of (62) and (58) yields, in view of (37),

(63)

By II.2.3, remark 8 the inequality (63) implies (57). Finally, (55), (57) and (56) imply (54), and thus (45) is proved. An important inference is the inclusion

P,~ (T (int Q(am)) , (64)

which follows from (54) by II.2.3, theorem 2. Next we assert that U o by itself is an m.m.c. for (xo, T, D) (see II.3.1, definition 1). Indeed, since Tuo=xo' there exists a component C of T-1xO such that uoEC. We must have

C ( int Q (am) . (65)

Otherwise, we should have (see 1.1.3, exercise 17) C n Ir Q (am) =1= 0, and hence also

(TC) n (TlrQ(rx",)) =1=0. (66)

However [see (46)]

T C = Xo EP:', T Ir Q(rx",) (CP:'.


Thus (66) cannot hold, and hence (65) is established. As IXm -i>-O, clearly (65) implies that C reduces to the single point u o. Accordingly, U o by itself is an m.m.c. for (xo, T, D). Consider now any open set 0 such that uoEO(D. As IXm-i>- 0, the inclusion Q(IXm) (0 will hold for m large enough. Now since

.u (xo, T, int Q (IX".)) = sgn ](uo, T) = ± 1, (67)

by (45), the domain int Q (IX",) is an indicator domain for (xo, T, D), in the sense of 11.3.2. In view of 11.3.3, definition 1 it follows that U o by itself is an e.m.m.c. for (xo, T, D). Note that U o mayor may not be essentially isolated in the sense of 11.3.3, definition 2. Assume that U o is essentially isolated. We have just seen that if any open set 0 containing U o is assigned, then for m sufficiently large Q (IXm) (0. In view of (67) and 11.3.4 it follows that

ie(uo, T) = sgn J(uo, T) = ± 1, (68)

provided that Uo is essentially isolated. Note next that U o mayor may not possess a neighborhood clear of relatives (see 11.3.7, definition 2). Assume that U o does possess such a neighborhood. Since we have already shown that U o by itself is an e.m.m.c. for (xo, T, D), it follows from 11.3.7 (3) that is (uo, T) =ie(uO' T). In view of (68) there follows the formula

is (uo, T) = sgn ](uo, T) = ± 1. (69)

Recall that in deriving (69), we had to assume that U o possesses a neighborhood clear of relatives. If this condition is not fulfilled, then (69) is false, since then is(uo,T)=O by 11.3.7 (1).

Situation 3. T possesses a total differential at U o and ](uo, T) =1=0. Consider any sequence

subject only to the condition that 1X1 is so small that Q (IXI) is contained in D [see (11), (12), (13)]. Then (see definition 1)


r(IXm) - Y (IX",) :;;;;; 0,

Y(lXm) -i>- 0 for m -i>- 00.

Let us put 1

1]", = - + r(IX",) - Y(IXm) , m = 1, 2, .... rn

(70)

(71)

(72)

(73)


Then, by (70), (71), (72), 'YJm> 0, 'YJm--+O. Thus all the assumptions used in the study of Situation 2 are fulfilled, and hence the terminology employed and the results derived there are again available. The numbers Am [see (36)] are now given, in view of (73), by the formula

(74)

The cubes

and the parallelotopes P';', Pm, P:;' are defined as in Situation 2. The new feature in the preseht Situation 3 is the validity of (70). On this basis we shall derive presently the following relations

P';' ( T(int Q (am)) (P'~,

is (uo, T) = sgnJ(uo, T) = ± 1.

(75)

(76)

The first inclusion in (75) is merely a re-statement of (64). The second inclusion in (75) is equivalent to the statement that

u E int Q (am) implies T u E P:;'. (77)

In turn, (77) is equivalent to the statement that

u E int Q (am) implies 5:\ -1 T u E 5:\ -1 P:;' = Q (a;;.) •

In view of (39), this last statement will be established if we show that

115:\-lTu-ull;;;;Amam if uEintQ(am). (78)

To verify (78), note that

115:\-1 Tu - ull;;;; A (5:\-1) IITu - 5:\ull

by V.2.t, lemma 2. Since uEint Q(am ), we infer from (10) and (15) the inequality

IITu- 5:\ull ;;;;llu-uollr(am ) ;;;;nrJ..mr(rJ..m )·

Hence, in view of (74)

IIB-1 T u - ull ;;;; nA (B-1) r(rJ..lII ) rJ..m < Am rJ..m.

Thus (78) is verified, and hence (75) is proved. As regards (76), in view of the comments made in deriving (69) we have only to show that U o possesses a neighborhood clear of relatives. In turn, this fact will be established if we show that if rJ.. > 0 is sufficiently small, then

(79)


Observe that r(rx) -+0 for rx-+O, since by assumption T possesses a total differential at Uo. Accordingly, we can choose rx > 0 so that

Consider any point u such that

uE Q(rx) , u =f= Uo·

Then, by (10), (15), and (SO),

IITu - £ull:;;;; Ilu - uollr(rx) < I~(;~)II .

On the other hand, by V.2.1, lemma 2,

lIu - uol! ;S A (£-1) !!£u - xol!.

(SO)

(S1)

There follows the inequality lI£u-xoll> I!Tu-£ull. Hence, by the triangle inequality,

IITu - xo!! ~ I!£u - xo!! -1!Tu - £u!! > o.

Thus (S1) implies that TU=f=Xo, and the proof of (76) is complete. We noted above that all the results derived for Situation 2 are available in the present Situation 3. In particular [see (43)J, we have again

1· LP';' IJ( T)I 1· LP';; un --- = U o = 1m --~ m_oo LQ{lXm) , m_oo LQ{lXm} •

(S2)

V.2.3. Preliminary study of Lipschitzian transformations in R'". Let U =f= 0 be an arbitrary subset of R". Consider a transformation

(1 )

We shall also use for T the representation

T: xi = Ii (uI , ... , Un), i = 1, ... , n, (2)

where 11' ... , In are real-valued functions of the coordinates ul, ... , un of the point uE U, and x!, ... , xn are the coordinates of the image point x=Tu.

Delinition 1. T is said to be Lipschitzian on U if there exists a (finite) constant M~O such that

for every pair of points uI , u 2E U.


Lemma 1. T is Lipschitzian on U if and only if the coordinate functions 11, ... , I" of Tare Lipschitzian on U in the sense of llI.1.3.

Prool. Consider any two points

of U, and let

T U l = Xl = (xl, ... , x~), T u2 = X 2 = (x~, ... , X;)

be their images under T. Then clearly

n

IITu2 - Tulll ~L Ix~ - xii· i~l

(4)

(5)

Assume first that T is Lipschitzian on U, and let M be a constant such that (3) holds for every pair of points ul , u2E U. From (2), (3), (4) it follows then that

and thus 11, ... , In are Lipschitzian on U. Assume, conversely, that 11' ... , In are Lipschitzian on U. Then there exist (finite) constants 111;;;:;: 0 such that

1/,(u2) - Ii (U1) I :;;;;111, IIU2 - ulll, i = 1, ... , n, (6)

for every pair of points u1 , u2E U. On setting

M = Ml + ... + 111,,,

there follows from (5) and (6) the inequality

IITu2 - Tu1 11:;;;; Mllu2 - ull!, and thus T is Lipschitzian on U.

Let us now consider a bounded continuous transformation

T:D --+Rn , (7)

where D is a bounded domain in Rn. Corresponding to (2) we have the alternative representation

T:xi = Ii (ul , ... , un), i = 1, ... , n, (8)

where 11' ... ' In are now real-valued, bounded, continuous functions of the coordinates u1, ... , un of the point U ED.


Lemma 2. Assume that T, given as in (7) and (8), is Lipschitzian in D. Then the first partial derivatives of the coordinate functions 11' 00', In of T exist a.e. in D, and T possesses a total differential a.e. in D.

Prool. By lemma 1 the coordinate functions of Tare Lipschitzian in D. Accordingly, by 111.1.3 the first partial derivatives of the coordinate functions exist a.e. in D, and each one of the coordinate functions possesses a total differential a.e. in D. Thus the lemma will be proved if we verify the following statement: if U o = (u~, ... , u~) is a point in D such that each one of the coordinate functions of T possesses a total differential at uo, then T also possesses a total differential at u o. To prove this statement, put

aij= 8t~~~oL, i,i=1,oo.,n.

For each integer i = 1, ... , n, put

n

C{Ji(U) = Ilu~ ul [/i(u) -li(uo) - .2.: aij(ui -- ul)] (9) o. ,-I

C{Ji(UO) = O. (10)

Assign S > O. Since Ii possesses a total differential at U o, there exists (see 111.1.3) a number 'Y}i=rli(s»O such that

Define 'Y}* by the formula

Then clearly

* _ 1 . ( ) 'Y} --mm 'Y}1' ... ,'Y}" . n

(11 )

For each positive number !x, denote by Q (!X) the oriented n-cube with center at U o and side length 2rx. Choose !Xo>o so small that

Consider the linear transformation [see V.2.2 (6)J

i=1, ... ,n.


Assume that O<<X<<Xo• and let u=l=uo be any point in Q(<x). Then

Ilu-uoll~n<x<n<Xo<n'fJ*:;;;''fJi' i=1 •...• n. (12)

Clearly the i-th component of the vector [see V.2.2 (8)]

Tu-i\u I u = -nu=-UJI

is equal to CPi(U). Hence. by (11) and (12).

" Il/ull ~ L ICPi (u) I < n B. ;=1

In view of V.2.2 (15) there follows the inequality

r(<x) ;;;;;nB if O<<X<<Xo.

As n is fixed and B> 0 was arbitrary. it follows that r(<x) -+0 if <x-+O. and thus T possesses a total differential at Uo (see V.2.2. definition 1).

Lemma 3. Assume that T. given as in (7). is Lipschitzian in D. Then the following holds.

(i) T satisfies the condition (N) in D (see IV.1.4). (ii) If S is any L-measurable set in D. then the image set TS is

L-measurable. (iii) There exists a (finite) constant K ~ 0 such that

LTS~KLS

for every L-measurable set SeD.

(iv) Tis ACB in D (see IV.S.2).

( 13)

Proal. Take any L-measurable set S (D. Assign B> O. By 111.1.1. lemma 14 there exists a sequence of oriented n-cubes Q1.···. Qk • ... such that

S ( U Qk ( D. L L Qk < L S + B. (14)

Since T is Lipschitzian in D. there exists a (finite) constant M;;;;:O such that

( 15)

for every pair of points u1 • U 2 in D. Let Uk be the center and 2IXk the side length of Qk. Then

Ilu -- ukll ;;;;;nlXk for uE Qk.

and hence. by (15).

IITu-Tukll;;;;;nMO:k for UEQk. Rado and Reichelderfer, Continuous Transformations. 22


Thus clearly T Qk is contained in the oriented n-cube with center at T Uk and side length 2 nM (Xk . Hence

L T Qk ::;;: (2nM (Xk)n = nn M n L Qk'


TS( U TQk'

L: LTQk::;;: nn M"(LS + s).


{16}

(17}

(18)

where L* denotes n-dimensional exterior measure in R n (see 111.1.1). Since s> 0 was arbitrary, it follows that

L*TS ;:;;;nnM"LS. (19)

To prove (i), assume that L 5 = O. Then T 5 is of L-measure zero by (19), and (i) follows. Once we know that (i) holds, (ii) follows directly from IV.1.4, corollary 2. Let now 5 be any L-measurable set in D. Then, since TS is L-measurable by (ii), we can replace (see 111.1.1) L* by L in (19), and (13) follows. To prove (iv) , consider any finite system of n-intervals II' ... , 1m in D such that

int I, n int I k = 0 for j =f= k.

By (13) we have then

L: LT(int I k ) ;:;;; KL: L(int I k ) ;:;;; K LD < 00,

and thus T is B VB in D (see IV.S.l, definition 1). Since T satisfies in D the condition (N), as we already proved, it follows by IV.S.2, theorem 3 that T is A C B in D.

Lemma 4. Assume that T, given as in (7), is Lipschitzian in D. Then

I](U, T)I ;:;;;DB(u, T) a.e. in D.

Proof. Note first that the ordinary Jacobian ](u, T) exists a.e. in D and T possesses a total differential a.e. in D by lemma 2. Note also that the derivative DB (u, T) exists a.e. in D by part (iv) of lemma 3 and IV.S.l, theorem 3 (see also IV.S.2, theorem 1). Accordingly, we have a decomposition D=sUS, such that Ls=O, and at every point u E 5 the ordinary Jacobian ](u, T) exists, the derivative DB (u, T) exists, and T possesses a total differential. Thus it is sufficient to


show that (20)

Now (20) is obvious if ](uo, T) =0. So we can assume that ](uo, T) =1=0. Then we have the Situation 3, studied in V.2.2, before us. Consider the sequence of oriented n-cubes Q (oc",) and the parallelotopes p'~ described there. According to V.2.2 (75) we have the inclusion

P';' (T(int Q (am)).

For the crude multiplicity function N(x, T, int Q(am)) there follows the inequality

N(x,T,intQ(am)):;;;;1 for xEP';'.

Hence

Since DB (uo, T) exists, the term on the left in (21) converges to DB (uo, T) for m--+ 00 [see IV.5.1 (8)]. By V.2.2 (82) the term on the right in (21) converges to 1](uo,TJI for m--+oo. Thus (21) yields, for m--+oo, the inequality (20), and the lemma is proved.

Lemma 5. Assume that T, given as in (7), is Lipschitzian in D. Let S be any L-measurable set in D. Then the relation

LTS=O (22)

holds if and only if

](u, T) = 0 a.e. on S. (23)

Proof. Note that T is A C B in D by lemma 3. Hence T is also B VB in D by IV.5.2, theorem 1. Assume now first that (22) holds. Then

DB(u, T) = 0 a.e. on S

by IV.5.1, theorem 5, and (23) follows by lemma 4. Assume, conversely, that (23) holds. Since T possesses a total differential a.e. in D by lemma 2, we have a decomposition

S = sUS', (24)

such that Ls = 0, (25)

and J(u, T) vanishes and T possesses a total differential at every point u E S'. Thus we have before us the Situation 1, described in V.2.2, at every point u E S'. Denote by Q (a, tt) the oriented n-cube with

22*


center at u E 5' and side length 2et. Then (see V.2.2, Situation 1)

lim LTQ(OI:,u) =0 for uE5'. " .... 0 L Q (01:, u)

Accordingly, for every point u E 5' we can select an oriented n-cube Qu such that

E · t Q QeD L T Qu u In u' u ,- L Qu < s,

where s> 0 is arbitrarily assigned. For fixed s> 0, denote by FE the family of all those oriented n-cubes Q which satisfy the conditions

QeD, LTQ --CQ- < s.

From the preceding remarks we conclude that the cubes QE Fe cover 5' in the manner required in the VITALI covering theorem. Accordingly (see 111.1.1, lemma 6), we can select from FE a sequence of cubes QI' ... , Qk' ... , such that

int Q; n int Qk = 0 for j =l= k,

L(S' - Uint Qk) = o.

Thus we have a decomposition

5' = s' US", such that

Ls' =0 and

Note that

(26)

(27)

(28)

(29)

since Qk E FE. Furthermore, since T satisfies the condition (N) in D by lemma ), the relations (25) and (27) imply that

LT s = 0, LT s' = o. ()O)

Combining (28), ()O), (26), (24), (29), we obtain

L T 5;;:;;; L L T Qk;;:;;; s L L Qk ;;:;;; s L D.

As s> 0 was arbitrary, (22) follows, and the lemma is proved.

V.2.4. Local Lipschitzian approximations in Rn. In V.2.2 we have shown that if the transformation T:D-+Rn satisfies certain conditions at a point uoE D, then there exists a linear transformation 5:$ (which


depends upon both T and u o) which yields a useful approximation to T. In the present section we shall show that if T satisfies a certain condition on a set SeD, then a useful approximation to T may be obtained in terms of a Lipschitzian transformation.

Lemma 1. Let U =l= fJ be a subset of R", and let g(u) be a real-valued function defined for u E U. Assume that g is Lipschitzian on U (see 111.1.3). Then there exists a real-valued function g*(x) with the following properties.

(i) g*(x) is defined and finite-valued for every point x ERn. (ii) g* (x) is Lipschitzian in RH. (iii) g* (u) = g (u) for u E U.

Briefly, if g is Lipschitzian on U, then there exists a Lipschitzian extension g* of g to all of R".

ProojI. By assumption, there exists a (finite) constant A:;;;; 0 such that

(1 )

for every pair of points ul , u2 E U. Define, for xE Rn ,

g*(x) = l.u.b. [g (u) - A II u - xii]. uEU

(2)

We proceed to verify that g* satisfies the requirements (i), (ii) , (iii). Choose a point 11,0 E U. Let x be any point in RH. For 11, E U we have then [see (1)]

g(u) - A Ilu - xii = g(uo) + (g(u) - g(uo)) - A Ilu - xii ;;;;; ;;;;; g(uo) + A 1111, - uoll- A 1111, - xii·

Since A:;;;; 0 and

by the triangle inequality, it follows that

g(u) - Allu -- xii ;;;;;g(uo) + Allx - uoll for 11,E U. (3)


(4)

On the other hand, (2) yields

(5 )

1 This method of proof. is due to E. J. MCSHANE, Extension of range of functions [BulL Amer. Math. Soc. 40, 837-842 (1934)]. The paper just referred to contains comments on further literature.


From (4) and (5) we see that g*(x) has a finite value at each point x E R", and (i) follows. Next we verify (ii) by showing that

(6)

for every pair of points Xl' X2 E R". Select any two points Xl> X2 E R". Let the notation be so chosen that

(7)

Assign 8>0. In view of the defining formula (2), there exists a point u* E U such that

g*(XI) ::;;; g (u*) - A jju* - xIII + 8.

On the other hand, (2) yields

g*(x2) :2:g(u*) - A IIu* - x21j·

From (7), (8), (9) we obtain the inequalities

0;;:;: g*(xl ) - g*(x2) ;;:;: 8 + A(jjU* - x 2 11 -jju* - XIII).

Observe that

by the triangle inequality. Hence, since A ~O,

(8)

(9)

(10)

As 8>0 was arbitrary, clearly (10) implies (6), and thus (ii) is proved. Consider now any point uoE U. Replacing X by U o in (4) and (5), we obtain the inequalities

and (iii) follows.

Definition 1. Let there be given a bounded continuous transformation

(11)

where D is a bounded domain in Rn. Take a point uED and a (finite) constant M~ O. Then E(u,M) denotes the subset of D defined by the

. formula

E(u,M)={u*ju*ED,IITu*-Tull;;:;:Mllu*-ulj}. (12)

Note that E(u,M) depends upon T also. However, T is thought of as fixed, and thus the notation E(u,M) is adequate.

Definition 2. Given T as in (11), we shall say that T is of appl'oximately bounded linear distortion at a point u ED if there exists a (finite)

§ V.2. Local approximations in RI<. 343

constant M :;:;: 0 such that u is a point of density (see III .1.1, definition 1) of the corresponding set E(u,M).

Lemma 21. Given T as in (11), let 5 be a subset of D such that T is of approximately bounded hnear distortion at every point uE S. Then there exists a sequence of sets 5 k' k = 1, 2, ... , such that 5 = US k

and T is Lipschitzian on each of the sets Sk'

Prool. Let us recall that if u is a point in Rn and r is a positive number, then L1(u, r) denotes the closed spherical neighborhood of u with radius r. Explicitly:

L1(u, r) = {xlxE Rn, Ilx - ull :s;: r}. ( 13)

Let us also recall (see 111.1.1, lemma 15) that if u1 , U 2 are any two distinct points in RH, then the number

oc = L[LJ(u1 ,lu2 - ulill nLJ(u2 , iiu2 - ulll)] LLl(u1 , IIu2 - ul!l + LLl(u2 , iiu2 - ulll)

(14)

is independent of the choice of u1 , U 2 and satisfies the inequalities

O<oc<1. ( 15)

Let u be a point in D, and let M, r be two numbers such that O:s;:M < co, O<r<oo. Let us put [see (13) and definition 1J

E(u, M, r) = E(u, M) n L1(u, r). ( 16)

Observe that since T is continuous, E(u,M) is closed relative to D. As D is open, clearly E(u, M) is a BOREL set and hence it is L-measurable. Hence E(u,M, r) is also L-measurable. For each positive integer k, let Sk be the set of those points u which satisfy the following conditions.

(i) u E S.

(ii) IITull:S;:k. (iii) If r is any number such that O<r:S;:1jk, then [see (14)J

L E(u, k, r) > (1 - oc) L L1(u, r).

We proceed to verify that the sets Sk so defined possess the desired properties. Consider any point uoE S. Since T is of approximately bounded linear distortion at uo, there exists a (finite) constant M';:;;;. 0 such that U o is a point of density of the corresponding set E(uo,M). Accordingly (see 111.1.1, definition 1), in view of (15) and (16) there

1 See the proof of theorem 5.2 in the paper by H. FEDERER, Surface area II. [Trans. Amer. Math. Soc. 55, 438-456 (1944)]. This paper of FEDERER contains many important results concerning Lipschitzian transformations.


exists a number ro>O such that

LE(uo, M, r) > (1 - oc.) LL1(uo, r} if 0 < r :S;;ro. (17)

of k is a sufficiently large positive integer, then

(18)

Let k be so selected that (18) holds, and let r be any number such that O<r:;;;'1/k. Then clearly

E(uo, k, r) ) E(uo' M, r) and 0 < r:;;;' ro,

and hence by (17) we have the inequalities

Thus it is established [see (i), (ii) , (iii)] that uoE Sk' As U o was an arbitrary point of S, it follows that

(19)

We shall now verify that Tis Lipschitzian on each one of the sets Sk' This fact will be proved if we show that for any two points u1 , u2 E Sk we have the inequality

Since (20) is obvious if U 1 =u2 , we can assume that U 1 =j=u2 • If

IIu2 - u111 > ~ ,

then (since liT ull -;;;;. k for u E Sk)

II T u2 - T u11l -;;;;. 2k = 2k2 • ~- < 2k2 1!U2 - u11!.

and thus (20) holds in this case. So we can assume that

Let us put

(20)

(21)

El = E(u1, k, lIu2 - Utll), E2 = E(u2, k, lIu2 - u11!) , (22)

(23)

§ V.2. Local approximations in Rn.

In view of (14), (16), (21) and (iii) we have then the relations

LEI> (1 - IX) LJ1 , LE2 > (1 - IX) LJ2,

L (E U E ) :s;; L (J U J ) IX = L (31 n 32) • 1 2 1 2 , L11 + L Ll2

On the other hand, by llL1.1, lemma 2 we have

Hence

L(EI n E2) + L(EI U E2) = LEI + LE2,

L(JI nJ2) + L(J1 UJ2) = LJl + LJ2·

345

L(ElnE2) = LEI + LE2- L(EI UE2) > (1 --'-(X) (LJ1 + LJz) - L (J1 U J 2)

= (1-(X) (LJl + LJ2) - [LJl + LJ2- L(J1 nJ2)]

= L(JI n J 2) -(X(LJl + LJ2) = o.

Thus L (El n E 2 ) > 0, and hence E1 n E2 =l= 0. So we can select a point u*EElnEz. Then [see (12), (16), (22), (23)J

IITu* -Tulll:s;; kllu* -u111, IITtt* -Tu211:s;; kllu* -u211, Ilu* - u111 :s;; IIu2 - u111, Ilu* - u2 11 :s;; IIu2 - ulll,

and hence

IITu2 - Tu111:s;; IITu2 - Tu*11 + IITu* - Tulll:s;; :s;; k(lluz - u*11 + !lUI - u*11) :s;; 2k IIu2 - u111 :s;; 2 k2 11 uz - ulll·

Thus (20) holds, and the proof is complete.

Assume now that the set 5, occurring in the statement of lemma 2, is L-measurable. Using the sets 5 k whose existence has been just established, we define

Note that since 5 is L-measurable and Sk is closed, clearly S~ is Lmeasurable. We assert that T is Lipschitzian on each one of the sets S~. Indeed, let u, v be any two points of S~. Since S~(Sk' we can select two sequences of points utn ' vtn E Sk such that Um --* U, Vm --* v. By (20), we have then

For m--* 00 there follows (in view of the continuity of T) the inequality

II T u - T v II :s;; 2 k2 11 u - v II '


and thus T is Lipschitzian on S~. Consider now the sets

Clearly, Sr, ... , Sf, ... are pair-wise disjoint L-measurable sets whose union is S. Furthermore, since Sf (S~ and T is Lipschitzian on S~, clearly Tis Lipschitzian on Sf (a vacuous statement if Sf = 0). These remarks yield the following result.

Lemma 3. Given T as in (11), let S be an L-measurable subset of D such that T is of approximately bounded linear distortion at every point uE S. Then there exists a decomposition

s = U Sk> k = 1, 2, ... ,

where the sets Sk are pair-'wise disjoint L-measurable sets and T is Lipschitzian on each one of the sets Sk.

In the following series of lemmas, we shall make use of the reprEsentation

T:xi = fi(ul, ... , un), i = 1, ... , n, (24)

for the bounded continuous transformation given as in (11) (see V .2.2).

Lemma 4. Let T be given as in (11) and (24). Assume that the first partial derivatives of the coordinate functions of T exist a.e. in D. Let S be an L-measurable subset of D such that Tis Lipschitzian on S. Then there exists a transformation T*: D---+Rn such that the following holds.

(i) T* is Lipschitzian in D. (ii) T*u=Tu for uES. (iii) ](u, T*) = ](u, T) a.e. on s. Proof. By V.2.3, lemma 1 the coordinate functions fl' ... , fn of T

are Lipschitzian on S. Accordingly, by lemma 1 each one of the functions fi admits of a Lipschitzian extension to R" and hence a fortiori to D. Let It be a Lipschitzian extension of Ii to D, and let T*: D---+Rn be the transformation defined by the equations

T* : xi = It (ul , ... , un) , i = 1, ... , n.

Then (ii) is obvious, and (i) holds by V.2.3, lemma 1. Observe now that the ordinary Jacobian ](u, T) exists a.e. in D by assumption, while the ordinary Jacobian ](u, T*) exists a.e. in D by V.2.3, lemma 2. Take a point

uo=(u~, ... ,u~)E S,


such that ](uo, T), ](uo, T*) both exist and U o is a point of linear density of 5 in the direction of each one of the coordinate axes (note that these conditions are satisfied a.e. on 5 in view of 111.1.1, lemma 10). For each integer j = 1, ... , n there exists then a sequence of points

such that

u;k=(uh, ... ,U'/k)ES, k=1,2, ... ,

. i _ i lim Uik - uo' k->oo

In view of (ii) it follows that

i, j = 1, ... , n.

Hence ](uo, T) =](uo, T*), and the lemma is proved.

Lemma 5. Given T as in (11) and (24), assume that the first partial derivatives of the coordinate functions of T exist a.e. in D. Then T is of approximately bounded linear distortion a.e. in D.

Proal. Let U o = (u~, ... , u~) be a point in D where (a) the first partial derivatives of the coordinate functions of T exist, and (b) each one of the coordinate functions of T possesses an approximate total differential. Since in view of 111.1.3 these conditions are satisfied a.e. in D, it is sufficient to show that T is of approximately bounded linear distortion at such a point uo. For each integer i = 1, ... , n, denote by E, the set of those points

for which n

!Ii(u) - Ii (Uo) - L f)l~~~o) (Ui - ui)i ~JIU - uoll· (25) i~l

Then, since Ii possesses an approximate total differential at Uo by assumption, U o is a point of density of E j , i = 1, ... ,n (see 111.1.3). Hence (see 111.1.1, lemma 11) U o is also a point of density of the set

E= nEi . (26)

Accordingly, in view of definition 2, the lemma will be proved if we can exhibit a (finite) constant M:;;;;O such that

(27)

348 Part V. Differentiable transformations in Rff.

Let us put _ I 81.(uo) I .. -a-max ~' 2,J-1, ... ,n.

From (25) and (26) we obtain the inequality n

Iii (u) - Ii (uo) I :s: lI u - uoll + L I 8f~oll!ui - uil :s: i=l

:s:lIu ---,- uoll + nallu - uoll = (1 + na) lIu - uoll,

for u E E. Hence, for u E E, n

II T u - T uoll :s: L 1/;(u) - I; (uo)! :s: n(1 + n a) lI u - uoll· ;=1

Thus (27) holds with M =n(1 +na), and the lemma is proved.

Lemma 6. Given T as in (11) and (24), assume that the first partial derivatives of the coordinate functions of T exist a.e. in D. Let 5 be an L-measurable subset of D. Then there exist pair-wise disjoint L-measurable sets s, 51' ... , 5k , •.• in D such that the following holds.

(i) 5 = s U 51 U· .. U 5 k U .... (ii) Ls= O. (iii) T is Lipschitzian on each one of the sets 5k •

Proal. From lemma 5 there follows the existence of a decomposition 5=sU5*, where s, 5* are disjoint L-measurable sets such that Ls=O and T is of approximately bounded linear distortion at every point u E 5*. By lemma 3 there exists a decomposition of 5* into pair-wise disjoint L-measurable sets 5 k , k = 1,2, ... , such that Tis Lipschitzian on each one of the sets 5k , and the present lemma follows.

Lemma 7. Given T as in (11) and (24), assume that the first partial derivatives of the coordinate functions of T exist a.e. in D. Let 5 be an L-measurable subset of D such that

LT5 = o. (28) Then

J(u, T) = 0 a.e. on 5. (29)

Proal. By lemma 6 we have a decomposition

5 = s U 51 U ... U 5k U ...

with the properties stated there. Thus clearly it is sufficient to show that

J(u, T) = 0 a.e. on 5 k (30)

§ V.3. Special classes of differentiable transformations in R". 349

for each positive integer k. Now since T is Lipschitzian on Sk> by lemma 4 (applied to Sk) there exists a Lipschitzian transformation T,,:D-'?Rn such that

](u, Tk ) = ](u, T) a.e. on Sk.

(31)

(32)

Since Sk( 5, it follows from (28) that LT Sk = o. Since TkSk = T Sk by (31), we conclude that L Tk 5 k = 0. As Tk is Lipschitzian in D, by V.2.3, lemma 5 (applied to Tk and 5,,) we infer that ](u, T k) = ° a.e. on Sk' and (30) follows in view of (32).

Lemma 8. Given T as in (11) and (24), assume that the first partial derivatives of the coordinate functions of T exist a.e. in D. Let 5 be an L-measurable subset of D such that

](u, T) = 0 a.e. on S.

Then there exists a decomposition 5 = sUS' where sand 5' are disjoint l-measurable sets such that

Ls=O, LTS'=O. (34)

Proof. By lemma 6 we have a decomposition

5 = s U 51 U ... U Sk U· ..

with the properties stated there. Thus clearly the present lemma will be proved if we show that LT US" = o. In turn, this last relation will be proved if we show that

(35)

for every positive integer k. Now since T is Lipschitzian on Sk' by lemma 4 (applied to Sk) there exists a Lipschitzian transformation Tk :D-'?Rn such that (31) and (32) hold. Since 5" (5, (33) and (32) imply that ](u, Tk) = 0 a.e. on 5". As Tk is Lipschitzian in D, by V.2.3, lemma 5 (applied to 1k and Sk) we conclude that LTkS" = o. As TkS" = T Sk by (31), the relation (35) follows, and the lemma is proved.

§ V.3. Special classes of differentiable transformations in Rn.

V.3.1. Preliminary comments. For continuous transformations in Rn we have developed in § IV.4 and § IV.5 two general theories which may be referred to as the e-theory and the B-theory respectively. In the present § V.3, we shall study various relationships between these


two theories for the case of a general Euclidean space R". The case n = 1 has been already discussed in detail in § V.1. Special features of the case n = 2 will be studied in Part VI.

The results to be derived in the present § V.3 may be classified as inclusion theorems and equivalence theorems. An inclusion theorem is a statement to the effect that a certain class of transformations is included in some other class of transformations. For example, theorem 1 in IV.S.3 is an inclusion theorem stating that the class of sA CB transformations is included in the class of eA C transformations. An equivalence theorem is a statement to the effect that under certain conditions corresponding concepts in the e-theory and the B-theory become equivalent to each other or to an analogous concept in classical Analysis. For example, in classical Analysis one operates with the ordinary Jacobian I(u, T), while in the e-theory and in the B-theory one uses the generalized Jacobians 1. (u, T) and IB (u, T) respectively. Each one of these three Jacobians (see V.2.2, IV.4.3, IV.S.3 for the explicit definitions) is meant to represent the local rate 01 change 01 volume under the transformation T. Accordingly, one may expect that under certain appropriate conditions these three Jacobians will have the same value. In V.3.2 we shall derive two fundamental equivalence theorems which confirm this expectation. Similarly, the e-theory and the B-theory give rise to the concepts eEV, EVB, sEVB and eA C, A CB, sA CB relating to bounded variation and absolute continuity respectively, and in turn these two sets of concepts depend upon two different multiplicity functions. Inclusion and equivalence theorems concerning these basic concepts will be discussed in V.3.3.

As regards the scope of results relating to equivalence theorems, it is natural to expect that the e-theory and the B-theory will yield equivalent information if applied to transformations of a more or less elementary character. Section V.3.4 contains results of interest from this point of view. The term elementary is of course a vague one, but it may be assumed that the class of generalized Lipschitzian transformations, defined and discussed in V.3.4, is sufficiently comprehensive to account for all those transformations which should be thought of as constituting the elementary range in the present context.

V.3.2. Comparison ofthe generalized Jacobians Je(u, T}, JB(u, T) with the ordinary Jacobian J(u, T). We shall use for the bounded continuous transformation T the alternative representations

as explained in V.2.2.

(1)

(2)

§ V.3. Special classes of differentiable transformations in Rn. 351

Theorem 1. Given T as in (1) and (2), assume that (a) T is eBV in D, and (b) T possesses a weak total differential a.e. in D (see V.2.2, definition 2). Then the essential generalized Jacobian ]e(u, T) is equal to the ordinary Jacobian ](u, T) a.e. in D.

Proof. The assumption (a) implies that the essential multiplicity function K(x, T, D) is L-summable in R n (see IV.4.1). Hence (see 111.1.1, lemmas 32, 43) there exists a BOREL set X in Rn such that the following holds.

(IX) LX =0. (p) K(x, T, D) < 00 if xE ex. (y) If Xo E ex and {Gm } is any regular sequence of closed sets such

that Xo E Gnp m = 1,2, ... , and oGm-+O,' then

lim _1~ jK(X, T, D) dL = K(xo, T, D) < 00. m-? 00 L Gm

Gm

We first verify the relation

]. (u, T) = 0 = ](u, T) a.e. on T-l X. (3 )

Note that since X is a BOREL set, T-l X is also a BOREL set, and hence T-IX is L-measurable (see IV.1.1, theorem 1, and III.1.1, lemma 19). Furthermore, TT-IX(X, and thus LTT-IX =0 by (IX). Accordingly, ](u,T) =0 a.e. on T-IX by V.2.4, lemma 7, while ]e(u,T) =0 a.e. on T-IX by IV.4.3, lemma 5, and (3) follows. Denote now by 50 the set of those points uE D where ](u, T) exists and is equal to zero. Thus

50 = {uluE D, ](Zt, T) exists, ](u, T) = o}. (4)

We assert that ]e(u, T) = 0 = ](u, T) a.e. on 50' (5)

Indeed, by V.2.4, lemma 8, we have a decomposition 50 = So U 5~, where So, 5~ are L-measurable sets such that L So = 0 and LT 5~ = O. This last relation implies (by IV.4.3, lemma 5) that ].(u,T) =0 a.e. on 5~, and (5) follows in view of (4). Let us now denote by 5* the set of those points u ED where one at least of J(u, T), ].(u, T) fails to exist or T fails to possess a weak total differential. In view of the assumptions (a), (b) and IV.4.3, theorem 1, we have then

.L 5* = O. (6)

In view of 0), (5), (6) the theorem will be proved if we show that

(7)


provided that U o E D - (5* U So U T-l X). (8)

Let us first observe that if (8) holds, then the following conditions are satisfied at U o.

(i) J(uo,T) and J.(uo,T} both exist. (ii) J(uo, T}:::j=O and T possesses a weak total differential at uo. (iii) TUoE CX.

Let us put Tuo = xo. In view of (ii), we have before us the Situation 2 discussed in V.2.2. Using the terminology employed there, let us consider the sequence of oriented n-cubes Q (cxm) with center at U o and side length 2cx"p where cxm---+O. By V.2.2 (45) we have

( T . Q( )) = {sgnJ(uo, T) if xEP~, } p- x, ,mt CXm 0 l·f " xE CPm ·

(9)

Observe that since

K(x, T, int Q(OCm)) ~ K(x, T, D) < 00 a.e. in R:',

we have K(x, T,intQ(cxm)) <00 a.e. in Rn. (10)

Furthermore [see V.2.2 (46)]

T tr Q (oc"J ( P~ - P~ . (11 )

In view of (10) and (11) we conclude from 11.3.4, theorem 2 [applied to the domain int Q (cxm ) ]

P-.(x, T, intQ(cxm)) = /-l(X, T, intQ(cxm)) a.e. on P~, (12)

p-.(x,T,intQ(cxm))=p-(x,T,intQ(cxm)) a.e. on CP,:. (13)

From (9), (12), (13) it follows that

( . ) {Sgn J(uo, T) fl. x, T, mt Q (cxm) = 0

a.e. on P~, } a.e. on CP~.

(14)

Consider now a point x ERn where

K(x, T,intQ(ocm )) < 00.

From 11.3.2, theorem 1 and 11.3.4, theorem 4 [applied to the domain int Q(ocm)] we obtain

Ifle(X, T,intQ(ocm)1 = IK+(x, T,intQ(OCm)) -K-(x, T,intQ()'"tn))I:s;;

~ K+( x, T, int Q (ocm )) + K-( x, T, int Q (ocm)) = K( x, T, int Q (cxm)).

§ Y.3. Special classes of differentiable transformations in Rn. 353

By (10) there follows the inequality

l,ue(x, T,intQ(oc",))1 :s;;,K(x, T,intQ(oc",)) a.e. in Rn.

Since K( x, T, int Q (oc",)) :s;;, K(x, T, D),

we conclude that

l,u'e(x, T,intQ(oc",))1 :s;;,K(x, T,D) a.e. in Rn. (15)

Let us put J ,ue(x, T, intQ(()(m)) dL = ;.~,

p-m

J K(x, T, D) dL = a;n, P';

J K(x, T, D) dL = d~,. P;';

Note that since Je(uo, T) exists, we have

by (4), (18), (19), in IVA.3. From (14), (16), (17) it follows that

(16)

( 17)

( 18)

(19)

J ,ue(x, T, int Q(a",)) dL = ;.:" (21)

;.: = }.;" + (;'~n - }.~) = (L P~,) sgnJ(uo, T) + ;.;" - }.;". (22)

From (15) to (19) we obtain

(23)

In view of V.2.2 (iii) and (44), the sequences of parallelotopes P,;" P,~ constitute regular sequences of closed sets, containing the point Xo = T 110 ,

such that

Hence, by (18), (19), (iii), ({3), (y) above,

a;" . K(' T D) --, ---;0- Xo" < 00 . LP",

(24)

By V.2.2 (43) we have

LP,~ . IJ( T)I -L Q (Cl",) -i> u o, , L P,~ I I ~ ](uo, T) .

L Q (ot",) (25)

Rado and Reichelderfer, Continuous Transformation~. 23



0';'" - a;" ---LQ(cc",)

Hence, by (23),

Combining (20), (21), (22), (25), (26), we obtain

].(uo, T) = lim L~~m) = (sgnJ(uo, T))1im L~~:m) = (sgn J(uo, T)) IJ(uo, T)I = J(uo, T),

and the proof is complete.

(26)

Theorem 2. Given T as in (1) and (2), assume that (a) T is BVB in D, and (b) T possesses a total differential a.e. in D (see V.2.2, definition 1). Then the generalized Jacobian JB (u, T) is equal to the ordinary Jacobian J(u, T) a.e. in D.

Prool. Let a point uoE D be assigned. We have to show that

(27)

except perhaps for certain exceptional points uoED which constitute a set of L-measure zero. Note that condition (a) implies (see IV.5.l, theorem 3) that the derivative DB (u, T) exists a.e. in D. Furthermore (see IV.5.3)

JB (u, T) = iB (u, T) DB (u, T) a.e. in D. (28)

By IV.S.l, theorem 4, we have DB (u, T) =D (u, G) a.e. in D, where G stands for the function of open intervals

G (inti) = L T (inti), inti ( D, (29)

and D (u, G) denotes the derivative of G. In view of condition (b) we can therefore assume, in proving (27), that the following holds.

(i) J(uo, T), DB (uo, T), D(uo, G), JB(UO' T) exist. (ii) DB (uo, T) =D (uo, G). (iii) JB(UO, T) =iB(uo, T) DB(uO' T). (iv) T possesses a total differential at uo.

Let us introduce the set

So={uluED,J(u,T) exists, J(u,T)=O}. (30)


We assert that JB(U, T) = 0 = ](u, T) a.e. on So.

Indeed, in view of condition (b), we can applyV.2.4, lemma 8, obtaining a decomposition So = sUS', where s, 5' are L-measurable sets such that Ls = 0 and

LTS'=O. (32)

Thus it is sufficient to show that

JB (u, T) = 0 = J(u, T) a.e. on 5'. (33)

Now (32) implies (see IV.5.t, theorem 5) that DB(u,T)=O a.e. on 5'. Thus (33) follows in view of (28) and (30), and (31) is established. From (31) we conclude that in proving (27) we can assume that

(v) ](uo, T) =f= o. In view of (i), (iv) , (v) we have before us the Situation 3 discussed in V.2.2. Using the terminology employed there, consider the sequence Q (ex",) of oriented n-cubes with center at U o and side length 2exm , where 0:.",-+0. By V.2.2 (75) and (76) we have the relations

p'~ ( Tint Q (exm) ( p,;;, iB (uo, T) = sgn ](uo, T).


Since D (uo, G) exists [see (i)], we have the relation

lim G (int Q (O(m)) = D (uo, G) . L Q (0(",)

By V.2.2 (82) we have

lim - LP,~ _ = lim LP~; = iJ(uo, T)I. L Q (O(m) L Q (0(",)

From (36), (37), (38) we see that

D(uo, G) = IJ(uo, T)I.

The relations (ii) , (iii), (39), (35) yield

(34)

(35)

(37)

(38)

(39)

JB(UO' T) = iB(uo, T) D(uo, G) = (sgn J(uo, T)) IJ(uo, T)I = J(uo, T),

and the proof is complete. 23*


V.3.3. Miscellaneous inclusion and equivalence theorems. As in V.3.2, we shall use the alternative representations

T:D --+Rn,

T:x i =f;(uI, ... ,u"), i=1, ... ,n,

for the bounded continuous transformation T.

(1 )

(2)

Lemma 1. Given T as in (1) and (2), assume that T possesses a weak total differential a.e. in D. Then there exist three pair-wise disjoint L-measurable sets s, 5', 5* in D such that the following holds.

(i) D= sU5'U5*. (ii) Ls =0. (iii) L T 5' = o. (iv) The ordinary Jacobian exists and is different from zero at

every point u E 5*, and T possesses a weak total differential at every point uE 5*.

(v) N(x,T,D)=K(x,T,D) for xEET(sU5'), where Nand K are the multiplicity functions defined in 1.1.2 and II.3.2 respectively.

Proof. By assumption, there exists a decomposition

where $1' 51 are L-measurable sets such that

(4)

and T possesses a weak total differential at every point u E 51. Let us put

52 = {uJuE 51' ](u, T) = O},

5* ={uluE 51' J(u, T) =FO}.

(5)

(6)

Since ](tt, T) is L-measurable in D (see III.1.3), it is clear that 52' 5* are L-measurable sets, and

52 n 5* =0. (7)

Furthermore, the following holds.

(0::) T possesses a weak total differential at every point uE 5*. (fJ) ](u, T) exists and is different from zero at every point uE 5*.

In view of (5) and V.2A, lemma 8 (applied to 52), we have a decomposition

(8)


where 52' 5' are L-measurable sets such that

LTS' =0.

(9)

(10)

Setting s = Sl U S2' we assert that the sets s, 5', S* possess the desired properties. Obviously s, 5', 5* are pair-wise disjoint L-measurable sets such that (i), (ii), (iii) hold [see (3), (4), (5), (6), (7), (8), (9), (10)]. Furthermore, (iv) holds in view of (ex) and (13). To verify (v), consider any point

Xo (f T(s US') . (11)

Then clearly T-1 Xo ( S*. ( 12)

If T-1 Xo = 0, then (v) is obvious (see II.3.2, remark 2). Hence ,ve can assume that T-1 Xo =F 0. Consider any point uoET-1 XO ' Then uoE S* by (12), and in view of (oc) and (13) it follows that at tlo we have before us the Situation 2 discussed in V.2.2. Accordingly, as we have shown there, U o by itself is an e.m.m.c. for (xo, T, D). Thus every point of T-1 Xo is an e.m.m.c. for (xo, T, D), and hence (see II.3.3)

N(xo, T,D) =k(xo, T,D). (13 )

It is convenient to divide the rest of the proof into two parts.

Case 1. k(xo, T, D) = 00. Then

K(xo, T,D) = k(xo, T,D)

by II.3A, theorem 3, and (v) follows in view of (13).

Case 2. k(xo,T,D)< 00, Note that N(xo,T,D»O since T1xo=F0, and hence [in view of (13) ]

0< k(xo, T, D) < 00. (14)

Applying theorem 4, remark 2, and remark 5 in II.3A, we obtain the relation

K(xo, T,D) =Lii.(C,T)i, CECf(xo,T,D). (15)

We observed above that each point of the set T-l X o is an e.m.m.c. for (xo, T, D). Thus the elements of the class G: (xo, T, D) are now the individual points of T-1 X o' In view of (13) and (14) it follows that each point of T-1 Xo is an essentially isolated e.m.m.c. for (%0' T, D). Since flu, T) exists and is different from zero and T possesses a weak total differential at every point uET-1 xo [see (12), (oc), (f3)], we can


apply V.2.2 (68), obtaining the formula

ie(u, T) =sgnJ(u, T) for uE T-1xO'

It is now clear that the summation in (15) contains N(xo, T, D) terms, each of which has the value 1. Thus (15) yields K(xo, T, D) =N(xo,T, D), and the lemma is proved.

Theorem 1. Given T as in (1) and (2), assume that (a) T possesses a weak total differential a.e. in D, and (b) T is BVB in D. Then

(a:) /JB(u,T)/=DB(u,T) a.e. in D (see IV.5.l, IV.5.3), (fJ) T is sBVB in D (see IV.5.3, definition 1).

Proof. By IV.5.l, theorem 3, the derivative DB (u, T) exists a.e. in D and is L-summable there. In view of the assumption (a) we can apply lemma 1, and thus we have at our disposal the sets s, 5', 5* with the properties (i) to (v) described there. Since Ls = 0, (a:) will be verified if we show that

/JB(U, T)I = 0 = DB(u, T) a.e. on 5',

IJB(u, T)I =DB(u, T) a.e. on 5*.

Recall tha t (see IV.5 .3)

JB (11" T) = iB (u, T) DB (1(, T) a.e. in D.

( 16)

('l7)

(18)

Now since LT 5' =0, we have DB (u, T) =0 a.e. on 5' by IV.5.l, theorem 5, and (16) follows in view of (18). To prove (17), we make use of the set I = I (T, D) of those points u ED which have a neighborhood clear of relatives (see 1I.3.7). Let us put

Sf = 5* n I, 5: = 5* n (D - I). ( 19)

Since S* is L-measurable and I is a BOREL set (see 1I.3.7, lemma 1), Sf and S: are L-measurable. As Ds(u, T) = 0 a.e. on D - I (see IV.5.l, theorem 6) and Si (D - I, we conclude by (18) that

IJB (u, T) I = 0 = DB (u, T) a.e. on Sr (20)

Observe now that since Sf (S*, the ordinary Jacobian exists and is different from zero for u ESt and T possesses a weak total differential at every point uE st. Thus we have before us, at every point uE Si, the Situation 2 discussed in V.2.2. Also, since St (I, every point uESt has a neighborhood clear of relatives. By V.2.2 (69) it follows that

iB(u, T) =sgnJ(u, T) = ± 1 for ~tE Sf. (21)



)Js(u, T)) = Ds(u, T) a.e. on Sf. (22)

As (19), (20), (22) imply (17), part (IX) of the theorem is proved. Since (as we noted above) DB (u, T) is L-summable in D, ((X) implies that Js (u, T) is also L-summable in D. As T is BVB in D by assumption, part (fJ) of theorem follows (see IV.5.3, definition 1).

Theorem 2. Given T as in (1) and (2), assume that (a) T possesses a weak total differential a.e. in D, and (b) T satisfies the condition (N) in D (see IV.1.4). Then

N(x, T, D) = K(x, T, D) a.e. in R". (23)

Proof. Lemma 1 yields the relation

N(x, T, D) = K(x, T, D) for x Ef T(s US'), (24)

where s, 5' are L-measurable subsets of D such that

Ls=O, (25)

LTS'=O. (26)

In Vlew of the assumption (b), (25) implies that LTs=O. Hence LT(sUS')=O by (26), and (23) follows now by (24).

Theorem 3. Given T as in (1) and (2), assume that (a) T possesses a weak total differential a.e. in D, and (b) T is A CB in D. Then the following holds.

(i) T is both sA CB and eA C in D.

(ii) Js(u,T)=J.(u,T)=J(u,T) a.e. in D. (iii) N(x, T, D) = K(x, T, D) a.e. in R".

Proof. By IV.5.2, theorem 1, Tis BVB in D. Thus the assumptions of theorem 1 are fulfilled, and it follows that Tis sBVB in D. Since T is A CB in D by assumption, it follows further that T is sA CB in D (see IV.5.3, definition 1 and definition 2). By IV.5.3, theorem 1 we conclude that T is eA C (and hence also eBV) in D, and

JB(U, T) = J.(u, T) a.e. in D.

In view of the assumption (a) we can apply V.3.2, theorem 1, obtaining

J.(u, T) = J(u, T) a.e. in D.

Thus (i) and (ii) are proved. Observe now that T satisfies the condition (N) in D by IV .5.2, theorem 3. Hence (iii) follows directly from theorem 2, and the proof is complete.


Lemma 2. Given T as in (1) and (2), assume that the first pC).rtiai derivatives of the coordinate functions of T exist a.e. in D. Let 5 be an L-measurable subset of D such that T is Lipschitzian on S. Then

J II(u, T)I dL = J N(x, T, 5) dL. 5

Proof. By V.2.4, lemma 4 there exists a transformation

T*:D ->-Rn,


(i) T* is Lipschitzian in D. (il) T*u=Tu for uES. (iii) I(u, T*) = I(u, T) a.e. on S.

(27)

By V.2.3, lemma 3, T* is ACB in D. Accordingly, by IV.5.2, theorem 5 (applied to T* and 5) we have the formula

.r DB (u, T*) dL = J N(x, T*, 5)dL. (28) 5

Since (ii) implies that N(x, T*, 5) =N(x, T, 5), we have

J N(x, T*, 5) dL. = J N(x, T, 5) dL. (29)

Note that T* possesses a total differential, and hence also a weak total differential, a.e. in D by V.2.3, lemma 2. Thus we can apply theorem 3 and theorem 1 to T*, obtaining

IB (u, T*) = J(it, T*) a.e. in D,

!IB (1£, T*) 1 = DB (11, T*) a.e. in D.

Clearly the relations (iii), (28) to (31) imply (27).

(30)

(31)

Lemma 3. Given T as in (1) and (2), assume that (a) the first partial derivatives of the coordinate functions of T exist a.e. in D, and (b) T satisfies the condition (N) in D. Let 5 be any L-measnrable subset of D. Then

.r 11(11, T)I dL = J N(x, T, 5) d L (32) s


Proof. The assumption (a) implies (see V.2.4, lemma 6) the existence oi a decomposition

5 = s U 51 U ... U Sk U ... ,

where s, 5k are L-rr.easurable sets such that

Ls =, 0,

sn 5k =0, Sin 5k =0 for j =1= k,

(33)

(34)

(35)

§ V.3. Special classes of differentiable transformations in RI!. 361

and Tis Lipschitzian on each one of the sets 5k • Application of lemma 2 (with 5 replaced by 5k ) yields

f I](u, T)I dL = J N(x, T, 5k ) dL, k = 1, 2,.... (';6) Sk

Let us put f(x) = L N(x, T, 5k )·

The relations (33), (35), (37) yield

N(x, T, 5) = f(x) + N(x, T, s).

(37)

(38)

Since T satisfies the condition (N) in D, (34) implies that LTs = o. Hence N(x, T, s) = 0 a.e. in Rn , and thus (38) shows that

N(x, T, 5) = t(x) a.e. in Rn. (39)

Note that N(x, T, 5), N(x, T, 5k ) are L-measurable in R" by IV.1.4, corollary 2. Assume now first that ](u, T) is L-summable on 5. Then (see III.1.t, lemma 37) the relations (33) to (35) yield

L f I](u, T)I dL = f I](u, T)I dL. (40) Sk 5

From (36) and (40) we conclude that the series

(41)

is convergent. Since N(x, T, 5k ) ~O, the convergence of the series (41) implies, in view of (37) and III.1.t, lemma 39, the L-summability in R" of t(x), as well as the relation

f f(x) dL = L f N(x, T, 5k ) dL. (42)

Clearly, (40), (36), (42), (39) imply (32). Assume next that N(x, T, S) is L-summable in R". By (39) it follows that t (x) is L-summable in R". Since N(x, T, 5,J ~ 0, (37) implies therefore (by III.1.t, lemma 40) that

L fN(x, T, 5k ) dL = f f(x) dL < 00. (43)

From (43) and (36) we conclude that the series

L f I](u, T)I dL Sk

is convergent. In view of (33), (34), (35) it follows, by IIL1.t, lemma 39, that ](u, T) is L-summable on 5. This fact implies, as we have already shown, that (32) holds, and the proof is complete.


Theorem 4. Given T as in (1) and (2), assume that (a) the first partial derivatives of the coordinate functions of T exist a.e. in D, and (b) T satisfies the condition (N) in D. Then T is A CB in D if and only if the ordinary Jacobian ](u, T) is L-summable in D.

Proof. Assume first that J(u, T) is L-summable in D. By lemma :3 (applied with 5 =D) we conclude that N(x, T, D) is L-summable in Rn ,

and thus Tis BVB in D by IV.5.l, theorem 1. Since T satisfies the condition (N) inD, itfollows that T is A CB in D (see IV.5.2, theorem :3). Assume, conversely, that T is A CB in D. Then T is also BVB in D, and hence N(x, T, D) is L-summable in Rn (see IV.5.2, theorem 1 and IV.5.l, theorem 1). From lemma 3, applied with S=D, it follows that J(u, T) is L-summable in D, and the proof is complete.

Theorem 5. Given T as in (1) and (2), assume that (a) T possesses a weak total differential a.e. in D, and (b) T satisfies the condition (N) in D. Then Tis eA C in D if and only if the ordinary Jacobian J(u, T) is L-summable in D.

Proof. Assume first that J(u, T) is L-summable in D. By theorem 4 it follows that T is A CB in D. Thus the assumptions of theorem :3 are satisfied, and hence T is eA C in D. Assume, conversely, that T is eA C in D. Then Je(-u, T) is L-summable in D by IV.4.3, theorem 1-Since ].(u, T) = J(~t, T) a.e. in D by V.3.2, theorem 1, it follows that J (u, T) is L-summable in D, and the theorem is proved.

Theorem 6. Given T as in (1) and (2), assume that (a) T possesses a weak total differential a.e. in D, (b) the ordinary Jacobian J(u, T) is L-summable in D, and (c) T satisfies the condition (N) in D. Then the following holds.

(i) T is both sACB and eA C in D. (ii) JB (u, T) =]. (u, T) = J(u, T) a.e. in D. (iii) DB (u, T) = De (u, T) = IJ(u, T) I a.e. in D. (iv) N(x, T, D) = K(x, T, D) a.e. in W.

Proof. Recall that the assumption (a) implies the existence of the first partial derivatives of the coordinate functions of T a.e. in D (see V.2.2, definition 2). Thus theorem 4 applies, and hence Tis ACB in D. By IV.5.2, theorem 1 it follows that T is also BVB in D. Hence we can apply theorem 3 and theorem 1. From theorem 3 we infer that (i), (ii) , (iv) hold. Theorem 1 yields

IJB (u, T) I = DB (u, T) a.e. in D,

and hence, in view of (ii) ,

DB(u, T) = IJ(u, T)I a.e. in D.


Since Tis eA Gin D, it is also eBV there. Hence (see IV.4.3, theorem 2)

De(u, T) = ile(u, T)i a.e. in D.

In view of (ii) it follows that

De(u, T) = iI(u, T)i a.e. in D,

and the theorem is proved.

V.3.4. Transformation formulas for definite integrals in Rn in terms of the ordinary Jacobian J(u, T). We shall use, as in V.3.2, the alternative representations


(-!)

(2)

Theorem 1. Given T as in (1) and (2), assume that Tis eAG in D and possesses a weak total differential a.e. in D. Then

J iI(u, T)i dL = J K(x, T, D) dL, D

J ](u, T) dL = J fJe(x, T, D) dL. D

0)

(4)

More generally, if H(x) is a finite-valued, L-measurable function in R n ,

then J H(Tu) i](u, T)i dL = J H(x) K(x, T, D) dL, (5)

D


J H(Tu) ](u, T) dL = JH(x)Pe(x, T,D) dL, (6) D


Proof. Note that

Ie (u, T) = ](u, T) a.e. in D

by V.3.2, theorem 1. We can therefore replace Ie(u, T) by ](u, T) in IV.4.4, theorem 1, theorem 2, theorem 4, obtaining the formulas (3), (4), (5), (6).

Theorem 2. Given T as in (1) and (2), assume that (a) T is defined and continuous on D, (b) LTjrD=O, (c) Tis eAG in D, and (d) T possesses a weak total differential a.e. in D. Then

J I(u, T) dL = J fJ(x, T, D) dL. (7) D


More generally, if H(x) is a finite-valued, L-measurable function in Rn, then

J H(Tu) I(u, T) dL = J H(x)fl(x, T,D) dL, (8) D


Proof. We have again Ie (u, T) = I(u, T) a.e. in D by V.3.2, theorem 1, and hence (7) and (8) follow from IV.4.4, theorem 3, theorem 4 by replacing Ie(u, T) by I(u, T).

The preceding two theorems belong to the e-theory (developed in § IV.4). To obtain corresponding theorems for the B-theory (developed in § IV.5) , one should replace the assumption that T is eAG in D by the assumption that Tis AGB in D. The resulting statements, while true, yield no new information. Indeed, if T is AG B in D and possesses a weak total differential a.e. inD, then Tis eAG inD byV.3.3, theorem 3, and thus we have before us merely special instances of the situations considered in the preceding theorems 1 and 2.

There arises the question whether the essential multiplicity function K can be replaced by the crude multiplicity function N in the formulas (3) and (5). The following theorem is of interest in this connection.

Theorem 3. Given T as in (1) and (2), assume that (a) T possesses a weak total differential a.e. in D, (b) the ordinary Jacobian I(u, T) IS L-summable in D, and (c) T satisfies the condition (N) in D. Then

J lJ(u, T) dL = J N(x, T, D) dL. (9) D

More generally, if 5 is an L-measurable subset of D and H(x) is a finitevalued, L-measurable function in Rn, then

J H(T u) II(u, T)I dL = J H(x) l'l(x, T, 5) d L, (10) s

as soon as one of the two integrals involved exists. In particular [corresponding to the special case H(x) = 1J we have the formula

J lJ(u, T)I dL = J N(x, T, S) dL. (11) s

Proof. By V.3.3, theorem 6, T is eAC in D, and thus (3) holds by theorem 1. Furthermore, by the same theorem 6 in V.3.3,

K(x, T, D) = N(x, T, D) a.e. in R",

and hence (9) follows from (3) by replacing K by N. The formula (11) follows directly from V.3.3, lemma 3, since I(u, T) is now L-summable on 5 as a consequence of the assumption (b). There remains to discuss the formula (10). Observe that, by V.3.3, theorem 6, T is sAGB and


hence also ACB in D. Accordingly, by IV.S.2, theorem 4,

f H(Tu) DB(U, T) dl = f H(x) N(x, T, S) dl, (12) 5

as soon as one of the two integrals involved exists. By V.3.3, theorem 6, we have

DB(U, T) = I](u, T)I a.e. in D,

and hence

f H(T u) DB (u, T) dl = f H(T u) I](u, T)I dl, (13) 5 5

'as soon as one of the two integrals involved exists. Combining the preceding remarks, we conclude that if one of the three integrals occurring in (12) and (13) exists, then the other two exist also and the three integrals have then the same value. It follows that (10) holds as soon as one of the two integrals involved exists, and the proof is complete.

The preceding theorem admits of various generalizations. In particular, a similar theorem holds for transformations from subsets of Rn into RN, where N~n. However, the study of these generalizations is beyond the scope of this treatise.

V.3.S. Closure theorems in Rn in terms of the ordinary Jacobian J (u, T). The term closure theorem is used here in the sense explained in IVA.S. Accordingly we consider, as in IVA.S, bounded continuous transforma tions

T:D -+Rn,

Ij:Dj-+Rn, j=1,2, ... ,

(1 )

(2)

where D, D j are bounded domains in R". We assume that T, 1j satisfy the following conditions.

(IX) The sequence {Ij} converges to T uniformly on compact subsets of D (see 11.3.2, remark 9).

(fJ) T possesses a weak total differential a.e. in D, ami the ordinary Jacobian ](u, T) is L-summable in D.

(y) r;. is cAC in D j and possesses a weak total differential a.e. in D j , i=1, 2, ....

Observe that (y) implies that r;. is eBV in D j (see IVA.2). Hence, by V.3.2, theorem 1 it follows further that

flu, 1j) = ]Ju, r;.) a.e. in D j •

From (3) and IV.4.3, theorem 1 it follows that

(0) ](tt, r;.) is l-sllmmable in D j .

(3)


Consider now a domain D such that D(D. By (;x) we have then D (Dj if j is sufficiently large, say j > j (D). By IV .4.2, theorem 1 (applied to If) it follows that

(8) If is eAC in D if D(D and j>j(D).

Theorem 1. Given T, If as in (1) and (2), assume that the following holds in addition to the conditions (;x), ({3), (y).

(i) lim inf f IIcu, If) I dl < 00. 1 .... 00 Dj

(ii) If Q is any oriented n-cube in D, then

lim fII(u,If)ldL=fII(u, T)ldl. 1 .... 00 Q Q

Then T is eAC in D.

Proof. Note that in view of (3) the condition (i) is equivalent to the condition

lim inf file (u, If)! d L < 00. 1--+ 00 Vj

Hence by IV.4.5, lemma 1 it follows that Tis eBV in D. In view of ((3) we can now apply V.3.2, theorem 1, obtaining the relation

I(u, T) = Ie(u, T) a.e. in D.

L'sing (3), we can therefore re-vvrite condition (ii) in the form

lim f lfe(u, If) I dl = f Ife(lI, T) I dL. 1-+ 00 Q Q

Thus the assumptions of IV.4.5, theorem 3 are satisfied, and hence T is eAC in D.

Theorem 2. Given T, If as in (1) and (2), assume that the following holds in addition to the conditions (0:.), ((3), (y).

(i) If F is any compact subset of D, then

f(u, lj) ---+ J(u, T) a.e. on F.

(ii) If Q is any oriented n-cube in D, then

lim f If(u, If) I dl = f If(14, T) I dL. }-+oo Q Q

Then T is dC in D.

Prooj. Let F be any compact subset of D. We assert that

lim f If(u, If) I dl = f If(t!, T)I dL. J .... OO F F

(i)


Note first that (i) implies (see 111.1.1, lemma 30) the inequality

f If(~t, T) 1 d L ~ li;m inf f I](u, If} 1 d L. F 1-->00 F

(5)

Assign now 8> o. Then (see 111.1.1, lemma 34) there exists an open set 0 = 0 (8) such that

F(O(D, (6)

f 1](11, T}I dL < f [f(u, T}! dL. + 8. (7) o F

We can select a finite system Ql; ... , Q", of oriented n-cubes such that

intQi n intQk =0 if i =l= k.

In view of 111.1.1, lemma 42 the relations (8) and (9) yield

'" L f If(u, T)[ dL = f [f(u, T)[ dL::s;: f !f(u, T}\ dL. k~l Qk UQk 0

By (7) there follows the inequality

In

L f lJ(u, T}I dL < f If(u, T)I dL + 8. k~l Qk F

Furthermore, (8) yields m

f If(u, If) 1 dL ~ L f If(u If) 1 dL. F k~l Qk

From the assumption (ii) we infer that

m m

(8)

(9)

(10)

(11)

.lim L f If(u, If)l dL = L J \J(u, T)i dL. (12) 1-->00 k~l Qk k=l Qk

In view of (10) and (12) we obtain from (11) the inequality

lim sup f If(u, If) [ d L £ f lJ(u, T)I dL + 8. (13) 1-->00 F F

Since 8> 0 was arbitrary, clearly (13) and (5) imply (4). Consider now any domain D such that

D(D. (14)

Applying (4) with F = D, there follows the relation

.lim f If(u, 'If) \ dL = f If(u, T)I dL :S f Iflu, T)J dL. , .... 00 f5 f5 D


In view of (f3) we conclude that

lim inf J If(u, If) I dL < 00. 1-+ 00 D

(15 )

From (E) and (14) it follows that If is eAG in D for j>j(15). In view of (15) it is now clear that the transformations TID, IfID, where j > j (15), satisfy the assumptions of theorem 1, with D and Di replaced by D. Hence T is eAG in D, and hence also eBV in D (see IV.4.2). From (f3) and V.3.2, theorem 1 it follows now that

flu, T) = fe(u, T) a.e. in D.

Hence, using again (f3),

Jlfe(u, T)I dL = J If(u, T)I dL:;:;;; J lJ(u, T)I dL < <x-. (16) D D D

Consider now a sequence of domains Dr' r = 1, 2, ... , filling up D from the interior (see 11.3.2, remark 10). Then 15r (D, r= 1,2, ... , and thus the preceding remarks can be applied with D = Dr. It follows that T is eAG in Dr and [by (16)J that the sequence of integrals

J Ife(u, T)I dL, r = 1,2, ... , Dr

is bounded. Hence T is eAG in D by IVA.5, theorem 2.

Theorem 3. Given T, If as in (1) and (2), assume that the following holds in addition to the conditions (IX), (f3), (y).

(i) If F is any compact set in D, then

flu, If) -+ I(u, T) a.e. on F.

(ii) If Q is any oriented n-cube in D, then the sequence {J(u, 1;)} satisfies the condition (V) on Q (see 111.1.1, definition 6).

Then T is eAG in D.

Proof. The assumptions (i) and (ii) imply (see IIL1.1, lemma 36) that

lim J II(u, If) I dL = J If(u, T)l dL, 1-,,00 Q Q

and thus the present theorem follows from the preceding theorem 2.

V.3.6. Generalized Lipschitzian transformations in R" t. We shall use, as in V.3.2, the alternative representations

T:D -+Rn,

T:x i =/,(u.1, ... , un), i=1, ... ,n,


(1 )

(2)

t This class of transformations has been introduced by T. RADO and P. V. REICHELDERFER, On generalized LIPSCHITZIAK transformations [Riv. Mat. Univ. Parma 2, 289-301 (1951)J.


Definition 1. Given T as in (1) and (2), we shall say that T satisfies the condition (if>, D) if the following holds.

(cr.) D is a domain in D. (fJ) if> = if> (u) is a real-valued, non-negative, L-summable function

in D. (y) If Q is any oriented n-cube in D, then

(3 )

where bT Q denotes the diameter of the image T Q of Q (see 1.1.4, definition 4).

Note that the domain D, occurring in the preceding definition, may coincide with D.

Lemma 1. Given T as in (1) and (2), assume that T satisfies the condition (if>, D). Let Q be any oriented n-cube in D. Then

L T Q ;;;: 2n J if> d L . (4) Q

Proof. Clearly TQ is contained in an oriented n-cube Q* with side length 2bTQ. Hence


Lemma 2. Given T as in (1) and (2), assume that T satisfies the condition (if>, D). Then the following holds.

(i) T possesses a total differential a.e. in D. Hence (see V.2.2) T also possesses a weak total differential a.e. in D, and the ordinary Jacobian ](u, T) exists a.e. in D.

(ii) ](u, T) is L-summable in D. (iii) T satisfies the condition (N) in D.

Proof. Since if> is L-summable in D, there exists (see 111.1.1, lemma 43) a decomposition

with the following properties.

(a) s, S are disjoint L-measurable sets. (b) Ls=O. (c) if>(u) < 00 on S.

(5)

(d) If u E Sand {Gm } is any regular sequence of closed sets such that

u E Gm (D. m = 1, 2, ... , lim bG", = 0, I1L~OO


370

then

Part V. Differentiable transformations in RH.

lim _1_ j<PdL = <P(U) <00. m_CO LGm

Gm

In view of (5) and (b), clearly (i) will be proved if we show that T possesses a total differential a.e. on S. Choose a point uoE 5, and put

(6)

For oc> 0, denote by Q (oc) the oriented n-cube with center at U o and side length 2oc. Tn view of (d) there exists a constant oco> 0 such that Q(oc)(D for O<oc<oco and

f<PdL<[<P(uo)+1]LQ(oc) if O<oc<oco' (7) Q(o:)

Consider any point uE D such that

(8)

Then clearly uE Q (II u - uoll) (D. Hence, application of (3) to Q (II u-uoll) yields

II T u - T uoll :s;; 15 TQ(il u - uoll):S;; (Q{i~/""i)<P dLr'n. (9)

Observe that

Thus (9) yields, in view of (8) and (7), the inequality

IITu - TUoll:s;; [<P(uol + 1]1/n21Iu - uoll·

Hencf', by (6),

IITu-Tuoll:S;;Mllu-uoll if o<llu-uoll<oco' (10)

Consider now anyone of the coordinate functions !i of T [see (2)]. From (10) it follows that

Hence clearly

lim sup I/;(uL= li(Uo21 < (Xl if U o E S. "--+1l, Ilu - uoll

(11)

As L(D-S) =0 by (a) and (b), (11) implies (see 111.1.3) that!i possesses a total differential a.e. in D (i = 1, ... , n). In turn, this fact implies (as we have shown in the course of the proof of V.2.3, lemma 2) that T

§ V.3. Special classes of differentiable transformations in RH. 371

possesses a total differential a.e. in D, and (i) is proved. To verify (iii), observe first that since tJ> is L-summable in D, for every e> 0 there exists an rJ=rJ(e»O such that

ftJ>dL<e E

if E is any L-measurable set in D which satisfies the inequality

LE <'Y).

Consider now any set F (D such that

LF=O.

(12)

(13)

(14)

Assign 8>0, and let 1} ='Y) (e) >0 be so chosen that (13) implies (12). By IIL1.t, lemma 14 the relation (14) implies the existence of a sequence {Qk} of oriented n-cubes such that

F( UQk(D, ( 15)

2: LQk < 'Y), (16)

int Qi n int Qk = 0 for j=t=k. ( 17) Let us put

E = UQk. (18)

Then E is an L-measurable set in D, and (16) and (17) imply (see IIL1.t, lemma 13) that LE=2:LQk<rJ. Hence [since (13) implies (12) due to the choice of rJJ

ftJ>dL<e. (19) E

On the other hand, (17) and (18) imply (see III.1.t, lemma 42) that

ftJ>dL=2:ftJ>dL. (20) E Qk

By lemma 1 we have

LTQk;;;;;2"ftJ>dL, k=1,2, .... (21) Qk

From (15), (18), (21), (20), (19) we conclude that

L T E;;;;; 2: L T Qk;;;;; 2" 2: f tJ> d L = 2" f tJ> d L < 2n e. Qk E

As T F ( T E, and e> 0 was arbitrary, it follows that

LTF=O. (22) 24*


Thus (14) implies (22), and (iii) is verified. To prove (ii), consider any point

Uo = (u~, ... , u~) E D

such that the first partial derivatives of the coordinate functions of T exist at U o' Take any sequence of real numbers hm' m = 1, 2, ... , such that

hm > 0, Itm -+ o.

Let Q", denote the oriented n-cube determined by the inequalities

u~ ;;;;; ui ~ u~ + It"" i = 1, ... , n.

Let j be anyone of the integers 1, ... , n. Consider the point

where i _ i 'f . ..!.. • i _ i + '

Umj-UO 1 t-rJ, Umj-UO 11m'

Clearly Qm (D if m is sufficiently large. Discarding, if necessary, a finite number of the terms of the sequence {ltm }, we can therefore assume that Q", (D for m = 1, 2, .... Since olJoui exists at U o by assumption, we have

Also, since obviously u o, umj E Qm' we have

Iii (Umj) - Ii (UO) I ;;;;; II T Umj - T Uoll ;;;;; 0 T Qm·


Iii (Umj) - Ii (UO) I ;;;;; (l (j) d L tn.

Noting that

we conclude that


(23)

(24)

Observe that in view of (i), which we already proved, the first partial derivatives of the coordinate functions of T exist a.e. in D. Further-

§ V.3. Special classes of differentiable transformations inRn. 373

more, since IjJ is L-summable in D, by III.1.i, lemma 43 we have

except for certain points U o which constitute a set of L-measure zero in D. Thus (24) implies that

1'·afio-l~ljJl/n a.e. in D, i,j=1, ... ,n. , au1

As J(u, T) is a sum of n! terms each of which is a product of n of the first partial derivatives of the coordinate functions of T, it follows that

i](u, T)i ~ n! IjJ (u) a.e. in D. (25)

Since IjJ is L-summable in D, (25) implies (ii), and the lemma is proved.

Definition 2. Given T as in (1) and (2), T is termed generalized Lipschitzian in a domain D(D if (a) for every n-interval I(D there exists a real-valued, non-negative function IjJr (u) such that T satisfies the condition (IjJr, int I) in the sense of definition 1, and (b) the ordinary Jacobian J(u, T) is L-summable in D.

Note that the domain D, occurring in the preceding definition, may coincide with D.

As regards the condition (b) in the preceding definition, observe that by 1.2.2, exercise 8 there exists a sequence {Ik} of n-intervals in D such that D = U int I k • The condition (a) implies, in view of lemma 2, that T possesses a total differential a.e. in int Ik , 1< =1,2, .... Clearly it follows that T possesses a total differential (and hence also a weak total differential) a.e. in D. Hence the first partial derivatives of the coordinate functions of T exist a.e. in D. Thus J(u, T) exists a.e. in D, and hence the condition (b) in definition 2 is meaningful. Note that the condition (a) in definition 2 implies, by lemma 2, that T satisfies the condition (N) in intlp" k=1, 2, .... By IV.1.4, lemma 1 it follows that T satisfies the condition (N) in D. These remarks yield the following statement.

Theorem 1. Assume that T, given as in (1) and (2), is generalized Lipschitzian in a domain D (D. Then (a) T possesses a total differential (and hence also a weak total differential) a.e. in D, (b) the ordinary Jacobian J(u, T) is L-summable in D, and (c) Tsatisfies the condition (N) in D.

In view of this theorem (in which D may coincide with D) a number of previously established general results can be applied to generalized Lipschitzian transformations. The following statement summarizes the principal conclusions obtained in this manner.


Theorem 2. Assume that T, given as in (1) and (2), is generalized Lipschitzian in D. Then the following holds.

(i) T is both sACB and eAC in D. (il) Js (u, T) = Ie (u, T) = J(u,T) a.e. in D. (iii) Ds (11, T) = De (u, T) = IJ(u, T) I a.e. in D. (iv) N(x, T, D) = K(x, T, D) a.e. in R". (v) If 5 is an L-measurable subset of D, then

J I](u, T)I dL = J N(x, T, 5) dL. 5

(vi) More generally, if 5 is an L-measurable subset of D and H(x) IS a finite-valued, L-measurable function in R", then

IH(Tu) IJ(u, T)ldL=IH(x)N(x, T, S)dL, s


(vii) I J(u, T) dL = I ,Ue (x, T, D) d L. v

(viii) More generally, if H(x) is a finite-valued, L-measurable function in R", then

I H(T u) ](u, T) dL = I H(x)Pe(X, T, D) dL, v


(ix) If T is defined and continuous on Jj and LT Ir D = 0, then

J J(u, T) dL = Ildx, T, D) dL. D

More generally, if H(x) is a finite-valued, L-measurable function in R", then

I H(T II) J(u, T) dL = I H(x)!t (x, T, D) d L, D


Proof. In view of theorem 1, the statements (i), (ii), (iii), (iv) follow directly from V.3.3, theorem 6, and the statements (v) and (vi) follow directly from V.3.4, theorem 3. Since Tis eAe in D by (i), the statements (vii), (viii), (ix) follow (in view of theorem 1) directly from V.3.4, theorem 1 and V.3.4, theorem 2.

We proceed to discuss some significant special instances of generalized Lipschitzian transformations in R" (a more detailed study of this topic will be made in Part VI for the case n = 2).

Lemma 3. Given T as in (1) and (2), assume that T satisfies the condition (f[>, D) in the sense of definition 1. Then T is generalized Lipschitzian in D.


Proof. The assumption implies (see lemma 2) that J(u, T) exists a.e. in D and is L-sumn able in D. Furthermore, if f is any n-interval in D, then clearly T satisfies the condition ((/J, int f), and the lemma follows.

Lemma 4. Given T as in (1) and (2), assume that Tis Lipschitzian in a domain D(D. Then there exists a (finite) constant K;;;.O such that on setting (/J (u) = K, T satisfies the condition ((/J, D).

Proof. By assumption there exists a (finite) constant M;;;.O such that

(26)

for every pair of points u1 , u 2 ED. Define

(/J(u) _M" n" for uED. (27)

Consider any oriented n-cube Q (D, and let h be the side length of Q. Then [see (27)]

J (/JdL =J1vr n" LQ =J11." nn it". Q

If u1 , u 2 are any two points in Q, then clearly

II U 2 - ttl II ::::: n h,

and hence, by (26) and (28),

IITu 2 - Tu 1 11::::: M nk = (I (/JdLt"·

(28)

Since 211 , 212 were arbitrary points of Q, it follows (see 1.1.4, definition 4) that

oTQ:::::(J(/JdL)lfn, ,Q '


Lemma 5. Assume that T, given as in ('I) and (2), is Lipschitzian in D. Then T is also generalized Lipschitzian in D.

Proof. By lemma 4 (applied with D = D) T satisfies the condition (W, D) for an appropriately chosen W. Hence T is generalized Lipschitzian in D by lemma 3.

Lemma 6. Assume that T, given as in (1) and (2), satisfies the following conditions. (a) The first partial derivatives of the coordinate functions of T exist a.e. in D, (b) the ordinary Jacobian J(u, T) is L-summable in D, and (c) T is Lipschitzian in every n-interval f (D. Then T is generalized Lipschitzian in D.

Proof. Consider any n-interval f (D. The condition (c) implies that T is Lipschitzian in int f. Hence by lemma 4 (applied with D = int I) T satisfies the condition ((/J1' int I) for an appropriately


chosen (/>[. Since J(u, T) is L-summable in D by assumption, the lemma follows.

Lemma 7. Assume that T, given as in (1) and (2), satisfies the following conditions. (a) The first partial derivatives of the coordinate functions of T exist and are continuous in D, and (b) ](u, T) is Lsummable in D. Then T is generalized Lipschitzian in D.

Proal. In view of lemma 6 it is sufficient to show that T is Lipschitzian on every n-interval I (D. Let such an interval I be assigned. Since I is compact (see 1.2.2, exercise 3), and the first partial derivatives of the coordinate functions 11, ... , In of T exist and are continuous on I, by 1.1.3, exercise 38 there follows the existence of a (finite) constant K = K(I) ~O such that

I, 0 f~ I' :s;; K on I, i, i = 1, ... , n.

cuI

Take any two points

U 1 = (ui, ... , tt~), U 2 = (u~, ... , u~)

(29)

in I, and let i be one of the integers 1, ... , n. For o;;;;t;;;; 1, the point

u (t) = [t u~ + (1 - t) ui, ... , t u~ + (1 - t) un lies in I (see 1.2.3, exercise 3). Accordingly the function

is well defined. Since the first partial derivatives of Ii exist and are continuous in D, we can evaluate the difference

by the rules of elementary Calculus, obtaining

In view of (29) there follows the inequality·

n

Iii (U2) - Ii (U1) I ;;;; K L: IU~ - uil ;;;; nKllu2 - u111· i~l

Hence clearly

" liT u 2 - T u111 ;;;; L: Iii (U 2) - Ii (U1) I ;;;; n2 KIIU2 - U 111, i~l

and thus T is Lipschitzian in I.

§ VI. 1. The topological index in RO. 377

The lemmas 5, 6, 7 yield examples of generalized Lipschitzian transformations. As noted above, more detailed information on this point will be provided in Part VI for transformations in R2. Let us observe that in view of lemma 7 we can apply theorem 2 to the class of the so-called transformations ot class C', obtaining relevant information about the transformation of multiple integrals and the geometrical meaning of the ordinary Jacobian J(u, T) for this classical elementary case.

Part VI. Continuous transformations in R2 t.

§ VI.I. The topological index in R2.

VI.t.t. Preliminaries. Throughout Part VI, we shall be concerned with bounded continuous transformations

(1 )

where D is a bounded domain in R2. It will be convenient to use certain alternative representations for T. Note that the points of R2 are ordered pairs (x, y) of real numbers x, y. Hence we can associate with the point (x, y) of R2 the complex number z=x+iy. In the sequel, the terms" complex number" and" point of R2" will be used interchangeably. Using complex numbers, the transformation (1) can be represented in the form

T:z=T(w), wED, (2)

where z is the image point of wunder T. Thus T is thought of nmv as a bounded, continuous, real or complex-valued function of the complex number wED. If we set w=u +iv, Z= x + iy, where u, v, x, y are real numbers, then we obtain for T the representation

T:x=x(u,v), y=y(u,v), (u,v)ED, (3)

where the coordinate functions x (u, v), y (u, v) are bounded, continuous, real-valued functions in D.

t For historical comments and further details (including applications in Surface Area Theory and Calculus of Variations) see the treatise on Length and Area listed in the Bibliography, and the expository paper by L. CESARI and T. RADO, Applications of Area Theory in Analysis, Proc. Internat. Congr. of Math. 2, 1 i4-179 (1950). For the equivalence (in the case n = 2) of the basic concepts used in this volume with those occurring in alternative approaches, see T. RADO, Twodimensional concepts of bounded variation and absolute continuity [Duke Math. ]. 14,587-608 (1947)]. It is easy to see that the essential multiplicity function K used in this volume corresponds to the multiplicity function '1'* occurring in the paper just referred to.

378 Part VI. Continuous transformations in R2.

The two-dimensional Euclidean space R2 possesses various properties not shared by the general Euclidean ·spaces R". Furthermore, the field of complex numbers possesses various special algebraic properties which play an important role in the sequel. Due to these circumstances, the theory of continuous transformations in R2 presents several remarkable special features as compared with the general theory developed in Part IV and Part V. Our main objective in Part VI is to discuss some of these special features.

In view of the availability of several excellent and completely rigorous treatises on the topology of R2 it seems unnecessary to give a detailed presentation of the topological theorems for R2 which we shall have occasion to use in the sequel (even though many such theorems follow directly from the general results derived in § 1.7). Accordingly, we shall merely state here some basic definitions and theorems concerning R2, and throughout Part VI we shall make free use of figures to illustrate statements that are both intuitive and true.

A s£mple are y in R2 is a set which is the homeomorphic image of the closed unit interval 0;;;;: u;;;;: 1. A simple arc is compact and connected,

and hence it is a continuum. A simple closed curve (or JORDAN curve) in R2 is a

c,

o c::,

Fig. 45.

set which is the homeomorphic image of the unit circle u 2 + v2 = 1. A simple closed curve is also a continuum. A bounded, finitely connected JORDAN region R in R2 is obtained as follows (see the figure). One starts with a simple

closed curve Co, and one selects a finite number m;;;;;O of simple closed curves C;, j = 1, ... , m, such that each Cj is interior to Co, and Cj

and Ck are exterior to each other if j =1= k. Then Co, Cr , ... , Cm constitute the frontier of a uniquely determined bounded domain D. The set

D U Co U ... U Cm

is termed a bounded, finitely connected JORDAN region R, and the set B = U Ck , Ii = 0, ... , m, is termed the boundary of R. A bounded, finitely connected JORDAN region R is a continuum, and we have the relations

'" fr R = fr in t R = B = U C k , k~O

where the symbols fr, in! are used in the sense of 1.1.3, definitions 4, 5. According as m = 0, 1, 2, ... , the region R is said to be simply, doubly, triply, ... connected. Any two finitely connected, bounded JORDAN

regions Rr , R2 with the same number of boundary curves are homeomorphic. In fact, if wr , w2 are arbitrarily assigned points in Rr , R2

§ VI. 1. The topological index in R·. 379

respectively, such that either wi E int Ri , j = 1, 2, or wi E Ir ~, j = 1, 2, then there exists a homeomorphism h from RI onto R2 which carries WI into w2 •

If we have occasion to consider a continuous transformation T from a bounded, finitely connected JORDAN region R into R2, then we use [in analogy with (1), (2) (3)] the alternative representations

T:z=T(w), wER,

T:x=x(u,v), y=y(u,v), (u,v)ER.

(4)

(5)

(6)

If I is a real or complex-valued function on a set 5 (R2, then we shall use anyone of the alternative notations

f(u, v), f(w), f(x, y), f(z),

where w = u + iv, z = x + i y (and of course u, v, x, y designate real numbers). Similarly, definite integrals will be written in one of the forms

fff(u,v)dudv, fff(w)dudv, fJt(x,y)dxdy, Jff(z)dxdy.

Thus we write now dudv or dxdy instead of dL, in conformity with general usage.

As. we noted above, a point of R2 is an ordered pair (x, y) of real numbers x, y. In conformity with the practice in Analytic Geometry, we interpret x and y as the first and the second coordinates of the point (x, y) with respect to a Cartesian coordinate system.

The matter of orientation enters the picture in this connection. Let us observe that we have at our disposal a positive orientation in R2 according to the abstract pattern developed in § 11.1. On the other hand, we have at our disposal the intuitive conception of orientation in R2. According to this intuitive conception (which of course can be formulated in an entirely rigorous manner), one speaks of a counterclockwise orientation, while the opposite orientation is referred to as the clock-wise orientation. To avoid circularity, we shall use the term positive orientation solely to refer to the abstractly defined orientation described in 11.1.4.

Let us recall a few definitions and facts concerning complex numbers. If x and yare real numbers and z = x + i y, then x and yare termed the real part and the lmagmary part respectively of the complex number z. In symbols


The absolute value of z, denoted by Izl, is given by the formula

Izl = (XZ + yZ)~. If z =F 0, then one has for z the trigonometric representation

Z = Izi (cosq? + isinq?L (7}

where q? is a real number which represents the radian measure of a certain angle in a well-known manner.

Definition 1. If z =F 0 is a complex number, then every real number rp which satisfies (7) is termed an argument of z. No arguments are assigned to z = o.

Lemma 1. If Z=FO is a complex number, and r>O and q? are real numbers such that

z = r (cosq? + i sinq?) ,

then r = Izl and q? is an argument of z.

Lemma 2. If Zl =F 0, ... , z" =F 0 are complex numbers and q?l' ... , q?n

are arguments of Zl' ... , zn respectively, then q?l + ... + q?" is an argument of the product Zl'" z".

Lemma 3. If z is a complex number such that Iz -11 < 1, then ffiz>o.

Lemma 4. If Z=FO is a complex number and q?, "p are any two arguments of z, then "p = q? + 2n 'J'l, where n is an integer.

These lemmas (which state well-known elementary facts) are listed merely for convenient reference.

Consider now the half-plane ffi z> 0 (that is, the set of those complex numbers z whose real part is positive). If ffi z > 0, then clearly z has a unique argument IX(Z) such that

_ ~ < IX (z) < Jl • 2 2

(8)

Clearly IX (z) is a continuous function of Z III the half-plane ffi z> O.

Definition 2. For complex numbers z such that ffi z> 0, the uniquely determined argument satisfying (8) will be denoted by arg z.

Thus arg z is a single-valued, continuous function of z which is defined only in the half-plane m z > O.

VI. 1.2. The argument of a complex-valued function. Let r be a continuum in R2 (see 1.1.3, definition 15). Consider a continuous (real or complex-valued) function f (w) on r. Put

M(j,r)=maxlf(w)l, wEr, (1)

m(j,F)=minlf(w)l, wEr, (2)

w(j, r) = max If(wz) -f(w1)1, WI' wzE r. (3)

§ VI. 1. The topological index in R2. 381

Since r is compact and I is continuous on r, the use of the max and min in (1), (2), (3) is justified in view of 1.1.3, exercises 38 and 39.

Assume now that I (w) =1= 0 on r. We have then (see VI.1.1, definition 1) for each point wEr infinitely many arguments for the complex number I(w) =1=0. If for each wEr we select a definite one of the arguments of I(w), then we obtain a single-valued function g:>(w) on r which is termed a single-valued argument for I(w) on r. It mayor may not be possible to make the selection in such a manner that g:> (w) is continuous on r.

Delinition 1. Let I (w) be a continuous, real or complex-valued function on a continuum r(R2, such that I(w) =1=0 on r. Then a function g:> (w) which is real-valued and continuous on r is termed a singlevalued continuous argument of I (w) on r if

I(w) =1/(w)1 (cosg:>(w) + i sin g:> (w)) , wEr.

Delinition 2. Let t (w) be a continuous, real or complex-valued function on a continuum r(R2 such that I(w) =1=0 on r. If there exists a single-valued continuous argument of I (w) on r, then I is said to satisfy the condition (arg, r).

Lemma 1. Assume that I satisfies the condition (arg, r). Then the following holds.

(i) If g:> (w) is a single-valued continuous argument of I (w) on r and n is an integer, then g:> (w) + 2n n is also a single-valued continuous argument of I (w) on r.

(ii) Conversely, if g:> (w) and 'tjJ (w) are any two single-valued continuous arguments of I (w) on r, then

'tjJ(w) = g:>(w) + 2nn, wE r,

where n is an integer independent of w. In particular, if 'tjJ(wo) = g:>(wo) for some point woEr, then 'tjJ(w) = g:>(w) on r.

Prool. (i) is obvious. To prove (ii), set

A (w) =1£(111) - cp (w), wE r. 2n

Clearly A(W) is continuous on r, and from VI.1.1, lemma 4 we conclude that A(W) is integral-valued. Since r is connected, by 1.1.3, exercise 37 it follows that A(W) is constant on r. Hence A(W) = n, where n is a constant integer, and (ii) follows.

Lemma 2. Let f (w) be a continuous, real or complex-valued function on a continuum r( R2, such that ffif(w) > 0 on r. Then I satisfies the condition (arg, F).

382 Part VI. Continuous transformations in RO.

Proof. Using the function arg z introduced in VI.Ll, definition 2, we note that the function

lJl(w) = arg /(w), wE F,

is clearly a single-valued continuous argument of / on F. Lemma 3. Let /(w) be a continuous, real or complex-valued function

on a continuum F(RZ, such that [see (2), (3)]

w(j, F) < m(j, F). (4)

Then / satisfies the condition (arg, F) . Proof. Note that the assumption implies that / =F 0 on r. Select

a point woE F and let lJlo be an argument of / (wo). Then

/ (wo) = II (wo) I (cos lJlo + i sin rpo)· (5 )


(6)

Then g(w) is continuous on F, and [by (4)J

Ig(w) -11 = if(w) - f(u'ill ~ w(f, r) < 1, Ole' E F. If(wo)1 - m(f, T)

By VI.Ll, lemma 3 it follows that mg (w) > 0 on r. Hence, by lemma 2 (applied to g) there exists a single-valued continuous argument "P (w) of g on r. In view of (5) and (6) it is clear that lJlo+"P(w) is a realvalued continuous argument of / (1U) on F, and the lemma is proved.

Lemma 4, Let 11 (w), "" In (w) be continuous, real or complexvalued functions on a continuum FCR2, such that Ij=FO on F and Ii satisfies the condition (arg, F), j = 1, 0 •• , n. Then the product 1=/1" 0 In also satisfies the condition (arg, F). Furthermore, if lJll (w), .. " rpll (w) are single-valued continuous arguments of 11 (w), '0', In (w) respectively on F, then rpl + ... + rpn is a single-valued continuous argument of I = 11 ... In on r.

Proof, By assumption, there exists a single-valued continuous argument rpi (w) of Ii (w) on F, j = 1, .'" n, Then lJll + ... + lJl" is a single-valued continuous argument of / = /1 ., '/" on F in view of VI.Ll, lemma 2, and the present lemma follows.

Lemma 5. Let 1;., rz be two continua in R2 such that there exists a homeomorphism h:1;. -7-r2 from I;. onto r2 • Let /1 (w) be a continuous, real or complex-valued function on r1 , such that /1 (w) =F 0 on r1 and there exists a single-valued continuous argument rpl (w) of 11 (w) on 1;.,

Put

§ VI. 1. The topological index in R2.

Iz(w) = 11 (h-l(w)) , wE rz,

IPz(w) = IPl(h-l (w)) , wEr2 •

383

(7)

(8)

Then IPz(w) is a single-valued continuous argument of Iz(w) on r2 •

Proof. Clearly IP2 (w) is a well-defined, real-valued, continuous function on~. If wEr2 , then h-l(W)E~, and by assumption

11 (h-l(w)) = III (h-l(W)) I [COSIPI (h-l(W)) + i sin IPl (/t-1(w))] .

In view of (7) and (8) this last formula yields

f2(W) = I/z(w)1 (COSIP2(W) + i sinIP2(w)), wE ~,'

and the lemma follows.

Lemma 6. Let r,~, ~ be three continua in RZ such that r=!;. u ~ and ~ n r z is non-empty and connected. Let I(w) be a continuous, real or complex-valued function on r, such that f (w) =F 0 on rand f(w) satisfies the conditions (arg,~) and (arg, rz). Then I(w) satisfies the condition (arg, r).

Prool. Take a point woE!;. n rz, and let Po be an argument of I(wo). By assumption, there exists a single-valued continuous argument IPt (w) of I (w) on ~ and also a single-valued continuous argument IP2*(W) of I(w) on rz. By VI.1.1, lemma 4 we have

IPt(wo) = Po + 2nl :7t, IP:(wo) = Po + 2n2 :7t,

where 1tl , n2 are integers. On setting

by lemma 1 it follows that IPl (w), IPz(w) are single-valued continuous arguments of f (w) on ~ and rz respectively. Clearly

(9)

Furthermore, since 1;. n r2 is a subset of both!;. and r2 , it is clear that IPI (w) and IP2 (w) are single-valued continuous arguments of I (w) on ~ n 1;. Now since ~ n 1; is connected, by lemma 1 (applied to .z;.nrz) it follows that IP2(W)-tpl(W) is constant on .z;.n1;. Hence, in view of (9),

(10)

Define now a function IP (w) on r by the formulas


Then g:;(w) is single-valued on F as a consequence of (10). Obviously g:;(w) is an argument of I(w) for every wEF. Since 4 and F2 are both closed, it follows readily that the continuity of g:;l (w), g:;2 (w) on 4 and r; respectively implies the continuity of g:;(w) on F= 4 U F2 • Thus g:;(w) is a single-valued continuous argument of I(w) on F, and the lemma is proved.

Delinition 3. A continuum F (R2 is said to satisfy the condition (arg) if every continuous, real or complex-valued function I(w) on F which is different from zero on F satisfies the condition (arg, F).

Lemma 7. Let Q be the unit square 0;;;; u;;;; 1, 0;;;; v;;;; 1 in R2. Then Q satisfies the condition (arg).

Proal. Consider any continuous, real or complex-valued function I(w) in Q which is different from zero in Q. We have to show that I(w) has a single-valued continuous argument in Q. Deny this assertion. Divide Q into two congruent rectangles R1 , R~ by a vertical segment. If I(w) would satisfy both of the conditions (arg, R1) and (arg, R~), then by lemma 6 it would follow that I(w) satisfies the condition (arg, Q), which we denied. Hence I(w) must fail to satisfy one of the conditions (arg, R1), (arg, R~), say (arg, Rl). On subdividing Rl into two congruent rectangles R2 , R~ by a horizontal segment, we obtain in the same manner the conclusion that I (w) must fail to satisfy one of the conditions (arg, R2 ), (arg, R~), say (arg, R2). Repeating this argument, we obtain an infinite sequence {Rn} of oriented rectangles m Q such that

(i) R1 )···) Rn) ... , (ii) oRn--,>-O, and (iii) I(w) fails to satisfy the condition (arg, R,,), n=1, 2, ....

Now since l(w)=l=O in Q, we have [see (2)J m(l,Q»O, and hence

m (I, R,,) ;;::;; m (I, Q) > 0, n = 1, 2, .... (11 )

Since I(w) is continuous in Q, and Q is compact, I(w) IS uniformly continuous in Q. Hence (ii) implies [see (3)J that

w(j, R,,) ---7-0. ( 12)

From (11) and (12) we conclude that if N is a sufficiently large positive integer, then

But then, by lemma 3, I(w) satisfies the condition (arg, RN ), in contradiction with (iii). Thus denial of the lemma leads to a contradiction, and the proof is complete.

§ VI.1. The topological index in R·. 385

Lemma 8. If I is the unit interval O;S;; u;S;; 1 on the number-line, then I satisfies the condition (arg).

The proof, based on successive subdivisions of I, is entirely analogous to that of lemma 7.

Lemma 9. If R is a bounded, simply connected JORDAN region in R2, then R satisfies the condition (arg).

Proof. Let Q be the unit square O;S;; u ;S;; 1, 0 ;;;;; v:;:;: 1 in R2. Then there exists (see VI.1.1) a homeomorphism h: Q-+R from Q onto R. Assign now any continuous, real or complex-valued function g(w) in R which is different from zero in R. We have to show that there exists a single-valued continuous argument of g(w) in R. Define

f(w)=g(h(w)), wEQ.

By lemma 7, there exists a single-valued continuous argument r:p (w) of f(w) in Q. Since

g(w) = f(h-I(W)) , wE R,

we conclude from lemma 5 that r:p (h-I (w)) is a single-valued continuous argument of g (w) in R, and the proof is complete. v

Lemma 10. If y is a simple arc in R2, then y satisfies the condition (arg).

Noting that y is homeomorphic to the unit in terval 0:;;;: u ;;;;; 1, the proof is made in the same manner as in lemma 9, except that lemma 8 is now used instead of lemma 7.

Consider now a simple arc in R2 with endpoints A, B. Such an arc admits of two orientations which are indicated by the arrows in

~' A

u Fig. 46.

the figure. By an oriented simple arc y we mean a simple arc with assigned orientation. In the figure, if one assigns the orientation indicated by the upper arrow, then A is the first end-point and B is the second end-point of the oriented simple arc y. The same simple arc with the opposite orientation is denoted by -yo The first endpoint of -y is then B and the second end-point of -y is A.

Definition 4. Let y be an oriented simple arc in R2 with first endpoint WI and second end-point w2 • Let f(w) be a continuous, real or complex-valued function on y which is different from zero on y. By lemma 10 there exists a single-valued continuous argument r:p (w) of f (w) on y. The quantity

Vy[arg f(w)] =r:p(w2) -r:p(w l ) (13)

is termed the variation of the argument of f(w) on the oriented simple arc y.



To justify this definition, we have to show that if tp (w) is any other single-valued continuous argument of I (w) on y, then

(14)

Now by lemma 1 we have

tp(w) =cp(w) + 2nn, wE y,

where n is a constant integer, and thus (14) is obvious. Lemma 11. Let y be an oriented simple arc in RZ with first end

point w1 and second end-point w2 • Then the following holds.

(i) If I (w) is a continuous, real or complex-valued function on y which is different from zero on y, then

Y,,[arg/(w)] = - L~[arg/(w)]. (15 )

(ii) Given I(w) as under (i), select a point w* of y which is different from both W 1 and w2 , and denote by Yl' yz the oriented sub-arcs of y with first and second end-points w1, w* and w*, Wz respectively. Then

Y,,[arg I(w)] = v;,,Carg I(w)] + Y".[arg I (w)J. (16)

(iii) If 11 (w), ... , 1m (w) are continuous, real or complex-valued functions on y such that Ij(w) =f=. 0 on y, j = 1, ... , m, then

Y" [arg £1 Ij (W)] = j~1Y" [arg Ij (w)] . (17)

Proal. To verify (i) and (ii), select a single-valued continuous argument cP (w) of I (w) on y. Noting that W 2 is the first end-point and WI

is the second end-point of -y, we have by definition the formula

V_y[arg I(w)] = cp(wl ) -cp(wz)'

and (15) follows in view of (13). Furthermore, by definition,

Vy,[arg I(w)] = cp(w*) - cp(wl ) ,

Y".[arg I(w)] = cp (w2) - cp (w*) ,

and (16) follows in view of (13). To prove (17), let CPj(w) be a single-valued continuous argument of Ij(w) on y, j = 1, ... , m. By definition

v;,[arglj(w)]=CPj(wz)-CPj(wl ), j=1, ... ,m. (18)

On the other hand, the function

'" cp(w) = LCPj(w), wEy, (19) i=1

§ VI.1. The topological index in RI. 387

is a single-valued continuous argument of fl ... fm on 'Y (see lemma 4). Hence by definition

v" [argiIT fi(W)] = q;(w2) -q;(w1),

and (17) follows in view of (18) and (19). Lemma 12. Let wo, w', w" be the vertices of a non-degenerate

triangle '1: in R2. Denote by A- the radian measure of the angle of '1:

at Wo and by 'Y the oriented segment with first end-point w' and second end-point w". Then '"

if the arrangement wo, w', w" corresponds to the counter-clockwise orientation of the perimeter of '1:,

and v" [arg (w - wo)] = - A-

if the arrangement wo, w', w" corresponds to the clock-wise orientation of the perimeter of '1:.

w"

w'

u Fig. 47.

Proof. The reasoning is indicated in the figure which illustrates the case when the arrangement wo, w', w" corresponds to the counterclockwise orientation of the perimeter of T. The radian measure of the angle q; =q; (w) is selected as a single-valued continuous argument of w - Wo on the oriented segment 'Y with first end-point w' and second end-point w". Clearly

v;, [arg (w - wo)] = q; (w") - q; (w') = A-.

Consider now a simple closed curve in R2. Such a curve admits of a counter-clockwise orientation and of a clock-wise orientation (in the figure, these orientations are indicated by " the upper and the lower arrow respectively). By an oriented simple closed curve C we mean a simple closed curve with assigned orientation. The same simple closed curve with the opposite orientation is then denoted by -C.

Definition 5. Let C be an oriented simple closed curve in R2. Consider a continuous, real or complex-valued function f(w) on C which is

u Fig. 48.

different from zero on C. Take a finite number of distinct points wo, WI' ••• ,w" on C (where n;;;:; 1) which follow upon each other in conformity with the orientation of C. Let 'Yi' j = 0, ... , n -1, be the oriented

25*


sub-arc of C with first end-point wi and second end-point Wi+l' and let y,. be the oriented sub-arc of C with first end-point w,. and second end-point W O' where these sub-arcs are oriented in conformity with v the assigned orientation of C (see

w.~

Fig. 49.

the figure). Then the quantity

.. (20) Vc [arg I(w)] }

=i~ v"j[arg I(w)]

is termed the variation of the argument of I(w) on the oriented simple closed curve C.

To justify this definition, it U must be shown that the value

of the summation in (20) is independent of the particular choice of the points of division on C. However, this fact follows readily by repeated application of part (ii) of lemma 11.

Lemma 13. Let C be an oriented simple closed curve in R2. Then the following holds.

(a) If f(w) is a continuous, real or complex-valued function which is different from zero on C, then

VcCarg I(w)] = - V_cCarg f(w)J.

(b) If 11 (w), ... , 1m (w) are continuous, real or complex-valued functions on C such that li(w)=f:O on C, j=1, ... , m, then

Vc [arglI Ii (W)] = ~1 VcCarg fi(w)].

(c) Given I(w) as under (a), the quantity

1 - VcCarg I(w)] 2:n

is equal to an integer.

(d) Given I(w) as under (a), assume that f(w) satisfies the condition (arg, C). Then

Vc [arg I(w)] = O. (21)

(e) Given I(w) as under (a), select another oriented simple closed curve C* in R2, and let h be a homeomorphism from C onto C*, such that the assigned orientation of C* corresponds under h to the assigned

§ VI. 1. The topological index in RO. 389

orientation of C. Define

f*(w) = f(h-I(w)) , wEC*. Then

VcCarg f(w)] = Vc.[arg f*(w)].

Proof. The statements (a), (b), (e) follow, in view of definition 5, readily from parts (i) and (iii) of lemma 11 and from lemma 5 respectively. To prove (c), select a single-valued continuous argument q;j(w) of t(w) on each one of the oriented arcs Yi' j = 0, ... , n, occurring in definition 5. Then, definition 4 and definition 5 yield [after re-arranging the summation in (20)] the formula

Vc [arg f(w)] = [q;o (WI) - q;1 (WI)] + ... } + [q;,,-I(Wn) -q;n(Wn )] + [q;n(Wo) -rpo(Wo)].

(22)

Each one of the differences appearing in (22) is the difference of two arguments of the same complex number. Thus, by VI.1.1, lemma 4, each one of these differences is equal to 2:n times some integer, and (c) follows. Assume now that there exists a single-valued continuous argument q; ('1£') of t (w) on C. Then we can choose

q;j(w)=q;(w), wEYi' j=O, ... ,n,

and (21) follows from (22).

Lemma 14. Given an oriented simple closed curve C in R2, let F(w), G(w), g(w) be three continuous, real or complex-valued functions on C such that

F(w) =G(w) +g(w), IG(w)1 >lg(w)1 on C. (23) Then

VcCarg F(w)] = VcCarg G(w)J. (24)

Proof. Note that the assumptions imply that F(w)=f=O, G(w)=f=O on C. Consider the auxiliary function

f( F(w) w)= G(wl' wECo (25)

Clearly f(w) is continuous and different from zero on C. From part (b) of lemma 13 we conclude that

VcCargF(wl)] = VcCarg G(w)] + VcCarg f(w)J. (26)

On the other hand, (23) and (25) yield

If(w) - 11 = IF(w) - G(wll = Jg(wll_ < 1 E C IG(wll IG(w)I' w .


Hence, byVI.1.1, lemma 3, 'iJt/(w) >0 on C. By lemma 2 it follows that I(w) satisfies the condition (arg, C), and hence by part (d) of lemma 13 we have

VcCarg I(w)] = O. (27)

Since (26) and (27) imply (24), the lemma is proved.

Lemma 15. Given an oriented simple closed curve C in R2, let I(w), In(w) be continuous, real or complex-valued functions on C such that l(w)=I=o, In(w) =1=0 on C, n=1, 2, ... , and

In (w) -+ I (w) uniformly on C. (28)

Then there exists an integer N such that

VcCarg I(w)] = VcCarg In (w)] for n>N. (29)

Proal. Since I (w) is continuous and different from zero on C, we have [see (2)] m (f, C) > O. In view of (28) there follows the existence of an integer N such that [see (1)]

M(f-In, C)< m(f, C) for n>N. (30)

On setting, for fixed n > N,

we have [in view of (30)]

1,,=G+g, IGI>lgl on C,

and thus (29) follows from lemma 14 (applied with F = In' n > N).

Lemma 16. Given a bounded, simply connected JORDAN region R in R2, let C be the arbitrarily oriented boundary curve of R, and let I(w) be a continuous, real or complex-valued function in R which is different from zero in R. Then

Vc [arg I (w)] = o. (3 1 )

Proal. By lemma 9 there exists a single-valued continuous argument cp (w) of I (w) in R. Clearly cp (w) yields a single-valued continuous argument of I(w) on C, and thus (31) follows from part (d) of lemma 13·

Delinition 6. Let R be a bounded, finitely connected JORDAN region in R2. By the oriented boundary B of R we mean the boundary of R with the standard orientation obtained as follows: the exterior boundary curve of R is oriented counter-clockwise, while the other boundary curves (if present) are oriented clock-wise. We shall write CE B to


state the fact that C is a boundary curve of R, oriented in the manner just described.

Delinition 7. Let B be the oriented boundary of the bounded, finitely connected Jo .. mAN region R(Ra (see definition 6), and let I(w) be a continuous, real or complex-valued function in R such that l(w)=l=O on B. Then the quantity

VB [arg I (w)] = L Vc [arg I(w)], C E B, (}2)

is termed the variation of the argument of I (w) on the oriented boundary B of R.

Lemma 17. Let B be the oriented boundary of the bounded, finitely connected JORDAN region R in R2, and let I(w) be a continuous, real or complex-valued function in R such that I(w) =l=0 on B. Then the following holds.

(a) If I (w) is different from zero in R, then

VB [arg I(w)] = o. (33)

(b) If G(w), g(w) are continuous, real or complex-valued functions in R such that l(w)=G(w)+g(w) on Band IG(w)!>lg(w)1 on B, then

(c) If {In (w)} is a sequence of continuous, real or complex-valued functions in R such that In(w)~/(w) uniformly on B, then there exists an integer N such that v

VB [arg I .. (w)] = VB [arg I(w)] for n> N.

Prool. (b) and (c) follow, in view of (}2), directly from lemma 14 and lemma 15 respectively. The proof of (a) follows readily by considering the figure (which illustrates the case when R is doubly connected). The auxiliary dotted arcs decompose R into two simply connected

u Fig. so.

JORDAN regions Rl , R2• On denoting by 1;., r 2 the counter-clockwise oriented boundary curves of Rl , R2 respectively, we have

Vr,[arg I (w)] = 0, Vr, [arg I(w)] = 0

by lemma 16. Clearly

(}4)

VB [arg I(w)] = Vr, [arg I(w)] + Vr.[arg I(w)], (}5)

392 Part VI. Continuous transformations in R".

since the contributions of the dotted arcs cancel out by part (i) of lemma 11, and thus (33) follows from (34) and (35).

Lemma 18. Let ZI' Z2' Z3 be the vertices of a non-degenerate triangle Lt in R2. Denote by C the perimeter of Lt, oriented in conformity with v the arrangement ZI' Z2' Zs of the

u Fig. 51.

vertices (see the figure). If Zo

is an interior point of Lt, then

Vc [arg (z - zo)] = 2n

if the orientation of C is counterclockwise, and

VcCarg (z - zo)] = - 2n,

if the orientation of C is clockWIse.

Proof· On denoting by Yl' Y2' Ys the oriented sides of Lt (see the figure, which illustrates the case when the orientation of C is clockwise), we have in view of definition 5

s Vc [arg (z - zo)] = L Vy; [arg (z - zo)]. (36)

i=1

Let A.1, )'2' A.s be the radian measures of the angles at the vertex Zo in the triangles Z2 Zo ZS, Z3 Zo ZI' ZI Zo Z2 respectively. Then by lemma 12

Vy,[arg(z-zo)]=-)'i' i=1,2,3· (37)

VcCarg (z - zo)] = - 2n.

A similar argument applies in the case when the orientation of C is coun ter -clockwise.

Consider now the following situation. Let WI' W2, W3 be the vertices of a non-degenerate triangle t in R2, such that the arrangement WI'

W 2 , Ws of the vertices corresponds to the counter-clockwise orientation of the perimeter r of t. Let S} be a non-singular linear transformation from R2 into R2 (see 11.2.5). On setting Zi = S}(Wi), i = 1,2,3, the points ZI' Z2' Zs are then the vertices of a non-degenerate triangle Lt in R2. Denote by C the perimeter of Lt, oriented in conformity with the arrangement ZI' Z2' Z3 of the vertices. Let Zo be a point in R2 which does not lie on C.

§ VI.t. The topological index in R2. 393

Lemma 19. Under the conditions just described, we have

Vr [arg (B(w) - Zo)] = 2nsgn det B (3 8)

if Zo is interior to LI, and

Vr [arg (B(w) - zo)] = 0 (39)

jf Zo is exterior to Lt.

Proof. Assume first that Zo is interior to LI, and consider for instance the case when det B > O. Then the arrangement Zl' Z2' Za corresponds to the counter-clockwise orientation of the perimeter of LI, and hence by lemma 18

Vc [arg (z - zo)] = 2n = 2n sgn det B, (40)

since det B > 0 by assumption. Also, since B is a homeomorphism, we have by part (e) of lemma 13

and (38) follows in view of (40). Assume next that Zo is exterior to LI. Then B(w) -zo=l=O in t, and (39) follows by part (d) of lemma 13.

Lemma 20. Let B be the oriented boundary of the bounded, finitely connected JORDAN region RCR2. Consider a continuous transformation T: R-+R2. Let Zo be a point in R2 such that zoEfTB. Then there exists a real number 'Yj> 0 such that

VB [arg (T(w) - zti)] = VB [arg (T(w) - zo)] if Izo - ztil < 'Yj. (41)

Proof. We assert that the number

'Yj = min IT(w) - zol, wE B, (42)

is adequate for our purposes. Note first that 'Yj>0 since T(w)-zo=l=O on B. Take now any point zt E R2 such that

(43) Define, for wE R,

f(w) = T(w) - Z6, G(w) = T(w) - zo, g(w) = Zo - Zo·

We have then, in view of (42) and (43),

f(w) = G(w) + g(w), IG(w)l> Ig(w)\ on B,

and (41) follows by part (b) of lemma 17.


Lemma 21. Let B be the oriented boundary of the bounded, finitely connected JORDAN region RCR2. Consider a continuous transformation T: R-+R2. Let Zo be a point in R2 such that zoEfTB. Take any finitely connected JORDAN region R* such that

R* C int R, T-I Zo C int R*.

Denote by B* the oriented boundary of R*. Then

Proal. The boundary curves of R* divide R into a finite number of (finitely connected) JORDAN regions Ro, RI , ... , Rm, one of which is R*, say

(45)

Let B j be the oriented boundary of Rj , j = 0, 1, ... , m (thus Bo = B*). If C* is a (properly oriented) boundary curve of R* = Ro, then - C* is a (properly oriented) boundary curve for precisely one of the regions RI , ... , Rm. Note also that zoEfTBj' j =0,1, ... , m since T-I zoC int R*. These remarks yield [in view of (45) and part (a) of lemma 13] the relation

'" VB [arg (T(w) -Zo)] = VB. [arg (T(w) -zo)] + 2.:VBJarg(T(w) - zo)]· (46) i~1

Now since T-I zoC int R*, clearly T(w) - zo=l=O for wE Rj , j = 1, ... , m. By part (a) of lemma 17 we conclude that

VB; [arg (T(w) - zo)] = 0, j = 1, ... , m,


VI.l.3. The k-th root of a complex-valued function. Let us recall that if z=l=O is a complex number and k is a positive integer, then there exist precisely k distinct solutions C of the equation C" = z, while the equation C" = 0 has the unique solution C = O.

Definition. Let I (w) be a continuous, real or complex-valued function on a continuum re R2, and let k be a positive integer. If g (w) is a continuous, real or complex-valued function on r such that g(w)"=f(w) on r, then g(w) is termed a single-valued continuous k-th root of I (w) on r.

Lemma 1. Let i (w) be a continuous, real or complex-valued function on a continuum r e R2 such that (i) i (w) =l= 0 on r, and (ii) I (w) satisfies the condition (arg, T) (see VI.l.2, definition 2). Then for every positive integer k there exists a single-valued continuous k-th root of I (w) on r.

§ VI. 1. The topological index in R". 395

Proof. By assumption there exists a single-valued continuous argument cp(w) of I(w) on r. Then

I(w) = I/(w)1 [coscp(w) +isincp(w)], wEr.

Thus clearly

g(w)=!/(wWlk(COSqJ~W) +isinqJ~W)), wEr,

is a single-valued continuous k-th root of I(w) on r. Lemma 2. Let I (w) be a continuous, real or complex-valued function

on a continuum r( R2, and let k be a positive integer. Assume that there exists a single-valued continuous k-th root go(w) of I(w) on r. Then the following holds.

(i) If A is a k-th root of unity (that is, Ak=1), then Ago(W) is also a single-valued continuous k-th root of I (w) on r.

(ii) If I(w) =l=0 on rand g(w) is any other single-valued continuous k-th root of I(w) on r, then g(w) = Ago (w), where A is a k-th root of unity independent of w. In particular, if g(w*) = go (w*) at some point w*Er, then g(w) = go(w) on r.

Proal. (i) is obvious. To verify (ii), note first that go(w) =l=0 on r [since I(w) =l=0 on r by assumption]. Hence, if g(w) is any singlevalued continuous k-th root of I(w) on r, then we can consider the auxiliary function h (w) = g (w)/go (w) on r. Since h (W)k = 1 on r, it follows that each value of h(w) is a k-th root of unity. Thus h(w) is a continuous function on r which takes on only a finite number of distinct values on r. Since r is connected, it follows (see 1.1.3, exercise 37) that h(w) is constant on r. As h(w)k= 1, the constant value of h(w) must be a k-th root of unity, and (ii) follows.

Lemma 3. Let I(w) be a continuous, real or complex-valued function on a continuum r( R2, and let k be a positive integer. Assume that there exists a single-valued continuous k-th root g(w) of I(w) on r. Let h:r--+r* be a homeomorphism from r onto a continuum r*(R2. Define

I*(w) =f(h-1(w)), g*(w) = g(h-l(W)) , wEr*.

Then I*(w) is continuous on r*, and g*(w) is a single-valued continuous k-th root of I*(w) on r*.

This statement is an obvious consequence of the definitions involved.

Lemma 4. Let R be a bounded, simply connected JORDAN region in R2, and let Wo be a point on the boundary curve of R. Consider a continuous, real or complex-valued function I (w) in R such that


(i) I (Wo) =0, and (ii) I(w) =1=0 if wo=1=wE R. Then for every positive integer k there exists a single-valued continuous k-th root of I(w) in R.

Prool. Without loss of generality we can assume (in view of lemma 3) tha t R is the region

where u+iv=w, and wo=O. For each positive integer n;;;;;2, let us denote by R,. the region

--;'-;;;;:u2 +v2 ;;;;:1, v:;;;;O. n

Clearly Rn is simply connected, and I (w) =1=0 in R,.. Note that the point w=1lies in R,.. Let (f.. be a k-th root of 1(1). Since R,. is simply connected and I(w) =1=0 in R,., by VI.1.2, lemma 9 it follows that I(w) satisfies the condition (arg, R,.). By lemma 1 it follows further that there exists a single-valued continuous k-th root gn(w) of I(w) in R,.. From lemma 2 we conclude that we can select gn (w) so that g" (1) = (f..

for every integer n ~2. Consider now two positive integers m, n such that m;;;;'n. Then clearly R".(R,.. We assert that

g,,(w) = gm(w) for wE Rm· (1 )

Indeed, in Rm the functions g" (w) and gm (w) are single-valued continuous k-th roots of I(w), and g,,(1) = g", (1) =(f... Thus (1) follows by part (ii) of lemma 2. We define now a function g (w) in R as follows:

g(w)=g,,(w) if wER", g(O)=O.

In view of (1) it is clear that g(w) is single-valued and continuous in R, except that the continuity of g (w) at w = 0 is perhaps not immediately obvious. To settle this point, note that since gn (wl = I (w) in R" and 1(0) =0, we have [in view of (1)]

g(W)k = I(w) for wE R. (2)

Now since I(w) is continuous in Rand 1(0) =0, we have I(w)-'?-o for w-'?-O, and hence by (2) also g (w) -'?-O for W-'?-O. In view of (2), it is now clear that g (w) is a single-valued continuous k-th root of I (w) in R, and the lemma is proved.

Lemma 5. Let R be a bounded, simply connected JORDAN region in R2, Wo an interior point of R, and j =1= 0 an integer. Consider a continuous, real or complex-valued function I (w) in R such that (i) f (wo) = 0, (ii) f (w) =1= 0 at every point w =1= Wo in R, and (iii) the variation ofthe argument of f (w) on the counter-clockwise oriented boundary curve C of R is equal to 2nj [see part (c) of VI.1.2, lemma 13]. Then there exists a single-valued continuous IjI-th root of f (w) in R.


Proof. In view of lemma 3 and part (e) of VI.l.2, lemma 13 we can assume without loss of generality that R coincides with the unit disc o ~ u2+ v2 ::;;; 1 and Wo = o. Introduce the auxiliary regions

RI : 0 S;; u2 + v2 :::;: 1 , v;;;;: 0,

R2 : 0;:;;:;: u2 + v2 S;; 1, v;:;;; 0,

and let ex: be an liI-th root of 1(1). By lemma 4 there exist single-valued continuous lil-th roots gl(W), g2(W) of I(w) in RI , R2 respectively, and in view of part (i) of lemma 2 we can select gl (w), g2(W) so that

Denote by CI the semi-circle u2+v2=1, v;;;;;:O, oriented from w=1 toward w=-1, and by C2 the semi-circle u2+v2=1, v;;;;;;O, oriented from w=1 toward w=-1. ByVI.l.2, definition 5 and part (i) of VI.l.2, lemma 11 we have then

Ve,[arg I(w)] - Ve, [arg f(w)] = Vc[arg f(w)] = 2:n;i· (4)

Let «Po be an argument of ex: [see (3)]. Since gl(W) =1=0 on CI , byVI.1.2, lemma 10 there exists a single-valued continuous argument «PI (w) of gl (w) on CI . In view of VI.l.l, lemma 4 we can select «PI (w) so that «PI (1) = «Po. Then (see VI.1.2, definition 4)

(5 )

Similarly it follows that we can select a single-valued continuous argument «P2(W) of g2(W) on C2 so that

(6)

Since gl(w)lil=/(w) on CI and g2(w)lil=/tw) on C2, we obtain from part (iii) of VI.1.2, lemma 11 the relations

Ve, [arg f(w)] = Iii Ve,[arg gl(W)],

Ve.[arg f(w)] = liIVe. [arg g2(W)].


2ni= Iii (<<PI(-1) -«P2(-1)),

and hence (since i =1= 0)

«PI(-1) -rp2(-1) = ± 2:n;. Observe that

(7)

(8)

(9)


Thus (9) implies that (10)

Consider now any real number Uo such that -1 < uo< 0. Then gl (w), g2(W) are single-valued continuous lil-th roots of j(w) on the segment -1~u:;;;uo, and j(w) of=. ° on this segment. In view of (10) it follows by part (ii) of lemma 2 that gl (w) = g2 (w) on this segnient. Since Uo was any real number such that -1 < U o < 0, we conclude [in view of the continuity of gl(W),g2(W)] that

(11)

A similar argument, using (3) instead of (10), yields

(12)

If we now define

then [in view of (11) and (12)] it is clear that g(w) is a single-valued continuous lil-th root of j(w) in R, and the lemma is proved.

VI. 1.4. The topological index in R2. We consider in this section a bounded continuous transformation

(1 )

where R is a bounded, finitely connected JORDAN region in R2. We shall also use for T the alternative representation (see VI.1.1)

T:z=T(w), wE R. (2)

Let B be the oriented boundary of R (see VI.1.2, definition 6). Select a point Zo in R2 such that

(3 )

Then T(w) - Zo is continuous in R and different from zero on B. Thus the variation of the argument of T(w) - Zo on B is available (see VI.1.2, definition 7). Observe that (3) is equivalent (see VI.1.1) to the relation

Zo Ef T jr int R. (4)

Accordingly (see 11.2.2) the topological index f-l (zo, T, int R) is available.

Theorem. Under the conditions just described, we have the relation

f-l (zo, T, int R) = _1 VB [arg (T(w) - zo)]. 2:n

(5 )

§ VI. 1- The topological index in R2.

Proof. To simplify notations, let us put

_1 VB [arg (T(w) - Zo)] =fh*(Zo, T, int R). 2:n:

We have to show that

fh (Zo, T, int R) = fh*(Zo, T, int R).

If zoEf T R, then fh (Zo, T, int R) = 0

by 11.2.3, theorem 2, and

fh*(zo' T, int R) = 0

399

(6)

(7)

by part (a) of VI.1.2, lemma 17. Thus (7) holds if zoEf T R, and hence we can assume, in proving (7), that zoET R. This inclusion implies, in view of (4), that

ZoE Tint R. (8)

It is convenient to divide the proof of (7) into several parts.

Case 1. R is a non-degenerate triangle, and T = B, a linear transformation from R2 into R2 (see 11.2.5). We assert that B is non-singular. Indeed, if B were singular, then we would have B R = B B and hence, by (8), ZoE BE, in contradiction with (3). Once we know that B is nonsingUlar, we can apply 11.2.5, theorem 4, obtaining in view of (8) the relation

fh(zo, B, int R) = sgn det B.

On the other hand, VI.l.2, lemma 19 yields

VB [arg (B(w) - zo)] = 2n sgn det B,

and (7) follows in view of (9) and (6).

(9)

Case 2. R is a polygonal region (that is, the boundary curves of R are simple closed polygons), and T is quasi-linear in R. This means that there exists a triangulation of R, consisting of a finite number of triangles tI , ... , tm (see the figure), such that T coincides with a linear transformation Bj in ii' j = 1, ... ,m. Let 5 be the union of all the edges of tI , ... , tm . It is convenient to consider now two sub-cases.

Case 2a. zoEfTS. Then we have

fh (zo, T, int til = fh*(zo, T, int ti ), } ( 10)

j= 1, ... ,m.

Indeed, if zoETinttj for a certainj, then (10) Fig. 52.

holds by Case 1. If zoEfT int ti for a certain j, then (since zoEfTS) we have zoEfTti' and hence both sides in (10) are equal to zero, as we noted


above. Thus (10) is verified. Denote by B j the counter-clockwise oriented perimeter of ti . Then

m

VB [arg (T(w) - zo)] = LVBj [arg (T(w) - zo)], (11) i=1

since the contributions of those edges which are not parts of B cancel out by part (i) ofVI.1.2, lemma 11. From (11) it follows, in view of (6), that

m

f-l*(zo, T, int R) = Lf-l*(Zo, T, int ti ). i=1

Observe now that the assumption zoEt TS implies that

m

T-1 Z0 (.U int ti . 1=1

(12)

Thus int t1 , ... , int tm is a (finite) sequence of domains in int R which is (zo' T, int R)-complete (see 11.2.1, definition 5). Hence

m

f-l (zo, T, int R) = Lf-l (zo, T, int til j=l

by 11.2.3, theorem 3, and (7) follows in view of (10) and (12).

Case 2b. Assume now that zoETS. In view of (6), (3) and (4) we conclude from 11.2.3, theorem 1 and VI.1.2, lemma 20 that there exists an 'YJ > 0 such that

f-l(zri, T, int R) =f-l(zo, T, int R) if Izo - zril < 'YJ, (13)

Il*(Zri, T, int R) =f-l*(zo, T, int R) if Izo - zril < 'YJ. (14)

Note that the set TS consists of a finite number of straight segments, some of which may reduce to single points. Thus clearly we can select a point zri such that

( 15)

The point zri satisfies the assumptions used in Case 2a, and hence

f-l(zri, T,int R) =f-l*(zt, T,intR). (16)

Clearly (13) to (16) imply (7).

Case 3. R is a polygonal region (as in Case 2), but T is not assumed to be quasi-linear. Take a triangulation 'r1 of R (as indicated in the figure used in Case 2). Subdividing each triangle t of 1'1 into six smaller triangles by drawing the three medians of t, we obtain a second triangulation 'r2 of R. Repetition of this process yields a sequence 'r1 , ... , "m' '"

§ VI. 1. The topological index in R". 401

of triangulations of R. For each positive integer m denote by Tm the quasi-linear transformation obtained as follows: in each triangle t of "tm, Tm coincides with the (uniquely determined) linear transformation which agrees with T at the vertices of t. Since T is uniformly continuous in R, clearly

Tm --+ T uniformly in R. (17)

From (3) and (17) we conclude that zoEfTmB if m is large enough, say m>M. By Case 2 we have therefore

.u(zo,Tm,intR) =.u*(zo,Tm,int R) for m>M. (18)

Part (c) of VI.1.2, lemma 17 yields, in view of (17) and (6),

.u*(zo, T, int R) = .u*(zo, Tm, int R) for m large. (19)

Note now that in view of (4) we have (see 1.1.4, exercise 3)

e(zo, T frint R) > o. From (17) it follows therefore that

e(T, Tm, frint R) < e(zo, T frint R) for m large. (20)

By 11.2.3, remark 8 we conclude from (20) that

.u (zo, T, int R) =.u (zo, Tm, int R) for m large,

and (7) follows in view of (18) and (19). We are now prepared to deal with the general case. In view of (4)

and (8), the set T-l Zo is a non-empty, compact subset of int R. Accordingly, we can select a polygonal region R* such that

T-l Zo ( int R* , R* ( int R. (21 )

Then R* presents Case 3, and hence

.u(zo, T, int R*) =.u*(zo, T, int R*). (22)

By 11.2.3, remark 4 it follows from (21) that

.u (zo, T, int R) =.u (zo, T, int R*). (23)

Denote by B* the oriented boundary of R*. From VI.1.2, lemma 21 we infer [in view of (21) and (6)J that

.u*(zo, T, int R) = .u*(zo, T, int R*),

and (7) follows by (22) and (23). Rado and Reichelderfer, Continuous Transformations. 26


§ VI.2. Special features of continuous transformations in R2.

VI.2.1. A countability theorem. The validity of the principal theorem to be derived in this section is restricted to R2. Simple examples show that the corresponding statement for R" is generally false if n>2.

Our first lemma is concerned with a continuous transformation T from a finitely connected, bounded JORDAN region R(R2 into R2. We shall use for T the alternative representations

T:z=T(U')' wE R,

in the sense of VI.1.1 (4), (5).

(1 )

(2)

Lemma 1. Given T as in (1) and (2), and given a point woE int R, assume that the following holds.

(i) R is simply connected. (ii) T(w) =I=T(wo) for wE R, w=l=wo' (iii) l.u(Two, T, int R)I ~1.

Then there exists a number 1] > 0 such that

N(z,T,intR)~I.u(Two,T,intR)1 for O<lz-Twol<1]· (3)

Proal. The assumption (ii) implies that T(wo) Ef Tlr int R, and thus .u (Two, T, int R) is defined (see 11.2.2). Let us put

Ip(Two, T, int R)I = k. (4)

Then by (iii) k~1. (5)

Let C be the countc!'-clockwise oriented boundary curve of R. By VI.1.4 (5) we have then

2n.u(Two, T, int Rj = J{[arg (T(w) - T(wo})J. (6)


f (w) = T(w) - T(wo) , w f R. (7)

From (i), (ii), (iii), (4), (5), (6) we conclude that f(w) satisfies the assumptions of VI. 1.3, lemma 5, and hence f (w) possesses a singlevalued, continuous k-th root. T*(w) in R. Thus

I(w) = T*(W)k, we: R. (8)

§ VI.2. Special features of continuous transformations in R2. 403

Consider the transfonnation

T*:z=T*(w), wE R. (9)

From (7), (8), (ii) we see that

T*(wo) = 0, T*(w) =1= 0 for wEe. (10)

Thus O\f T* fr int R, and hence fI (0, T*, int R) is defined (see 11.2.2). By VI.1.4 (5) we have

fI (0, T*, int R) = _1 Vc [Cj.Ig T*(w)J. 2:n:

(11)

On the other hand, (8) implies (see VI.1.2, lemma 13) that

• 1 Vc [arg T*(w)] = T Vc [arg t (w)J. ( 12)

From (11), (12), (7), (6), (4) it follows that

fI(O, T*, int R) = ± 1 =1= O. (13 )

In view of 11.2.3, theorem 1, the relation (13) implies that fI (z, T*, int R) is different from zero in a certain neighborhood of the point z = O. Thus there exists a number ~ > ° such that

fI(e, T*, int R) =1= ° for ° ~ lei <~. ( 14) Let us put

(15 )

We proceed to verify that this number 'YJ has the property required in our lemma. Since ~ > 0, clearly 'YJ> O. Consider any point z such that

0< Iz-T(woll <'fj. ( 16)

Then the complex number z - T(wo) has k distinct k-th roots Zl' ... , Zk'

Thus z;=z-T(wo), r=1,oo.,k,

z, =1= Zs if r =1= s .

Furthennore, from (17), (16), (15) it follows that

° < I zr I < ~ , r = 1, 00 • , k.


fI(z"T*,intR)=I=O, r=1,oo.,k. 26*

( 17)

( 18)

( 19)

(20)


By 11.2.3, theorem 2 the relations (20) imply that

z,ET*intR, r=1, ... ,k.

Hence there exist points WI' ... , Wk such that

w,EintR, r=1, ... ,k,

T*(w,) =Z" r= 1, ... , k.

(21)

(22)

From (22) and (18) it follows that W,=f=Ws for r=f=s. The relations (7), (8), (22), (17) yield

T(w,) = T(wo) + t (w,) = T(wo) + T*(Wr)k = T(wo) + (z - T(wo)) = z.

Hence the set (T-1z) n int R contains at least k distinct points, namely, at least the points WI"'" Wk' Thus it is established that (16) implies the inequality N(z, T, int R) ?:.k, and (3) follows in view of (4).

Lemma 2. Given a bounded continuous transformation

(23) as in VI.1.1 (1), let D be a domain in D, and let k"2:.2 be an integer. Denote by F(k, T, D) the set of those points ZoE R2 which satisfy the following conditions. (a) There exists precisely one point woED such that T(wo) =zo, and (b) at this point Wo we have (see 11.3.7)

lis (wo, T)I = k. (24)

Then the set F(k, T, D) is countable.

Proof. The assertion is obvious if F(k, T, D) = 0. So we can assume that F(k, T,D) =f= 0. Consider any point zoEF(k, T, D), and let Wo be the unique point in D such that T(wo) =zo [see condition (a)J. Then, in view of (24),

(25)

Note that condition (a) implies that D is clear of relatives of Wo (see 11.3.7). If R is a circular disc with center at Wo and a sufficiently small radius, then we have R (D, and hence int R is also clear of relatives of woo Therefore [see 11.3.7 (3) and 11.3.4, definition 1J

(26)

From (25) and (26) we conclude that the transformation TI R and the point Wo satisfy the assumptions of lemma 1. Hence there exists a number 'f}='f}(zo) >0 such that

N(z, T,intR)?:.k?:.2 if 0< Iz-zol<'f}(zo).


Since R(D, we have a fortiori

N(z, T, D) ~ 2 if 0< Iz - zol < 'I') (zo) . (27)

We assert that

0< Iz - zol < 'I') (zo) implies z Ef F(k, T, D). (28)

Indeed, if 0< Iz-zol <'I') (zo), then by (27) there exist at least two distinct points w', w" in D such that T(w') =Z, T(w") =Z, and thus z violates the condition (a) required of the points of the set F(k, T, D). From (28) it follows that every point Zo of F(k, T, D) possesses a neighborhood clear of further points of F(k, T, D). Thus F(k, T, D) is an isolated set. Hence (see 1.2.1, exercise 5) F(k, T, D) is countable, and the lemma is proved.

Theorem. Given T as in (23), let I be the set of those points wED where liB (w, T) I ;;:;;2. Then j is countable.

Proof. For each integer k let Ik be the set of those points wED where iB(w, T) =k. Then clearly 1= U Ik , Ikl ~2, and thus it is sufficient to prove that Ik is countable if Ikl ;;;;;;2. We divide the proof of this fact into two parts.

Part 1. Let k be an integer such that Ikl ~2. We assert that the set Tlk is countable. To see this, let 0 be the class of all those oriented squares Q (D which satisfy the following conditions. (a) The coordinates of the center of Q are rational numbers, and (b) the sidelength of Q is a rational number. Then clearly 0 is a countable class. We first verify the inclusion

Tlk (UF(lkl, T, intQ) , Q (0, (29)

where F(lkl, T, int Q) is defined as in lemma 2. To establish (29), note that if zoE Tlk , then there exists a point Wo such that

Since iB (wo, T) =t= 0, the point Wo possesses (see 11.3.7) a neighborhood clear of relatives. Thus there exists an open set 0 such that

woEO(D, T(w)=t=zo if wEO, w=t=wo.

Since 0 is open, there follows the existence if a square Qo EO such that woEintQo, Qo(O. Then clearly Wo is the only point in intQo whose image under T is ZOo Furthermore, iB (wo, T) = k. Thus, by the definition of the set F(lkl, T, intQo), we have zoEF(lkl, T,intQo). Since Zo was an arbitrary point of TI" and QoE 0, (29) follows. As


:0 is a countable class and F(lk[, T, int Q) is countable for Ikl:;:;;; 2 by lemma 2, the inclusion (29) implies that Tlk is countable for Ikl;;;;:; 2.

Part 2. We next show that if Ikl ~ 2, then the set Ik is countable. Since we already know that Tlk is countable if Ikl;;;;:; 2, it is sufficient to verify that for each point zoE Tlk the set

(30)

is countable. In tum, the countability of G (zo) will be established (see 1.2.1, exercise 5) if we show that each point woEG(zo) possesses a neighborhood clear of further points of G(zo). Consider a point woEG(zo). Then woElk' and thus liB(wo,T)I=lkl=l=o. By the definition of the local index iB (see 11.3.7) it follows that Wo possesses a neighborhood 0 clear of relatives. Hence if wE 0 and w =l= wo, then wEfT-lzo and [see (30)J a fortiori wEfG(zo). Thus 0 contains no further point of G (zoL and the proof is complete.

VI.2.2. Some applications of the countability theorem. We consider a bounded continuous transformation

T:z=T(w), wED, (1 ) as in VI. 1. 1 (2).

Theorem 1. Given T as in (1), we have

K(z, T, D) :;;'N(z, T, D) a.e. in R2. (2)

Proof. Lemma 6 in IV.5.3 yields -

K(z, T, D) :;;. N(z, T, D) if z Ef T I.

Since we are now operating in R2, the set TI is countable by the

countability theorem in VI.2.1. Hence the set TI is of L-measure zero, and thus (2) follows from (3).

Theorem 2. In R2 the concepts BVB and s BVB are equivalent. Explicitly, the transformation (1) is sBVB in D if and only if it is BVB in D.

Proof. Assume that Tis BVB in D. Then (see IV.5.1, theorem 3) the derivative DB (w, T) exists a.e. in D and is L-summable in D. Furthermore (see IV.5.3)

IB (w, T) = 1'B (w, T) DB (w, T) a.e. in D. (4)

The countability theorem in VI.2.1 yields

LI=O,

lis (w, T) 1:;;'1 for wED - I.

(5)

(6)



IIB(W, T)I :;;;:DB(w, T) a.e. in D. (7)

Since DB (w, T) is L-summable in D, (7) implies that IB (w, T) is also L-summable in D. In view of IV.5.3, definition 1 it follows that T is sBVB in D. Conversely, if Tis s BVB in D, then Tis BVB in D by definition.

Theorem 3. In R2 the concepts A C Band sAC B are equivalent. Explicitly, the transformation (1) is sA CB in D if and only if it is ACB in D.

Proof. Assume that T is A CB in D. Then Tis BVB in D by IV.5.2, theorem 1, and hence T is s BVB in D by the preceding theorem. Accordingly, T is sA CB in D as a direct consequence of IV.5.3, definition 2. Conversely, if T is sA CB in D, then by definition T is also A CB in D.

Theorem 4. In R2 the class of BVB transformations is included in the class of eBV transformations. Explicitly, if the transformation (1) is BVB in D, then it is also eBV in D.

Proof. If T is BVB in D, then N(z, T, D) is L-summable in R2 by IV.5.l, theorem 1. By (2) it follows that K(z, T, D) is L-summable in R2. Hence T is eBV in D in view of IV.4.l, definition 1.

Theorem 5. In R2 the class of AC B transformations is included in the class of eA C transformations. Explicitly, if the transformation (1) is AC B in D, then it is also eA C in D. Furthermore, if the transformation (1) is A CB in D, then

IB(W, T) = Ie(w, T) a.e. in D. (8)

Proof. Assume that T is ACB in D. Then T is sA CB in D by theorem 3. Hence Tis eA C in D and (8) holds by IV.5.3, theorem 1.

Remark. According to the preceding results, the concepts s BVB and sACB are irrelevant in R2, and the B-theory is included in the e-theory in R2.

VI. 2.3. On the existence of the weak total differential in H2. We shall use in this section the alternative representations

T:z=T(w), wED,

T:x=x(u,v), Y=Y(u,v), (u,v)ED,

(1 )

(2)

for the bounded continuous transformation T: D-+R2, as explained in VI.1.l. We shall establish the fact that if the first partial derivatives


of the coordinate functions of T exist a.e. in D, then T possesses a weak total differential a.e. in D, in the sense of V.2.2, definition 2. Simple examples show that the analogous statement for R n is generally false if n > 2.

Given T as in (1) and (2), assume that the first partial derivatives xu' xv, Yu' Yv exist a.e. in D. Let (uo, vol be a point in D where XU' XV'

Yu, Yv exist. To simplify notations, we put [for (u, v) E DJ

X(u, V,1to, vo)=x(u, v)-x(uo, vo)-(u-uo) Xu (u~, vo)-(v-vo) Xv (uo, vo), (3)

Y(u, V,Uo, vo)=Y(u, v)-Y (uo, vo)-(u-uo) Yu (uo, vo)-(v-vo) Yv (uo, vo)' (4)

Lemma 1. Given T as in (1) and (2), assume that the first partial derivatives xu, xv' Yu, Yv exist a.e. in D. Assign s>O, .>0. Then there exists a set E = E(s, .) CD and a number a = a (s, .) > Osuch that the following holds.

(i) E is closed. (ii) L (D - E) < s. (iii) Xu, Xv' Yu, Yv exist at every point (u, v)EE. (iv) Xu' XV' Yu, Yv are continuous on E (relative to E). (v) If (uo, vo)' (u, v) are any two points such that (a) (uo, vol E E,

(b) (u, v)ED, (c) at least one of the inclusions (uo, v)EE, (u, vo)EE holds, and (d) the distance of (uo, vol and (u, v) does not exceed a, then

IX(u, v, uo, vo)1 ::;;;:.[(u - UO)2 + (v - VO)2J~,

IY(u, v, uo, vo)1 ::;;;:.[(u - U O)2 + (v - vo)2]~.

(5)

(6)

(vi) Let Q be any oriented square in D, with center (uo, vo) E E and side-length 2h-:;;;'a, such that the points of intersection of the perimeter of Q with the lines u = U o and v = Vo lie in E. Then (5) and (6) hold for every point (u, v) on the perimeter of Q.

Proof· The first partial derivatives Xu, Xv, y", Yv exist a.e. in D by assumption and are L-measutable in D by 111.1.3. Accordingly (see 111.1.1, lemma 22), we can select in D a set E* such that the following holds.

(ex) E* is closed. ((3) L (D - E*) < s/2.

(y) Xu, Xv, Yu' Yv exist at every point (u, v)fE*. (v) Xu, Xv, Yu, Yv are continuous on E* (relative to E*).

Since E* is compact, (v) implies that x,,, Xv, y", Yv are uniformly continuous on E* (relative to E*). Hence we can select a number

§ VI.2. Special features of continuous transformations in RI. 409

0'* > 0 such that

whenever

IX .. (u", v") - X .. (u', v')1 :;;'./3, IxlI (u", v") - xlI (u', v')1 :;;;'./3,

I Y .. (u", v") - y .. (u', v')1 :;;;'./3,

IYlI(u", v") - YlI(u' , v')1 :;;;'./3,

(7)

(8)

(9)

(10)

(u", v") E E*, (u', v') E E*, [(u" - U')2 + (v" - v')2Ji:;;;;: 0'*. (11)

For each positive integer n, denote by E~ the set of those points (u*, v*)EE* for which the following statements (a), (b) hold simultaneously.

(a) If (u, v*)ED and lu-u*1 :;;;'1/n, then

Ix(u,v*) -x(u*,v*) -(u-u*)x .. (u*,v*)I:;;;;: ; lu-u*l, (12)

ly(u,v*)-y(u*,v*)-(u-u*)y .. (u*,v*)I;:;;;;; lu-u*!. (13)

(b) If (u*,v)ED and Iv-v*I~1/n, then

Ix(u*, v) - x(u*, v*) - (v - v*) Xv (u*, v*)1 s; 2. Iv - v*l, (14) 3

Iy(u*, v) - y(u*, v*) - (v - v*) Yv(u*, v*)I:::;; 2. Iv - v*l. (15) 3

Obviously (16)

Furthermore, since Xu, XV, Y .. , Yv exist at every point (u*, v*) EE*, clearly

E*=UE:, n=1,2, .... (17)

Finally, since X, y, Xu, XV' Y .. , Yv are continuous on E*, and E* is closed, it is easy to see that E~ is closed (and hence L-measurable). Hence (see 111.1.1, lemma 4) the relations (16) and (17) imply that

L (E* - E:) -+ 0 for n -+ 00 •

Thus we can select an integer N such that

(18) Let us put

(19)


We assert that (J and E, as defined in (19) , satisfy the conditions (i) to (vi). Note first that since E~ is closed, the conditions (i) to (iv) are obviously satisfied in view of (IX), (P), (y), (<5), (18), (19). There remains to verify (v) and (vi). Take any two points (uo, vol, (u, v) such that

(uo, vol E E, (u, v) ED, [(u - UO)2 + (v - VO)2]!;;;;;: (J, (20)

and assume that one at least of the inclusions (uo, v)EE, (u, vo)EE holds. Suppose, for instance, that

(u, vol E E. (21 )

The quantity X(u, v, uo, vol, defined by (3), can be re-written as follows.

X(u, v, Uo, vol = [x(u, v) - x(u, vol - (v - vo) xv(u, vol] + ) + [x(u, vol - x(uo, vol - (u - uo) Xu (uo, vol] +

+ (v - vol [xv(u, Vol - Xv (uo, vo)J·

Note that

(22)

lu-uol, \v-vol;;;;;:[(U-UO)2+(V-vo)2J];;;;;'(J;;;;;:U* (23)

by (19) and (20). In view of (21), (19), (17), (20), (23), (11) we can apply (8) with

(u", v") = (u, vo), (u' , Vi) = (uo, vo)'

I t follows that

I (v - vol [xv (u, Vol - Xv (uo, vol J\;;;;;: ;- [(u - UO)2 + (v - VO)2]~. (24)

Next, observe that

(u, vol E E~, (u, v) E D, Iv - vol;;;;;: 1IN ,

by (21), (19), (20), (23). Accordingly, we can apply (14) with (u*, v*) =

(u, vo), obtaining the inequality

Similarly, we have

(uo, vol E E~, (u, vol E D, \u - uo\ ;;;;;: 1IN,

by (21), (19), (20), (23). Hence we can apply (12) with (u*, v*) = (uo, vo), obtaining the inequality

§VI.2. Special features of continuous transformations in R2. 411

Clearly (5) follows from (22), (24), (25), (26) under the assumption (21). An entirely similar reasoning yields the verification of (6) under the conditions stated in (v).

Consider now any square Q with the properties described under (vi). The lines u = Uo and v = Vo intersect the perimeter of Q in the points

Let (u, v) be any point on the perimeter of Q. Then clearly

[(u - UO)2 + (v - VO)2Jfr;;;;: 2h ;;;;: (J. (28)

Furthermore, one at least of the relations u = U o ± h, v = Vo ± h holds. Thus at least one of the points (u, vol, (uo, v) is included among the points (27), and hence at least one of the points (u, vo)' (uo, v) lies in E. In view of (28) it follows that the points (u, v), (uo, Vol satisfy the assumptions made under (v), and hence (5) and (6) hold:

Lemma 2. Let S be a bounded L-measurable set in R2. Then there exists a decomposition

S =sU Sf (29)

such that the following holds. (i) Ls = 0. (ii) If (uo, vol E Sf, then there exists a sequence of oriented squares Qn' with center at (uo, Vol and side-length 2h" -+0, such that the points of intersection of the perimeter of Q .. with the lines u = Uo and v = Vo lie in S.

Prool. By 111.1.1, lemma 10 there exists a decomposition of S of the form (29) such that (i) holds and every point of Sf is a point of linear density of S in the direction of both of the coordinate axes. We assert that (ii) also holds for this decomposition of S. Indeed, let (uo, vol be a point of Sf. Denote by H~ the set of those real numbers h for which (uo + h, Vol E S, and by H;' the set of those real numbers h for which (uo - h, Vol E S. Similarly, denote by H;, H;; the set of those real numbers h for which (uo, vo+h)E Sand (uo, vo-h)E S respectively. Since (uo, vol is a point of linear density of S in the direction of both of the coordinate axes, it is clear that the point h = ° of the numberline is a point of density of each one of the sets H~, H;., H; , H;; . Hence (see 111.1.1, lemma 11) h = ° is also a point of density of the set

H = H: n H;. n H; n H;; .

Accordingly we can select a sequence of real numbers hn such hnE H, h" =l= 0, hn -+ 0. Now since the inclusion hn E H implies the inclusions hn E lJ~, hn E H;., hn E H;, h" E H;;, it follows that


Denote by Qn the oriented square with center at (uo, vol and sidelength 2lhn l. Then (30) shows that the points of intersection of the perimeter of Qn with the lines u = U o and v = Vo lie in 5, and thus the lemma is proved.

Theorem 1. Given T as in (1) and (2), assume that the first partial derivatives Xu, Xv, Yu' Yv exist a.e. in D. Then T possesses a weak total differential a.e. in D, in the sense of V.2.2, definition 2.

Proof. Assign a number 1] > 0. For each positive integer i select numbers Cj>O, 1'j>O such that

(31 )

Denote by E j , I:1j the set E and the number 1:1 > ° which correspond to C = Cj' l' = 1'j in the sense of lemma 1. Put

5 = n E j , j = 1, 2, .... (32)

Then 5, as the intersection of closed sets, is closed and hence is Lmeasurable. Furthermore, since

D - 5 ( U (D - Ejl ,

and L (D - Ej ) < Cj' there follows in view of (31) the inequality

L(D - 5) < 1]. (33)

On applying lemma 2 to the set 5 defined by (32), we obtain a decomposition

5 = s U 5', (34) such that

Ls=o (35)

and for every point (uo, volE 5' there exists a sequence of oriented squares Qn (uo, vol satisfying the following conditions. (a) The point (uo• vol is the center of Qn(uo, vol. (b) The side-length 2hn(uo, vol of Qn(uo, vol converges to zero. (c) The points of intersection of the perimeter of Q" (uo, vol with the lines u = U o and v = Vo lie in 5.

Consider a point (uo, vol E 5'. Keeping (uo, vol fixed, select an integer j> 0. In view of (32) it follows that (uo, vol E E j and that the points of intersection of the perimeter of Qn (uo, vol with the lines u=uo and V=Vo lie in E j • Since h,.(uo, vol -+0, we shall have 2h,,(uo,vo):;;;'l:1j for n sufficiently large, say for n>N(uo, t'o, f)· By part (vi) of lemma 1 it follows that

IX (u, v, U o, vol I:;;;. Tj [(u - UO)2 + (v - VO)2]~,

I Y(u, v, uo, vo)1 :;;;. 1'j [(u - UO)2 + (v - VO)2J!',

(36)

(37)


provided that (38)

In view of (3) and (4) the quantity y(hn(uo, vo)) , defined in V.2.2 (14), is now given by the formula

(h ( )) _ [X (u, v, uo, vo)2 + Y(u, v, UO, VO)2]; \ y n UO, Vo - max --"------'---_----"---"''----_-'-__ -''c,----'''---"--

[(u - UO)2 + (v - VO)2]! '

(u, v) E jr Qn(uo, vo).

From (36) to (39) we conclude that

y(hn(uo,vo))~V2Ti for n>N(uo,vo,j).

For n-+ 00 there follows the inequality

(39)

Since the positive integer i was arbitrary and Tr~O, Y (hn (uo, vol) ;:;;;:; 0, it follows finally that

Thus it is established (see V.2.2, definition 2) that T possesses a weak total differential at every point (uo, vol E 5'. Since L (D - 5') < 1] by (33), (34), (35) and 1]>0 was arbitrary, we conclude that T possesses a weak total differential a.e. in D.

In applying the general theorems derived in § V.3 to differentiable transformations in R2 we can replace, in view of the preceding theorem 1, the assumption that T possesses a weak total differential a.e. in D by the assumption that the first partial derivatives of the coordinate functions of T exist a.e. in D. From V.3.2, theorem 1 ; V.3.3, theorem 6; V.3.4, theorem 1; V.3.4, theorem 2; V.3.4, theorem 3 we obtain in this manner the following series of statements for bounded continuous transformations in R2 (see VI. 1. 1 for the special notations adopted in R2).

Theorem 2. Given T as in (1) and (2), assume that Tis eBV in D and the first partial derivatives xu' xV, Y .. , Yv exist a.e. in D. Then the essential generalized Jacobian Je(w, T) is equal to the ordinary Jacobian J(w, T) a.e. in D.

Theorem 3. Given T as in (1) and (2), assume that the first partial derivatives xu, xv' Yu, Yv exist a.e. in D, the ordinary Jacobian ](w, T) is L-summable in D, and T satisfies the condition (N) in D. Then the following holds.


(i) T is both sA CB and eA C in D. (ii) Is (w, T) = Ie (w, T) = I(w, T) a.e. in D. (iii) Ds (w, T) = De (w, T) = II(w, T) I a.e. in D. (iv) N(z, T, D) =K(z, T, D) a.e. in R2.

Theorem 4. Given T as in (1) and (2), assume that Tis eA C in D and the first partial derivatives XU' xv, y", Yv exist a.e. in D. Then

f II(w, T)I dudv = f K(z, T, D) dxdy, D

f ](w, T) dudv = f fle(z, T, D) dxdy. D

More generally, if H(z) is a finite-valued, L-measurable function in R2, then

f H(Tw) II(w, T)I dudv = J H(z) K(z, T, D) dxdy, D


f H(Tw) I(w, T) du dv = f H(z)p,(z, T, D) dx dy, D


Theorem 5. Given T as in (1) and (2), assume that (a) T is defined and continuous in 1), (b) LT!rD=O, (c) Xu, Xv, Yu, Yv exist a.e. in D, and (d) T is eA C in D. Then

f I(w, T) dudv = fp(z, T, D) dxdy. D

More generally, if H(z) is a finite-valued, L-measurable function in R2, then

f H(Tw) I(w, T) du dv = f H(z)p (z, T, D) dx dy, D


Theorem 6. Given T as in (1) and (2), assume that the first partial derivatives x,,, XV' Yu, yvexist a.e. in D, the ordinary Jacobian I(w,T) is L-summable in D, and T satisfies the condition (N) in D. Then

f II(w, T)I dudv = f N(z, T, D) dxdy. D

More generally, if 5 is an L-measurable subset of D and H(z) is a finitevalued, L-measurable function in R2, then

f H(Tw) I](w, T)I dudv = f H(z) N(z, T, 5) dxdy, s

§ Vr.3. Special classes of differentiable transformations in R2. 415

as soon as one of the two integrals involved exists. In particular, corresponding to the special case H(z) = 1,

J I](w, T)I du dv = J N(z, T, 5) dx dy. s

The closure theorems derived in V.3.5 can be improved in a similar manner in the special case of continuous transformations in R2. We state explicitly only the result obtained by combining theorem 1 with V.3.5, theorem 3.

Theorem 7. Let there be given bounded continuous transformations [see VL1.1 (2)J

T:z = T(w), wED,

If:z = If (w), wE D j , j = 1, 2, ... ,


(i) D, D j are bounded domains in R2. (ii) The sequence {If} converges to T uniformly on compact sub

sets of D. (iii) The first partial derivatives of the coordinate functions of T

exist a.e. in D, and the ordinary Jacobian ](w, T) is L-summable in D. (iv) If is eAC in D j , and the first partial derivatives of the co

ordinate functions of 1; exist a.e. in D j , j = 1, 2, .... (v) If F is any compact subset of D, then

](w, If) -l>- ](w, T) a.e. on F.

(vi) If Q is any oriented square in D, then the sequence {J(w, 1j)} satisfies the condition (V) in Q (see IIL1.1, definition 6).

Then T is eAC in D.

§ VI.3. Special classes of differentiable transformations in R2.

VI.3.1. Some elementary inequalities. We begin with a review of various facts relating to real-valued functions of two real variables. Many of the results discussed in the present § VL3 remain valid for functions of an arbitrary number of variables.

Our objective in this section is to prove some inequalities needed later on. Free use will be made of familiar devices in elementary Differential and Integral Calculus.

Lemma 1. Let g(u, v) denote a real-valued continuous function in a bounded domain D in R2, such that

g(u, v) ~ 1 for (u, v) E D. (1 )

416 Part VI. Continuous transformations in R2,

Take a square Q(D (it is not required that the sides of Q should be parallel to the coordinate axes). Select a point (uo, volE Q, and put

e = [(u - UO)2 + (v - VO)2]~, (u, v) E R2,

Let P be a real number such that

P>2. (2)

Then there exists a (finite) positive constant ~ (P), which depends solely upon p, such that

II g(~ v) du dv-;;, ~(P) [II g(u, v)P dUdVr Q Q

(3 )

Proof. Let h be the side-length of Q. Denote by K the circular disc with center at (uo, vol and radius V"Zh. Then clearly

Q(K. (4)

Introduce polar coordinates e, IX. with pole at (uo, Vol. Define the real number q by the equation

-~+~=1, P q

(5)


2-q= P-2 >0. P-1 (6)

Using (6) and (4), we obtain the relations

where 4-q

C1 (P) = 71:22=": . (8)

The inequality of HOLDER (see 111.1.1, lemma 29) yields, in view of (5) and (7),

IIg(~,V) dUdv~[Ifg(U'V)PdudvtP[fId:qdVrq ~ I Q Q Q

;;;:;; ~ (P) h2~q [If g (u, v)P du dvtP, Q

(9)

§ VL3. Special classes of differentiable transformations in R2. 417

where

From (1) it follows that

and hence, since p> 2,

~2 II g(u, v)P du dv;;;;'1, Q

(10)

[ ~2 II g(u, v)P dUdVrp :;;; f-k2 II g(u, v)P du dvt (11) Q Q

From (11) we conclude that 2

[ff g(u, v)Pdudvt'P;;;;:; [ff g(u, v)Pdudvl~ hP - l. (12)

Q Q

The inequalities (9) and (12) imply that

II ~ v) du dv;S; r;. (p) h 2~ q ++-1 [II g(u, v)P du dvt

Q Q

and (3) is established, since [in view of (5)]

2-q 2 --+--1 =0.

q p

Lemma 2. Let I(u, v) be a real-valued continuous function in a bounded domain D in R2, such that the first partial derivatives I .. , Iv of I exist and are continuous in D. Select a square Q (D (it is not required that the sides of Q should be parallel to the coordinate axes), and denote by Ii, P2 a pair of v

opposite vertices of Q. Let a real number p> 2 be assigned. Then there exists a (finite) positive constant r; (P), which depends solely upon p, such that

Proal. Denote by A, B the remaining two vertices of Q (see the figure). Introduce a first system of polar coordinates el' <Xl with

B

Fig. 53.

pole at PI' and a second system of polar coordinates e2' (X2 with pole at P2 , where the polar angles (Xl' <X2 are measured in the manner indicated



in the figure. Denote by .11 , .12 the triangles with the vertices PI' A, B and P2 , A, B respectively. Applying lemma 1 with

g = 1 + If,,1 + Ifvl, (uo, vol = PI

we obtain the relations

II (1 + Iful + Ifvl) del dIX1 = II 1 + I/"L + Ifvl du dv ;;;;: .1, .1,

~ II 1 + 1/;11 + Ilvl du dv ~ I;. (p) [11(1 + Iful + \fvl)P du dVy. Q Q

The same argument applies with ~ replaced by P2 • Accordingly, we have the inequalities

11 (1 + Iful + Ifv\) de1 dIX1;;;;: I;.(P) [11(1 + Iful + Ifvl)P dudv]!, (14) .1, Q

If (1 + Iful + Ifv\) de2 dIX2 ~ I;. (P) [If (1 + Iful + Ifvl)P du dV],\. (15) ~ Q

Denote by P a variable point on the diagonal AB of Q. Then

If(P2) - f(l~JI:::;: If(P) - I (P2) I + I/(P) - I(~)I· (16)

The polar angles lXI' IX2 corresponding to P are clearly equal to each other. On denoting by IX their common value, the absolute values on the right-hand side in (16) are continuous functions of IX for 0:;;'IX:;;'n/2, while the absolute value on the left hand side in (16) is independent of IX. Hence integration of (16), with respect to IX = IX1 =0(2

from 0 to n/2 yields the inequality ",/2 ",/2

~ II (P2) - I (PI) I ;;;;:JI/(P) - I(P2)1 dIX2 + fl/(p) - I (PI) I dIX1 • (17) 2 0 Q

For fixed 0(1 we have

Now since

and

I dt I(P) - I(~) = -d- del' (h

P,P

we have the inequality

I dd~ I:::;: Ilul + Ilvl·

(18)

§ VI.3. Special classes of differentiable transformations in R2. 419

Hence (18) yields

I!(P) - l(lDI:s;:; J(l/ul + Ilvi) del < J(1 + Ilul + Ilvl) del' P,P P,P

and consequently

~2 ~ .

I I/(P) - l(l~JI dCXI~ I 1(1 + Ilul + Ilvi) deldCXl) o 0 P,P

= II (1 + I lui + Ilvl) del dCXl' LI,

An entirely similar argument yields the inequality ,,/2

(19)

I II (P) - I (P2) I dCX2:S;;; II (1 + Ilul + I Iv \) de2 dCX2' (20) o ~


I/(~) - l(lDI:s;;; ~ I;.(P) ({f (1 + Ilul + IlvW dudv)~.

Thus (13) holds with


Lemma 3. Let I (u, v) be a real-valued, continuous function in a bounded domain D(R2, such that the first partial derivatives lu, Iv of I exist and are continuous in D. Let Q be a square in D, and let w(j, Q) be the oscillation of I in Q (see 1.1.2). Let a real number P>2 be assigned. Then there exists a (finite) positive constant r(p), which depends solely upon p, such that

w(j, Q):s;;;T(P) [11(1 + Ilul + IlvWdudvl~. (21) Q

Proof. Denote by Po the center of Q. Select any two points P', P" in Q (if the points Po, P', P" are not all distinct, then the following argument becomes simpler in an obvious manner). Denote by Q' the square for which Po and P' are opposite vertices. Obviously Q' (Q. Applying lemma 2 with Q, Ii, P2 replaced by Q', Po, P', there follows the inequality

Since Q' (Q, it follows further that

27*


An entirely similar argument yields the inequality

Now since

I/(Pll) - I (Pi)! ~ II (PII) - I (PO) I + I/(PI

) - I (Po) I ' we obtain from (22) and (23) the inequality

II (Pll) - I (Pi) I s:; 2rz (P) [II (1 + 1/ .. 1 + Ilvl)P du dv]k. (24) Q

Since pi, pIt were arbitrary points in Q, (24) implies that (21) holds with F(p) =2~(P).

VI.3.2. Approximation by integral means. It frequently happens that a certain statement relating to real-valued functions can be proved readily by elementary methods if the functions involved are sufficiently smooth. The proof of the same statement under more general conditions may then be obtained, in certain cases, by appropriate approximations. The method of approximation by integral means, to be discussed briefly in the present section, is often useful in this connection. Let I(u, v) be a real-valued, L-measurable function in a bounded domain D(Rz. Assume that I(u, v) is L-summable in every oriented rectangle in D. Select a domain D such that D(D. Then there exists a number £5> 0, which depends upon D, such that

(u, v) E D implies L1(u, v, £5) (D, (1 )

where L1(u, v, £5) is the circular disc (closed spherical neighborhood) with center (u, v) and radius £5. Let h be a real number such that

0< h < £5/12. (2)

If Q (u, v, h) denotes the oriented square with center (u, v) and sidelength 2h, then (1) implies that

Q(u,v,h)(D if (~t,v)ED, O<h<~/12· (3)

Accordingly, I (u, v) is L-summable in Q (u, v, h) if (u, v) E D and (2) holds. We can therefore define a function Ih (u, v) in D by the formula

h h

fh (u, v) = -{z ( (I (u + rt., V + (3) drt. d(3, (u, v) E D, 0 < h < ~/V2-. (4) 4 ••

-h -II

Thus A(u, v) is equal to the integral mean of I on Q(u, v, h).

§ Vl.3. Special classes of differentiable transformations in RO. 421

Lemma 1. For fixed h satisfying (2), Ih(U, v) is continuous in D.

Proal. Select a point (uo, vo)ED. Let {(un' v,,)} be any sequence of points in D such that (u",vn)~(uo,vo)' Put

Since Qo(D, we can select an oriented square Qt with center at (uo, Vol such that Qo(intQ~, Q:(D. Since (un,vn)~(uo,vo), we can choose a positive integer N such that Q" (Qt for n> N. In view of (4) we have the inequality

Ifn(uo, VO)-lh(U", v,,) I ~ 4~2 [ ffl/(u, v)1 dudv+ ffl/(u, v)1 du dV]. (5) Qn-Q. Q.-Qn

Now clearly

(6)

Since I is L-summable in Qt, and Qn - Qo, Qo - Q" are L-measurable subsets of Qri for n<N, the relations (6) imply (see 111.1.1, lemma 33) that the integrals occurring in (5) converge to zero for n~oo. Thus (5) yields (since h is fixed)

lim Ifh(UO' vol - fh(Un , v,,) I = 0, n-+oo


Lemma 2. If {hn } is any sequence of real numbers such that

o < h" < MVi hI> ~ 0, (7)

then fhn (u, v) ~ f (u, v) a.e. in D.

Proal. Since 15(D, it follows that I is L-summable in 15. Indeed, since 15 is compact, we can select a finite sequence R1 , •.. , Rm of oriented rectangles in D such that 15 ( U Rj , j = 1, ... ,m. Since I is L-summable by assumption in R j , j = 1, ... , m, it follows that f is L-summable in 15, and hence also in D. Accordingly (see 111.1.1, lemma 43) there exists a decomposition D=s U 5 such that Ls=O and the following holds for every point (uo, volE S. If {Gn } is any regular sequence of closed sets in D such that (uo, vol E G" and the diameter of G" converges to zero, then

!~ L~"fff(u,v)dudv=/(uo,vo). Gn


On selecting G" as the oriented square with center (uo, vol and sidelength 2h", in view of (4) it follows that

lim fhn (uo, vol = f(uo, vol for (uo, vol E S, ....... 00


Lemma 3. Assume that ItIP is L-summable in every oriented rectangle in D, where p ~ 1. Then the family of functions

satisfies the condition (V) in every oriented rectangle in D (see 111.1.1, definition 6).

Proof. Select an oriented rectangle R(D. Clearly we can choose an oriented rectangle R* such that R(intR*, R*(D, and such that

(u,v)ER, ICtI<!5/lI2, IPI<!5ll12 imply (u+Ct,v+P}ER*. (8)

Assign now 8> O. By assumption, ItIP is L-summable in R*. Hence (see 111.1.1, lemma 33) there exists a number 'YJ> 0 such that

1111 (u, v) IP du dv < 8, (9) S.

whenever S* is an L-measurable subset of R* such that

(10)

Select any L-measurable set S such that

S(R, LS <'YJ. (11)

Let Ct, P be any two real numbers such that

( 12)

and denote by Sa/J the set obtained by subjecting S to the translation by the vector with components Ct, p. Then clearly Sa/J is an L-measurable set which satisfies [see (8), (11), (12)J the conditions

L Sa./J < 'YJ, Sa./J ( R* . ( 13)

From (9) to (13) we conclude that

II I/(u, v)iP du dv < 8. (14) Sa./J

It is convenient to divide the rest of the proof into two parts.

§ V1.3. Special classes of differentiable transformations in R2. 423

Part 1. Assume that P>1. Then using (4) and the inequality of HOLDER, it follows that

h h

I/h(U,v)l;:;;: 4~2 J JI/(u+CI.,v+P)ldrxdP~ -h -h

where q satisfies the equation

~+~=1. p q

Thus it follows that h h

1/"(u,v)IP~ 4~!2 J JI/(u+rx,v+P)lPdCl.dp. -h -h

In view of the FUBINI-ToNELLI theorem (see the Saks treatise listed in the Bibliography) we conclude that

h h.

JJlth(U,v)IPdudv~ 4~2 J J[JJI/(u+rx,v+PWdudv]drxdfJ S -h -h S

h h

= 4~2 J J[JJI/(u,vWdudv]dCl.d fJ . -h -h soc{3

Using (14) it follows finally that

II I/"(u, v)IP dudv<e. ( 15) s

Thus it is established that (11) implies (15), and the lemma is proved for the case P> 1.

Part 2. Assume that P = 1. From (4) it follows that

~Jlth(U, v)1 du dv ~ IJ [ 4~!2 _I jl/(U + rx, v +fJ)1 drx dfJ] du dv h h

=4~2 J J[JJI/(u+rx,v+fJ)ldudv]drxdfJ -I> -h 5

h I>

= 4~2 J J[JJI/(u,v)ldUdV]dCl.dfJ. -h -h Sa.{3


In view of (14), applied with p = 1, there follows the inequality

ffltll(u,v)ldudv<e for SeQ, lS<'Yj, s


Returning to the function tll(U, v) given by (4), let us note that this function possesses various useful properties beyond those explicitly stated in the preceding lemmas. We shall discuss only a few additional properties which will be needed later on. The. following preliminary comments will simplify the discussion. Let a, b be two real numbers such that a < b, and let P (u) be a real-valued continuous function in the open interval a - h < u < b + h of the real number-line, where h>O. Then we can define, in analogy with (4), a function Pk(U) by the formula

II

p,,(u) = 21h f p(u + oc) doc, a < u < b. -h

(16)

We shall need only the fact that the derivative p~ (u) of Ph (u) exists for a < u < b and is given by the formula

'(u)= tp(u+h)-tp{u-h) Ph 2h' ( 17)

To verify (17), take a point Uo such that a<uo< b, and consider the difference quotient

tpll (uo + Llu) - tpk (uo)

Llu

If ILlul is sufficiently small, then this quotient is defined, and from (16) there follows the formula

(18)

Since P (u) is continuous, clearly

u.+h+.1u

lim -i-- f p(u) du =p(uo + h), L!u_OLJU

(19) u.+11

u.-h+.1u

lim _1_ f p(u) du =p(uo - h). .1u_o Llu

(20) uo-h


From (18), (19), (20) it follows that !p~(uo) exists and is given by the formula

'( )_ p(uo+k)-p(uo-k) !p" Uo - 2k

Thus (17) is proved.

Lemma 4. Assume that I (u, v) is continuous in D. Then the function A (u, v) given by (4) has continuous first partial derivatives

l ot" I" = of" ""=--au' v ov in D, and

" 11I .. (u, v) = 4~2 I[f(u+h,v+f3)-/(u-h,v+f3)]df3, (21) -;11

" IlIv(u,v) = 4~2 ![f(u+a.,v+h)-/(u+a.,v-h)]doc. (22) -II

Prool. Take a point (uo, vol E D. Consider the auxiliary function

II

f{J(u) =_1 !I(u, vo+f3) df3. 2k

-II

(23)

If a, b are two numbers, sufficiently close to uo' such that a<uo< b, then clead y !P (u) is defined and continuous for a - h < u < b + h (recall that I is continuous by assumption). Accordingly, on defining !p" (u) by (16), the formula (17) will hold. From (16), (23), and (4) it follows that !p,,(u) =A(u, vol, and hence, by (17) and (23),

I ( ) - '( ) - p(uo+h)-p(uo-h) II .. UO' Vo -!p" Uo - 211

= 4~2 L['(uo +h, Vo + f3) df3 -jl(uo - h, Vo + f3) df3] .

Thus (21) is proved. The formu,la (22) is verified in an entirely similar manner. Since I is continuous, clearly (21) and (22) imply the continuity of Ihu,/lIv in D.

Lemma 5. Assume that I(u, v) is continuous inDo Let D be a domain such that 15 CD, and consider the function I" (u, v) given by (4). Assign e> o. Then there exists a number 'Yj> 0 such that

IA(u, v) - f(u, v)1 < e if (u, v) ED, 0 < h < min('Yj, ~ ). (24)


Prool. Since 15 is compact and I is continuous in D, we have a number 'f}>0 such that

I/(u+oc,v+,8)-/(u,v)l<e if (u,v)E15, locl<'f), 1,81 <'f}. (25)

Let (u, v) be any point in D, and let h satisfy the inequalities

O<h<min(n, ~).


I 1>" I

Ifn(u, v) - f(u, v)1 = 4~2 _[ -f [j(u+oc, v +,8) - f(u, v)J docd,81 <

" h

< 4~2 f f e doc d,8 = e, -I> -I>


VI.3.3. Bounded variation and absolute continuity in the TONELLI

sense. Let f(u, v) be a real-valued continuous function in an oriented rectangle

R:a~u~b, c~v~d, (1 )

where a < b, c< d. For fixed v satisfying the inequalities c ~v :;;;,d, I(u, v) is then a (continuous) function of u for a:;;;'u:;;;,b, and accordingly we can consider the total variation of I(u, v), as a function of u for fixed v, in the closed linear interval a:;;;'u~b (see V.1.1, definition 1). This quantity will be denoted by

v.. (v, I, [a, bJ). (2)

Thus (2) designates a function of v defined for c:;;;' v ~d. Similarly, we denote by

v" (u, I, [c, dJ)

the total variation of I(u, v) with respect to v, for fixed u, in the closed linear interval c~v:;;;'d. Note that the total variations (2) and (3) may assume the value + 00.

Lemma 1. Vu(v, f, [a, bJ) is a lower semi-continuous function of v, and similarly v;, (u, I, [c, dJ) is a lower semi-continuous function of u. Hence (see 111.1.1, lemma 20) the total variations (2) and (3) are Lmeasurable functions for c ~ v -;;;;, d and a ~ u -;;;;, b respectively.


Proof. Since the argument is entirely analogous in either case, we consider explicitly only the total variation (2). Take vo, v" such that

To simplify notations, put

Since f is continuous in R, clearly gn(u)-+gO(u) for a:;;;;'u;;;;'b. Hence (see V.1.1, lemma 4)

V(go, [a, b]) :;;;;'lim inf V(gn' [a, b]) , n-->-oo

and the lemma follows, since clearly

V(go, [a, bJ) = v.. (vo, f, [a, b]), V(g", [a, b]) = v.. (vn' f, [a, bJ).

Definition 1. If f(u, v) is a real-valued continuous function in an oriented rectangle R given as in (1), then f(u, v) is said to be of bounded variation in R in the TONELLI sense (briefly, BVT in R) provided that the total variations v.. (v, I, [a, b]) and Y.(u, f, [e, dJ) are L-summable in the closed intervals [e, d] and [a, b] respectively.

Lemma 2. If f(u, v) is BVT in R, then the first partial derivatives lu, Iv exist a.e. in R and are L-summable in R.

Proof. We shall consider explicitly only fu, since an entirely similar argument applies to fv. Let E be the set of those points (u, v) E R where fu fails to exist. Since E is L-measurable (see 111.1.3), the fact that LE=O will be established (see 111.1.1, lemma 17) if we show that for a.e. voE [e, d] the intersection of E with the segment v=vo, a:;;;;'u:;;;;'b is of (linear) L-measure zero. Now if v is so chosen that

v.. (v, f, [a, b]) < 00, (4)

then I (u, v) is of bounded variation, as a function of u, in the closed interval [a, bJ. Hence, by theorem 5 in V.1.6, f,,(u, v) exists for a.e. u E [a, b] if (4) holds. Next, since v..(v, I, [a, b]) is L-summable in [e, d], by 111.1.1, lemma 31 it follows that (4) holds for a.e. vE [e, d], and thus it is established that lu exists a.e. in R. Consider again a number v E [e, d] such that (4) holds. By V.1.6, theorem 5 it follows that fu (u, v), as a function of u, is L-summable in [a, b] and

b

J I/u(u, v)1 du:;;;;' v.. (v, f, [a, b]). (5 ) a


Now since (4) holds for a.e. vE [c, d] and v,. (v, I, [a, b]) is L-summable in [c, d], (5) also holds for a.e. vE [c, dJ and yields the inequality

!f; I/u(u, v)1 dU] dv ~ Jvu(v, I, [a, b]) dv < 00. (6) cae

By the FUBINI-ToNELLI theorem, the existence of the iterated integral in (6) implies the L-summability of lu in R, as well as the inequality

d

II I/u (u, v)1 du dv ~ IVu(v, I, [a, b]) dv. (7) R c

An entirely similar argument yields the L-summability of Iv in Rand the inequality

b

II I/v (u, v)1 du dv ~ I v" (u, I, [c, dD du. (8) R a

Definition 2. If t (u, v) is a real-valued, continuous function in an oriented rectangle R given as in (1), then I(u, v) is said to be absolutely continuous in R in the TONELLI sense (briefly, A CT in R) provided that (i) Tis BVT in R, (ii) for a.e. vE [c, d], I (u, v) is absolutely continuous as a function of u in the closed interval [a, b], and (iii) for a.e. uE [a, b], t(u, v) is absolutely continuous as a function of v in the closed interval [c, dJ.

Lemma 3. Let f(u, v) be a real-valued, continuous function in a bounded domain D(R2, such that I(u, v) is ACT in every oriented rectangle in D. Let D be a domain such that 15 (D. Consider the function

" " I,,(u, v) = 4~2 -f jl(u+a.,V+P)da.dP, (u,v)ED, O<h<1~' (9)

which has been introduced in VI.3.2. Then the following holds for the partial derivatives

(a.) IlIu, Iltv exist and are continuous in D. (f3) Iltu, fltv are given in D by the formulas

It "

thu(U, v) = 4~2 J Jtu(u+a., v +P)da.dP, (10) -II -h

I> II

tllV(U,v)=4~2 J Jtv(u+rt.,v+P)da.dp. (11) -I> -II

§ VI.3. Special classes of differentiable transformations in RI. 429

(y) If {hll} is any sequence of positive numbers such that

t5 O<h,.< V2' h .. -+O,

then

Prool. (ex) follows directly from VL3.2, lemma 4, which also yields the formulas

h

lhu(U,V) = 4~2 JU(u+h,v+P)-/(u-h,v+P)JdP, (12) -h

11

Ihv(U,V) = 4~2 J[f(u+ex, v+h)-/(u+ex, v-h)]dex, (13) -Is

for (u, v)ED. Consider now a point (uo, vo)ED. Denote by Q the oriented square with center at (uo, vo) and side-length 2h. Then QCD [see VI.3.2 (2), (3)J, and hence I(u, v) is ACT in Q by assumption. Accordingly, for a.e. v E [vo - h, Vo + h J, t (u, v) is absolutely continuous as a function of u for uE [uo-h, uo+h], or equivalently, for a.e. PE [- h, h J, I (u, Vo + P) is absolutely continuous as a function of u for uE [uo-h, uo+hJ. By V.1.6, theorem 8 it follows that

I(uo+h, Vo +P) - I(Uls0-h, vo+P) u.u£:i .. (u, vo+P) du ) (14)

=f 1 .. (uo+ex,vo+P)dex for a.e. PE [-h,h]. -h

Note that lu is L-summable in Q by lemma 2. Accordingly, in view of the FUBINI-ToNELLI theorem there follows from (14) the formula

Is

4~2 J [f(uo + h, Vo + P) - I(uo - h, Vo + P)J dP

-Is Is "

= 4~2 J J t .. (uo + ex, Vo + P) dexdP, -Is -Is

and thus (10) is proved. In an entirely similar man,ner, (11) is seen to follow from (13). There remains to verify (y). Since I(u, v) is ACT in every oriented rectangle in D, by lemma 2 it follows that I .. exists a.e. in every oriented rectangle in D. Since D can be represented (see 1.2.2, exercise 8) as a countable union of oriented rectangles, it follows that I .. exists a.e. in D, and by IIL1.3 it follows further that I .. is L-measurable in D. Lemma 2 also yields the fact that I .. is L-summable


in every oriented rectangle in D. Thus we can apply VI.3.2, lemma 2 with I replaced by I .. , obtaining the relation

(f .. )"" -+ I.. a.e. in D. (15)

On the other hand

I"" .. = (f"h .. (16)

by (10). From (15) and (16) we conclude that 1" .... -+ I .. a.e. in D. An entirely similar argument yields the relation Ih"v-+ Iv a.e. in D, and (y) is proved.

Lemma 4. Let I(u, v) be a real-valued, continuous function in a bounded domain D(R2, and let p be a real number greater than or equal to one. Assume that I(u, v) is ACT in every oriented rectangle in D and 1/ .. 11', IlvlP are L-summable in every oriented rectangle in D. Let D be a domain such that 15 (D. Consider the functions (see VI.3.2)

h h d I,,(u, v) = 4~2 _f_fl(zt+a.,V+fJ)da.dfJ, (u,v)ED, O<h< V2 '

Then the family of functions

satisfies the condition (V) in every oriented rectangle in D.

Prool. We note again, as in the proof of part (y) of lemma 3, that I .. exists a.e. in D and is L-measurable in D, and I" is L-summable in every oriented rectangle in D. Furthermore, 1/ .. 11' is L-summable by assumption in every oriented rectangle in D. Accordingly, we can apply VI.3.2, lemma 3 with I replaced by I ... obtaining the conclusion that the family of functions

i(/"hIP, 0 < h < ~ ,

satisfies the condition (V) in every oriented rectangle in D. Similarly we find that the family of functions

I (fv)"IP, 0 < h < ~ ,

satisfies the condition (V) in every oriented rectangle in D. Since

by part (fJ) of lemma 3, the present lemma follows.


Lemma 51. Let t(u, v) be a real-valued, continuous function in a bounded domain D (R2, such that (a) t (u, v) is A CT in every oriented rectangle in D, and (b) there exists a real number P>2 such that ituiP, it.ip are L-summable in every oriented rectangle in D. Let Q be an oriented square in D and OJ (I, Q) the oscillation of t in Q. Then

0)(1, Q);;;;,r(p) [JJ(1 + ifui + ifvJ)Pdudv]!, (17) Q

where r(p) is the constant occurring in V1.3.1, lemma 3 [thus r(p) is positive, finite, and depends solely upon P J.

Proof. Note that (17) has been already established in VI.3.1, lemma 3 for the special case when t(u, v) has continuous first partial derivatives. The more general situation considered in the present lemma will be treated by the method of approximation by integral means, as follows. Select an oriented square Q (D. Since, by assumption, ifuiP and itviP are L-summable in Q, by 111.1.1, lemma 28 it follows that the integral appearing in (17) exists. Select a domain D such that

Q(D, D(D, ( 18)

and consider, as in VI.3.2, the functions

" " t,,(u, v) = 4:2 -f 1 t(u+('/., v+fJ) docdfJ, (u, v)ED, 0 < h < ;1· (19)

Select a sequence of real numbers hn such that

15 o < hn < V1' hn -+ o. (20)

We have then, by lemma 3,

and hence also [see (18)]

In view of (20), the sequences

(22)

1 The inequality (17) is a special case of an inequality due to A. P. CALbERON,

On the differentiability of absolutely continuous functions [Riv. Mat. Univ. Parma 2, 203-213 (1951)].


satisfy the condition (V) in Q by lemma 4. Take now any L-measurable set 5 (Q. By 111.1.1, lemma 28 we have the inequality

[ff (1 + I/hn,,1 + I/hnvW du dv ]l/P :;;: 1 5 :;;: [If lP du dv ]l/P + [If I/hRul P du dv rp + [If I/hnvl P du dV]l/P . (23)

Since the sequences (22) satisfy the condition (V) in Q, it is clear from (23) that the sequence

also satisfies the condition (V) in Q. In view of (21) it follows (see 111.1.1, lemma 36) that

Since I (u, v) is continuous in D, from VI.3.2, lemma 5 we infer that

Ilrn -'r I uniformly in Q. Hence clearly

(25)

Observe now that Ihn has continuous first partial derivatives in D by lemma 3. Accordingly, we can apply VI.3.1, lemma 3 to Ilrn' obtaining the inequality

Clearly (26), (25), (24) imply (17).

VI. 3.4. Special classes of differentiable transformations in R2. In this section we shall use the representation

T:x=x(u,v), Y=Y(u,v), (u,v)ED, (1 )

for the bounded continuous transformation T:D-'rR2 (see VI.1.1).

Definition 1. Given T as in (1), we shall say that T satisfies the hypothesis H ° in D if (i) the first partial derivatives xu' xv, Yu, yv exist a.e. in D, (ii) the ordinary Jacobian ](w, T) is L-summable in D, and (iii) x(u, v), y(u, v) are ACT in every oriented rectangle in D.

Practically all the special classes of differentiable transformations in R2 which have been studied in the literature satisfy the hypothesis Ho, and hence the clarification of the implications of this hypothesis is a matter of considerable interest. It has been conjectured that the

§ VI.3. Special classes of differentiable transformations in RO. 433

hypothesis Ho implies that T is eA C in D. Since the validity of this

conjecture is still an open question, we shall discuss only a few typical

results relating to the hypothesis Ho. The proofs will be based upon

the method 01 approximation by integral means.

Given T as in (1), take a sequence {Di} of domain"s which fill up D

from the interior (see 11.3.2, remark 10). Then DieD, j = 1,2, ....

Hence (see VI.3.2) for every positive integer j we can consider the

functions

" h

Xh(U, v) = 4~2 _[_I x(u+rx,v+P)drxdP, (u,v)EDi , O<h<V2"' (2)

h h

y,,(u,v)=4~!2 _[_[y(u+rx,V+P)drxdP, (1,f"v)EDi · o<h< 1%' (2*)

where {)i is an appropriately chosen positive constant. By VI.3.2,

lemma 5 we can select for each j a number hi such that

and furthermore

!Xhj(U,V)-X(U,V)!<-\ for (u,v)EDi , 1

!Yhj (u, v) - Y (u, v) I <-~- for (u, v) E Di . 1

To simplify notations, we put

Xhf(1-t, v) = Xi(u, v), (u, v) E Di ,

Yhf(U, v) = Yi(u, v), (u, v) E Di .

(3)

(4)

(5)

(6)

(7)

Lemma 1. Xi and Yi are continuous in Di , and the first partial

derivatives Xi'" Xiv' Yi", Yiv exist and are continuous in Di ·

Prool. Since x(u, v) and y(u, v) are continuous in D, this lemma

is a direct consequence of the lemmas 1 and 4 in VI.3.2.

Let us introduce now the auxiliary transformations

T;:x=Xi(u,v), Y=Yi(u,v), (u,v)ED;.

Lemma 2. T; is eA C in Di .

Prool. Since the first partial derivatives of the coordinate functions

of T; exist and are continuous in Di by lemma 1, in view of V.3.6,



lemma 7 (applied to:q it is sufficient to show that the ordinary Jacobian ](w,1f) is L-summable in Di . Now since T is bounded in D, we have a constant M> 0 such that

Ix(u,v)l, ly(u,v)1 <M, (u,v)ED. (8)

Furthermore, by VI.3.2, lemma 4 we have [in view of (1) and (6)] the formula

hi

Xiu(u, v) = 4~i J[x(u+hi,v+P)-x(u-hi,v+P)]dP, (u,v)EDi · -hi


IXiu (u, v) I < :.-' (u, v) E Dj for 1 large. 1

A similar argument yields the same bound for IXivl, IYiul. Ilf.l. Hence

2M2 I](w, 1f)1 <~, wE Di for j large.

1

Thus ](w.1f) is continuous and bounded in Dj • Hence ](w,1f) i£ L-summable in Dj , and the lemma is proved.

In the course of the following discussion we shall make statements about the behavior of the sequence {1f} on some compact subset F of D. Since the domains Dj fill up D from the interior, there exists an integer 10 = jo (F) such that F (Di for 1> io· Then 1f is defined on F for i > jo' and in the sequel it will be understood that our statements are meant to apply for 1> 10 only.

Lemma 3. The sequence {1f} converges to T uniformly on compact subsets of D, in the sense of 11.3.2, remark 9.

Proof. Take a compact set F(D. Since the domains Dj fill up D from the interior. we can select an integer 10 such that F(Dj for j> jo. From (4), (5), (6), (7) we infer that

e (T, 1f. F) < ~ for j> jo· 1

Thus clearly e(T, 1f,F)~O for j~oo, and the lemma follows.

Theorem 1. Given T as in (1), assume that the following holds. (a) T satisfies the hypothesis Ho in D. (b) If Q is any oriented square in D, then the sequence {J(w,1f)} satisfies the condition (V) in Q. Then Tis eAe in D.

§ Vr.3. Special classes of differentiable transformations in R!. 435

Proof. The assumption (a) implies that the first partial derivatives Xu, Xv' Yu' Yv exist a.e. in D and the ordinary Jacobian J(w, T) is L-summable in D. By lemma 3, the sequence {11} converges to T uniformly on compact subsets of D. The first partial derivatives of the coordinate functions of 11 exist and are continuous in D j by lemma 1, and 11 is eA C in Di by lemma 2. By assumption, the sequence {J(w,11)} satisfies the condition (V) in every oriented square Q(D. Accordingly, the present theorem will appear as a consequence ofVI.2.3, theorem 7 if we show yet that if F is a compact set in D, then

J(w, 11) -+ J(w, T) a.e. on F. (9)

Now since the domains Di fill up D from the interior, we have an integer io=jo(F) such that F(Di•· By (3) we have then O<hi < t5j.lV2 for j > io. Since X (u, v) and y (u, v) are A CT in every oriented rectangle in D, we can apply VI.3.3, lemma 3 to the functions x(u, v), y(u, v) and the domain Dio' In view of (7) and (6) it follows (since hi-+O) that Xi .. ' Xiv, Yiu, Yiv converge to xu, xV, Y .. , Yv respectively a.e. in Di•• Hence (9) holds, since F (Dio '

We proceed to consider a few of the numerous applications of the preceding theorem.

Theorem 2. Given T as in (1), assume that the following holds. (a) T satisfies the hypothesis Ho in D. (b) There exist two real numbers p, q such that

1 1 P>1 q>1 --+-=1 , 'p q , (10)

and Ix .. I!'>, Ixvl!'>, IYulq, IYvlq are L-summable in every oriented rectangle in D. Then T is eA C in D.

Proof. In view of theorem 1 it is sufficient to verify that the sequence {J(w, 11)} satisfies the condition (V) in every oriented square Q(D. In turn, this latter fact will be established if we show that the sequences

(11)

satisfy the condition (V) in every oriented square Q(D. Select such a square Q. Since the domains Di fill up D from the interior, we can select an integer io such that Q (D io ' By (3) we have then 0 < hi < ~j.lV 2 for j>jo' In view of the assumptions (a) and (b) we conclude that we can a ppl y VI.3.3, lemma 4 to the functions x (it, v), y (u, v) and the domain Dio' It follows that the sequences

{IXi .. I!'>} , {IXivl!'>}, {I~'ulq}, {IYivlq}, i > jo, Rado and Reichelderfer, Continuous Transformations. 28a


satisfy the condition (V) in Q. Hence if 13 > 0 is assigned, then there exists a number f/ > 0 such that (for j> jo)

IIIXiulP du dv < 13, IIIXivlP du dv < 13, ) 5 5

IIIYfulq dudv < 13, IIIYfvlq dudv < 13, 5 5

( 12)

whenever 5 is an L-measurable set such that

5(Q, L5<f/. (13)

Assume that (13) holds. Then .the HOLDER inequality (see 111.1.1, lemma 29) yields, in view of (12),

.1. + 1c I {I IXiu Yfv du dvl ~ [{I IXiulP du dvlllP [{I I Yjvlq du dvlllQ < eP q = 13

for j> io. An entirely similar reasoning shows that

I[IXivYjududvl<e for j>jo'

Thus the sequences (11) satisfy the condition (V) in Q, and the proof is complete.

Definition 2. A transformation T, given as in (1), is termed a DIRICHLET transformation in D if (i) T satisfies the hypothesis Ho in D, and (ii) for every oriented rectangle R(D the DIRICHLET integrals

II (x! + x;) du dv, II (y! + y~) du dv R R

of the coordinate functions of T exist.

Theorem 3. If T is a DIRICHLET transformation in D, then T is eAC in D.

This statement is merely a special case of theorem 2, corresponding to p = q = 2. However, theorem 3 is of interest for two reasons. First, DIRICHLET transformations play an important role in various applications. Second, relatively simple examples show that a DIRICHLET transformation is generally not absolutely continuous in the BANACH sense. Thus the important class of DIRICHLET transformations is beyond the scope of the B-theory, while on the other hand (in view of theorem 3), these transformations constitute a very special sub-class of the class of eA C transformations.

The following theorem is concerned with another very special but quite important class of transformations in R2.


Theorem 4. Given T as in (1), assume that the following holds. (a) T satisfies the hypothesis Ho in D. (b) One of the coordinate functions x(u, v), y(u, v) of Tis Lipschitzian in D. Then Tis eAC in D.

Proal. Assume, for instance, that y(u, v) is Lipschitzian in D. Having selected an oriented square QeD and a number 8>0, the proof begins as that of theorem 2, the modifications taking place when the inequalities (12) are reached. Instead of (12) we have now the following facts at our disposal. Since x(u, v) is A CT in every oriented rectangle in D, the partial derivatives x,,, Xv are L-summable in every oriented rectangle in D by VI.3.3, lemma 2. Applying VI.3.3, lemma 4 to x(u, v) with P=1, we obtain [corresponding to the first line in (12)] the conclusion that (for j>jo) .

(14)

whenever 5 is an L-measurable set such that

5eQ, L5<1]. ( 15)

Since y (u, v) is Li pschi tzian in D, there exists a constant M;;;;: 0 such that

ly (u 2 , v2) - y(ul , v])l :;;;: M [(uz - 1ll)2 + (V2 - VI)ZJ~ (16)

for any two points (u2 , v2), (ul , VI) in D. In view of (7) and VI.3.2, lemma 4 we have the formula

"I Yf .. (u, v) = 41h~ J [y(u + hi' v + (J) - y(u - hi' v+{J)] d{J, (u, v) E Dj •

1 -hJ


( 17)

A similar argument yields

IYfvi :;;;: M in Di . ( 18)

Consider now any L-measurable set 5 such that (15) holds. From (14), (17), (18) we conclude that

IjJXiuYfvdudvl:;;;:MjJIXiuldudv:;;;:M8 for j>jo,

and similarly


Since M is fixed and e was arbitrary, these inequalities imply that the sequences

{Xju Yiv}, {Xjv Yi,,}, j> jo,

satisfy the condition (V) in Q. Hence the sequence {J(w, 1f)}, j> jo, also satisfies the condition (V) in Q, and thus the present theorem follows from theorem 1.

The next theorem yields an important class of generalized Lipschitzian transformations in R2.

Theore'l1'} 6. Given T as in (1), assume that the following holds. (a) T satisfies the hypothesis Ho in D. (b) There exists a number p> 2 such that IxulP, IxvlP, ly"IP, IYvlP are L-summable in every oriented rectangle in D.

Then T is generalized Lipschitzian in D (and hence T possesses all the properties stated in V.3.6, theorem 2).

Proof. Take an oriented rectangle R (D and select an oriented square Q(R. We can then apply VI.3.3, lemma 5 to both x(u, v) and y(u, v), obtaining the inequalities

ro(x,Q);;;;;F(p)[II(1 + Ix,,1 + IXvl)PdudvJ~, (19) Q

ro (y, Q);;;;' r(p) [II (1 + IYul + IYvl)P du dv Jk. (20) Q

Let us put

cp = 1 + Ix,,1 + Ixvl + IYul + IYvl· (21)

The assumption (b) implies, by llI.1.!, lemma 28, thatcpP is L-summable in R. Since clearly

~ T Q;;;;; ro (x, Q) + ro (y, Q),

we obtain from (19), (20), (21) the inequality

~ T Q ;;;;; 2r(p) [fJ ql du dVP. Q

(22)

Hence on setting

<P(u, v) = 4F(P)2cp(U, v)P, (u, v) E R,

it follows that <P is a non-negative, L-summable function in R such that T satisfies the condition (<P, intR) [see (22) and V.3.6, definition 1J. Since the assumption (a) implies that the ordinary Jacobian J(w, T) is L-summable in D, it follows finally (see V.3.6, definition 2) that T is generalized Lipschitzian in D.

Index.

Absolute continuity III.1.2, III.2.2, IV.2.3, IV.3.3, IV.S.2, V.U.

ACb IV.2.3. ACB IV.S.2. ACM IV.3.3. ACT V1.3.3. Admissible multiplicity function IV.3.1. a.e. II!. 1. 1. ~e (D) IV.4.3. Analytic set 1.1.4. Approximate total differential III.1.3. arg z V!. 1. 1.

Base-function IV.2.1. Base-set IV.2.1. ba(u) IV.S.1. be (u) IV.4.1. b;(u), h';(u) IV.4.3. \B~, \B~+, \B~- IV.4.1, IV.4·3. Bi-measurable IV.4.6. Bi-unique 1.1.1. BOREL measurable II1.1.1. BOREL set 1.1.4. Boundedly compact 1.1.4. Bounded variation III.2.1, IV.2.2, IV.3.2,

IV.S.1, V.1.1. BVb IV.2.2. BVB IV.S.1. BVM IV.3.2. BVT V1.3.3.

Cartesian product I. 1. 1. CA UCHY sequence I.1.4. Characteristic neighborhood II.3.3. Closed set 1.1.3. Closure I.1.3. Coboundary homomorphism 1.4.3. Cochain mapping 1.4.2. Cohomology group 1.4.1, 1.6.1. Cohomology sequence 1.4.3, I.6.4. Compact I.1.3.

Complement C 1.1.1. Complete space 1.1.4. Completely additive family 1.1.4. Component 1.1.3. Condition (arg) V!. 1.2. Condition (arg, r) V1.1.2. Condition (N) IV.1.4. Condition (V) II1.1.i. Connected 1.1.3. Continuous 1.1.3, 1.1.5. Continuum 1.1.3. Convex I.2.2. Covering I. 1. 3. Crude multiplicity function 1.1.2.

Da(u, T) IV.S.1. De(u, T), D;(u, T), D;(u, T) IV.4.1,

IV.4·3. Deformation retract 1.1.5. Ll(n, m), Ll*(n, m) I.2.3. Dense 1.1.3. Derivative III.2.3. Determining system 1.1.4. Diameter I. 1.4. Differential III.i.3. Disconnected I.1.3. Domain I.1.3.

eA C IV.4.2. eBV IV.4.1. E(T. D), EP(T, D), Ei(T, D), Ef(T, D)

11.3.6. (};(x, T, D), (};i(x, T, D) II.3.3. En 1.2.2. e (x, E) 1.1.4. e.m.m.C. II.3.3. Essential absolute continuity IV.4.2. Essential bounded variation IV.4.1. Essential generalized Jacobian IV.4.3. Essentially isolated II.3.3. Essential local index Il.3.4.

440

Essential maximal model continuum II.3.3.

Essential sets II.3.6. Essential total variation IV.4.1. Exact 1.3.3.

Finite measure II1.1.2. Formal complex 1.5.2, 1.5.5. Frame II. 1. 1. Frontier (fr) 1.1.3. Fully normal 1.1.3.

Generalized Lipschitzian V.3·6.

HAUSDORFF space 1.1.3. Homeomorphism 1.1.5. Homotopy, homotopic 1.1.5, 1.4.5. Hypothesis Ho V1.3.4.

I 1.3.1. iB (u, T) 11.3.7. ie(C, T) II.3.4. Inclusion mapping 1.1.1-Indicator domain II.3.2. Indicator system 11.3.2. Interior (int) 1.1.3.

J(u, T) V.2.2. Is (u, T) IV.5.3. f e (u, T) IVA.3. JORDAN region VI. 1. 1.

k (x, T, D) II.3.3. K(x, T, D), K+(x, T, D), K-(x, T, D)

II.3·2. Sf (D), Sf+(D), Sf-(D) IVA.1, IVA.3.

LEBESGUE integral, measure II1.1.1-L-measurable III.1.1. L-summable III. 1. 1. Lipschitzian III.1.3, V.2.3. Locally connected 1.1.3. Lower semi-continuous 1.1.3.

M(X, A), M(X) 1.6.1. MAYER complex 1.4.1. Maximal model continuum II.3.1. Measurable homeomorphism IVA.6.

Index.

Metric space 1.1.4. m.m.c. II.3.1. fl- (x, T, D) II.2.2. fl-. (x, T, D) 11.3.4.

n-cell E" 1.2.2. n-interva~ 1.2.2. n-sphere Sn 1.2.2. N(x, T, 5) 1.1.2. Negative indicator domoain 11.3.2. Negative indicator system II.3.2. Normal 1.1.3. VB IV.5.1. ve IV.4.1. v-integral III. 1.2. v-summable III. 1.2.

Open covering 1.1.3. Open set 1.1.3. Ordinary Jacobian V.2.2. Oriented n-cube 1.2.2. Oscillation w 1.1.2, 1.1.5, V.1.1.

Pair 1.1.1. Parallelotope 1.2.2. Partition AlB 1.1.3. Point of density III. 1. 1. Point of linear density III. 1 .1. Positive indicator domain II.3.2. Positive indicator system 11.302. Preferred generator 11.1.3. p-coboundary 1.4.1. p-cochain 104.1. p-cocyde 1.4.1. p-function 1.5.1.

Rn 1.2.1. Reduced base-set IV.2.1. Refinement 1.1.3. Regular determining system 1.1.4. Relatives 11.3.7. Retract, retraction 1.1. 5. e(T1 , T2 , E) 1.1·5·

sn 1.2.2. sA CB IV.5.3. sBVB IV.5.3. @)(x, T, D), @)+(x, T, D), @)-(x, T, D)

11.3.2.

Segment 1.2.2. Separable 1.1.3. Simple arc VI. 1. 1.

Singular measure III. 1.2. Spherical frame I1.2.1. Standard triple 1.4.3. Star 1.1.3. Star refinement 1.1.3. Stereographic projection 1.2.2. Strongly adjacent 1.2.3. Strongly connected 1.2.3. Sub-additive 111.2.3. IIL2.4. Subcomplex 1.4.6. Sub-covering I. 1. 3. Subspace I. 1.3. Summatory function IV.i.3. IV.2.1.

Index.

Topological index II.2.2. Topological product 1.1.3. Topological space I. 1.3. Topologically similar 11.3.8.

441

Total differential 111.1.3. V.2.2. Total variation 111.2.1. IV.S.1. V.1.1.

V(f. 5) V.1.1. V(p. 0) 111.2.1. Vg(T.O) IV.S.1. Ve (T. D) IVA.1. Variation of argument VL1.2.

Weak total differential V.2.2.

Bibliography.

ALEXANDRO:fF, P., and H. HOPF: Topologie. Grundlehren der mathematischen Wissenschaften, Vol. 45. Berlin: Springer 1953.

ELLENBERG, S., and N. STEENROD: Foundations of Algebraic Topology. Princeton University Press 1952.

HAHN, H.: Reelle Funktionen, Part 1. Leipzig 1932.

HALMOS, P. R: Measure Theory. New York: D. Van Nostrand Co. 1950.

HAUSDORFF, F.: Mengenlehre. Leipzig 1935.

KURATOWSKI, C.: Topologie, I, II. Monografie Matematyczne, Warszawa, 1948, 1950.

LEFSCHETZ, S.: Algebraic Topology. Amer. Math. Soc. Colloquium Publications, Vol. 27, 1942.

MAYRHOFER, K.: Inhalt und MaB. Wien: Springer 1952.

RADO, T.: Length and Area. Amer. Math. Soc. Colloquium Publications, Vol. 30, 1948.

SAKS, S.: Theory of the Integral. Monografie Matematyczne, Warszawa, 1937.

SIERPINSKI, W.: Introduction to General Topology. University of Toronto Press 1934.

WHYBURN, G. T.: Analytic Topology. Amer. Math. Soc. Colloquium Publications, Vol. 28, 1942.

WILDER, R L.: Topology of Manifolds. Amer. Math. Soc. Colloquium Publications, Vol. 32, 1949.

Continuous Transformations in Analysis: With an Introduction to Algebraic Topology

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