Strange Functions in Real Analysis

STRANGE FUNCTIONS IN REAL ANALYSIS

PURE AND APPLIED MATHEMATICS A Program of Monographs, Textbooks, and Lecture Notes

EXECUTIVE EDITORS Zuhair Nashed University of Delaware Newark, Delaware

Earl J. Taft Rutgers University New Brunswick, New Jersey

EDITORIAL BOARD M. S. Baouendi University of California, Sun Diego Jane Cronin Rutgers University Jack K. Hale Georgia Institute of Technology

Anil Nerode Cornell University Donald Passman University of Wisconsin, Madison Fred S. Roberts Rutgers University

S. Kobayashi University of California, Berkeley

David L. Russell Virginia Polytechnic Institute and State University

Marvin Marcus University of California, Santa Barbara

Walter Schempp Universitat Siegen

W.S. Massey Yale University

Mark Teply University of Wisconsin, Milwaukee

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STRANGE FUNCTIONS IN REAL ANALYSIS A. B.Kharazishvili Tbllisi State University Tbilisi, Republic of Georgia

MARCLL

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Preface

At the present time, many strange (or singular) objects in various fields of mathematics are known and no working mathematician is greatly surprised if he meets some objects of this type during his investigations. In connection with strange (singular) objects, the classical mathematical analysis must be noticed especially. It is sufficient to recall here the wellknown examples of nowhere differentiable real-valued functions; examples of Lebesgue measurable real-valued functions nonintegrable on any nonempty open subinterval of the real line; examples of Lebesgue integrable realvalued functions with everywhere divergent Fourier series, and others. There is a very powerful technique in modern mathematics by means of' which we can obtain various kinds of strange objects. This is the socalled category method based on the classical Baire theorem from general topology. Obviously, this theorem plays one of the most important roles in mathematical analysis and its applications. Let us recall that, according to the Baire theorem, in any complete metric space E (or, in any locally compact topological space E ) the complement of a first category subset of E is everywhere dense in E, and it often turns out that this complement consists precisely of strange (in a certain sense) elements. Many interesting applications of the category method are presented in the excellent textbook by Oxtoby [117] in which the deep analogy between measure and category is thoroughly discussed as well. In this connection, the monograph by Morgan [110]must also be pointed out where an abstract concept generalizing the notions of measure and category is introduced and investigated in detail. Unfortunately, the category method does not always work and we somet i m eneed s an essentially different approach to questions concerning the existence of singular objects. This book is devoted to some strange functions in real analysis and their applications. Those functions can be met in various studies in analysis and play an essential role there, especially as counterexamples to numerous statements which seem t o be very natural but, finally, fail to be true in certain situations (see, e.g., [49]).Another important role of strange functions, with respect to given concepts of analysis, is to show that those concepts

iv

P REF A C E

are, in some sense, not satisfactory and hence have to be revised, generalized or extended in an appropriate direction. In this context, we may say that strange functions stimulate the development of analysis. The book deals with a number of important examples and constructions of strange functions (primarily, we consider functions acting from the real line into itself). Notice that many such functions can be obtained by using the category method (for instance, a real-valued continuous function defined on the closed unit interval of the real line, which does not possess a finite derivative a t each point of this interval). But, as mentioned above, there are some situations where the classical category method cannot be applied, and thus, in such a case, we have t o appeal to the corresponding individual construction. We begin with functions which can be constructed within the theory ZF & DC where ZF denotes the Zermelo-Fraenkel set theory without the Axiom of Choice and DC denotes a certain weak form of this axiom: the so-called Axiom of Dependent Choices which is enough for most domains of classical mathematics. Among strange functions whose existence can be established in ZF & DC the following ones are of primary interest: Can. tor and Peano type functions, everywhere differentiable nowhere monotone functions, Jarnik's nowhere approximately differentiable functions. Then we examine various functions whose constructions need essentially noneffective methods, i.e. they need an uncountable form of the Axiom of Choice: functions nonmeasurable in the Lebesgue sense, functions without the Baire property, functions associated with a Hamel basis of the real line, Sierpiriski-Zygmund functions which are discontinuous on each subset of the real line of the cardinality continuum, etc. Finally, we consider a number of examples of functions whose existence cannot be established without additional set-theoretical axioms. However, it is demonstrated in the book that the existence of such functions follows from (or is equivalent to) certain widely known set-theoretical hypotheses (e,g, the Continuum Hypothesis). Among other topics presented in this book, closely connected with strange functions in real analysis, we wish to point out the following ones: Egorov's and Mazurkiewicz's theorems on uniform convergence of measurable functions, some relationships between the classical Sierpiriski partition of the Euclidean plane and Fubini type theorems, sup-measurable and weakly sup-measurable functions with their applications in the theory of ordinary differential equations. In the final chapter of our book, we consider the family of all nondifferentiable functions from the points of view of category and measure. We present one general approach illuminating the basic reasons which necessarily imply that the above-mentioned family of functions has t o be large in

PREFACE

V

the sense of category or measure. Notice that, in connection with nondifferentiable functions, a short scheme for constructing the classical Wiener measure is discussed in this chapter, too, and some simple but useful statements from the general theory of stochastic processes are demonstrated. This book is based on the course of lectures given by the author at Institute of Applied Mathematics of Tbilisi State University in the academic year 1997 1998, entitled:

-

Some Pathological Functions in Real Analysis. These lectures (their role is played by the corresponding chapters of the book) are, in fact, mutually independent from the logical point of view but are strictly related from the point of view of the topics discussed and the methods applied (such as purely set-theoretical arguments and constructions, measure-theoretical methods, the Baire category method, and so on). The material presented in the book is essentially self-contained and, consequently, is accessible to a wide audience of mathematicians (including graduate and postgraduate students). For the reader's convenience, Chapter 0 plays the role of introduction to the subject. Here some preliminary notions and facts are given that are useful in our further considerations. The reader can ignore this auxiliary chapter, returning to it if the need arises. In this connection, the standard graduate-level textbooks and monographs (for instance, [53], [66], [91], [117], [127]) should be pointed out containing all preliminary notions and facts from set theory, general topology and real analysis. We begin with basic set-theoretical concepts such as: binary relations of special type (namely, equivalence relations, orderings, functional graphs), ordinal numbers, cardinal numbers, the Axiom of Choice and the Zorn lemma, some weak forms of the Axiom of Choice (especially, the countable form of AC and the Axiom of Dependent Choices), the Continuum Hypothesis, the Generalized Continuum Hypothesis, and Martin's Axiom as a set-theoretical assertion which is essentially weaker than the Continuum Hypothesis but rather helpful in various constructions of set theory, topology, measure theory and real analysis. Then we briefly present some basic concepts of general topology and classical descriptive set theory, such as: the notion of a first category set in a topological space, the Baire property (the Baire property in the restricted sense) of subsets of a topological space, the notion of a Polish space, Borel sets in a topological space, analytic (Suslin) subsets of a topological space, and the projective hierarchy of Luzin, which takes the Borel and analytic sets as the first two steps of this hierarchy. It is also stressed that Borel and

vi

PREFACE

analytic sets have a nice descriptive structure but this feature fails to be true for general projective sets (since, in certain models of set theory, there exist projective subsets of the real line which are not Lebesgue measurable and do not have the Baire property). The final part of Chapter 0 is devoted to some classical facts and statements from real analysis. Namely, we recall here the notion of a real-valued lower (upper) semicontinuous function and demonstrate basic properties of such functions, formulate and prove the fundamental Vitali covering theorem, introduce the notion of a density point for a Lebesgue measurable set, and present the Lebesgue theorem on density points as a consequence of the above-mentioned Vitali theorem. In addition, we give here a proof of the existence of a nowhere differentiable real-valued continuous function, starting with the well-known Kuratowski lemma on closed projections. Let us emphasize once more that the question of the existence of real-valued continuous nondifferentiable functions, with respect to various concepts of generalized derivative, is one of the central questions in this book. We develop this topic gradually and, as mentioned earlier, investigate the question from different points of view; however, the main focus is on its purely logical and set-theoretical aspects.

Contents

. . . . . . . . . . . . . . . . . . . . . . . . . iii Introduction: basic concepts . . . . . . . . . . . . . . 1 Cantor and Peano type functions . . . . . . . . . . . . 33 Singular monotone functions . . . . . . . . . . . . . . 55 Preface..

Everywhere differentiable nowhere monotone functions. ..

........... ............ Nowhere approximately differentiable functions . . .

69 83

I

Blumberg's theorem and Sierpiliski-Zygmund function.

95

Lebesgue nonmeasurable functions and functions without the Baire property

113

.... ....... .. .... . ........

... .... .. ...... Hamel basis and Cauchy functional equation . . . . ..

159

.................

181

Luzin sets, Sierpinski sets and their applications Egorov type theorems.

137

Sierpinski's partition of the Euclidean plane

. . . . . 197

Sup-measurable and weakly sup-measurable functions.

217

... .......... .. .. .... . .. .

Ordinary differential equations with bad right-hand sides

. . . . . . . . . . . . . . . . . . . . . . 237

viii

.

13

CONTENTS

Nondifferentiable functions from the point of view of category and measure

. . . . . . . . . . . . . . . . 251 Bibliography . . . . . . . . . . . . . . . . . . . . . . . 279 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291

0. Introduction: basic concepts

In this chapter we fix the notation and present some elementary facts from set theory, general topology and the theory of real functions. We shall systematically utilize these facts in our further considerations. The symbol ZF denotes the Zermelo-Fraenkel set theory which is one of the most important formal systems of axioms for the whole of modern mathematics (in this connection, see [88] and [91]; cf. also [13]). The basic notions of the Zermelo-Fraenkel system are sets and the membership between them. Of course, the system ZF consists of several relation axioms which formalize various properties of sets in terms of the relation We do not present here a list of these axioms and, actually, we shall work in the so-called "naive set theory". The symbol ZFC denotes the Zermelo-Fraenkel theory with the Axiom of Choice. In other words, ZFC is the following theory:

where AC denotes, as usual, the Axiom of Choice. At the present time, it is widely known that the theory ZFC is a basis of modern mathematics, i.e. almost all fields of mathematics can be developed by starting with ZFC.The Axiom of Choice is a very powerful set-theoretical assertion which implies many extraordinary and interesting consequences. Sometimes, in order t o get a required result, we do not need the whole power of the Axiom of Choice. In such cases, it is sufficient to apply various weak forms of AC.Some of these forms are discussed below. If x and X are any two sets, then the relation x X means that belongs to X. In this situation, we also say that x is an element of X. One of the axioms of set theory implies that any set y is an element of some set Y (certainly, depending on y). Thus we see that the notion of an element is equivalent to the notion of a set. The relation X Y means that a set X is a subset of a set Y . The relation X c Y means that a set X is a proper subset of a set Y . If R(x) is a relation depending on an element x (or, in other words,

2

CHAPTER

0

R(x) is a property of an element x), then the symbol

denotes the set (the family, the class) of all those elements x for which the relation R(x) holds. In our further considerations we always suppose that R(x) is such that the corresponding set {x : R(x)} does exist. For example, a certain axiom of ZF states that there always exists a set of the type {x : x X S(x)} where X is an arbitrarily given set and this case we write X : instead of {x : x X then we write instead of {x : The symbol

is an arbitrary relation. In

Also, if we have two relations

and

denotes, as usual, the empty set,

If X is any set, then the symbol denotes the family of all subsets we have of X , = {Y : Y The set is also called the power set of a given set X . If x and y are any two elements, then the set

is called the ordered pair (or, simply, the pair) consisting of x and viously, we have the implication ((x, y) =

(x =

y=

for all elements x , x', y'. Let X and Y be any two sets. Then, as usual, X U Y denotes the union of X and Y; X fl Y denotes the intersection of X and Y; X \ Y denotes the difference of X and Y; X A Y denotes the symmetric difference of X and Y, i.e. XAY = ( X \ Y ) u ( Y \X).

Ob-

INTRODUCTION

We also put X x Y = {(x,y)

:

XEX, y EY).

The set X x Y is called the Cartesian product of the given sets X and Y. In a similar way, by recursion, we can define the Cartesian product

of a finite family {XI, X2,... , X,) of arbitrary sets. If X is a set, then the symbol card(X) denotes the cardinality of X . Sometimes, card(X) is also called the cardinal number of X . w is the first infinite cardinal (ordinal) number. In fact, w is the cardinality of the set N = {0, 1, 2, , n, ,,.) ..#

of all natural numbers. Sometimes, it is convenient to identify the sets w and N. A set X is finite if card(X) < w . A set X is infinite if card(X) 1 w . A set X is (at most) countable if card(X) 5 w . Finally, a set X is uncountable if card(X) > w . For an arbitrary set E, we put: = the family of all finite subsets of E; [E]sW= the family of all countable subsets of E . w l is the first uncountable cardinal (ordinal) number. Notice that w l is sometimes identified with the set of all countable ordinal numbers (countable ordinals). Various ordinal numbers (ordinals) are denoted by a, p, y, E, C, ... . Let a be an ordinal number. We say that a is a limit ordinal if a = sup{@ :

p < a).

The cofinality of a limit ordinal a is the smallest ordinal 5 such that there exists a family {aS : C < E) of ordinals satisfying the relations (VC < E ) ( q < a ) ,

a = sup{ac :

C <El.

The cofinality of a limit ordinal a is denoted by the symbol cf (a). Clearly, we have the inequality cf ( a ) a for all limit ordinal numbers


w , there exists a model of ZFC in which we have the equality c = w,. Actually, if we start with an arbitrary countable transitive model for ZFC (strictly speaking, for a relevant fragment of ZFC) satisfying the Generalized Continuum Hypothesis, then the above-mentioned equality is true in a certain Cohen model for ZFC extending the original model (for details, see [88], Chapter 7). The Generalized Continuum Hypothesis holds in a special model of set theory, first constructed by Godel. This model is called the' Constructible Universe of Godel and usually denoted by L. Various facts and statements concerning L are discussed in [88], Chapter 6 (see also [55] and [56]). It is reasonable to note here that, in L, some naturally defined subsets of the real line are bad from the point of view of Lebesgue measure and Baire property (i.e. they are not measurable in the Lebesgue sense and do not have the Baire property). Let n be a fixed natural number. The symbol Rn denotes, as usual, the n-dimensional Euclidean space. If n = 0, then Rn is the one-element set consisting of zero only. If n > 0, then it is sometimes convenient to consider R n as a vector space V over the field Q of all rational numbers. According to a fundamental assertion of the theory of vector spaces (over arbitrary fields), there exists a basis in the space V (see, e.g., [31] where much more general statements are discussed for universal algebras). This basis is usually called a Hamel basis of V . Obviously, the cardinality of any Hamel basis of V is equal to the cardinality of the continuum. Notice also that the existence of a Hamel basis of V cannot be established without the aid of uncountable f o r m of the Axiom of Choice because the existence of such a basis immediately implies the existence of a subset of the real line R, nonmeasurable with respect to the standard Lebesgue measure given on R. Let X and Y be any two sets. A binary relation between X and Y is an arbitrary subset G of the Cartesian product of X and Y , i.e.

In particular, if we have X = Y , then we say that G is a binary relation on the basic set X. For any binary relation G C X x Y , we put

~7'2(G)= {y : (3x)((x, y) E G)). It is clear that G C prl(G) x prz(G).

6

CHAPT ER

0

The Axiom of Dependent Choices is the following set-theoretical statement: If G i s a binary relation on a nonempty set X and, for each element x E X , there exists an element y E X such that (x, y) E G, then ihere exists a sequence (xo, X I , ..., xn, ...) of elements of the set X , such that

The Axiom of Dependent Choices is usually denoted by D C . Actually, the statement D C is a weak form of the Axiom of Choice. This form is completely sufficient for most fields of classical mathematics: geometry of a finite-dimensional Euclidean space, mathematical analysis of the real line, Lebesgue measure theory, etc. We shall deal with the axiom D C many times in our further considerations and discuss some interesting applications of this axiom. It was established by Blair that, in the theory ZF, the next two assertions are equivalent: a) the Axiom of Dependent Choices; b) no nonempty complete metric space is of the first category (the classical theorem of Baire). Exercise 1. Prove the logical equivalence of assertions a) and b) in the theory Z F . Note that implication a) =+ b) is widely known in analysis. In order to establish the converse implication, equip a nonempty set X with the discrete topology and consider the complete metric space X W . Further, by starting with a given binary relation G on X satisfying

define a certain countable family of dense open subsets of X W and obtain with the aid of this family the desired sequence of elements from X . Let X be an arbitrary set. A binary relation G C X x X is called an equivalence relation on the set X if the following three conditions hold: 1) (x, x) E G for all elements x E X ; 2) (2, y) E G and (y, z) E G imply (8, z) E G; 3) (x,y) E G implies (y,x) E G. If G is an equivalence relation on X , then the pair ( X , G) is called a set equipped with an equivalence relation. In this case, the set X is also called the basic set for the given equivalence relation G.

7

INTRODUCTION

Obviously, if G is an equivalence relation on X , then we have a partition of X canonically associated with G. This partition consists of the sets

where G(x) denotes the section of G corresponding to an element x E X ; in other words, G(x) = {y : (a:,!I) E G I . Conversely, every partition of a set X canonically defines an equivalence relation on X . Let X be an arbitrary set and let G be a binary relation on X . We say that G is a partial order on X if the following three conditions hold: (1) (x, x) E G for each element x E X ; (2) (x, y) E G and (y, z) E G imply (x, z) E G; (3) (x,y) E G and (y,x) E G imply x = y. Suppose that G is a partial order on a set X . As usual, we write

0, the class Pr,-I(E) has already been defined. If n is an odd number, then, by definition, Pr,(E) is the class of all continuous images (in E ) of sets from the class P T , - ~ ( E ) . If n is an even number, then, by definition, Pr,(E) is the class of all complements of the sets from the class P T , - ~ ( E ) . Finally, we put

Pr(E)= u{Pr,(E)

: n

< w).

Sets from the class P r ( E ) are called projective subsets of the space E. The notion of a projective set was introduced by Luzin and, independently, by Sierpiliski. At the present time, there are many remarkable works devoted to the theory of projective sets. Elements of this theory are presented in the monograph by Kuratowski mentioned above (a more detailed discussion of this subject can be found in [97], [91], [55], [56] and [65]). Thus we conclude that Borel subsets of E (i.e, sets from the class Pro(E)) and analytic subsets of E (i.e. sets from the class P r l ( E ) ) are very particular cases of projective sets. Note that many natural problems concerning projective sets cannot be solved in the theory Z F C . For example, the following statements are true: (a) it cannot be proved, in Z F C , that each uncountable set from the class P r a ( R ) contains a subset homeomorphic to the Cantor discontinuum; (b) it cannot be proved, in Z F C , that each set from the class P r 3 ( R ) is measurable in the Lebesgue sense. Note that the statement analogous to (b) and concerning the Baire property of sets from the class P r 3 ( R ) is true, too. Now, let us recall some elementary facts about the Baire property of subsets of general topological spaces. Let E be an arbitrary topological space.

I N T R O D U C TI O N

19

We say that a set X E E is nowhere dense (in E) if int(cl(X)) = 0. For example, if V is an open subset of E, then the set bd(V) is nowhere dense in E. We say that a set X E E is a first category subset of E if X can be represented in the form

where all X, (n E w ) are nowhere dense subsets of E. The family of all first category subsets of E is denoted by the symbol I((E). If E is not a first category space, then K ( E ) is a a-ideal of subsets of E. We say that a set X C E has the Baire property (in E ) if X can be represented in the form X=(UUY)\Z where U is an open subset of E and both Y and Z are first category subsets of E. It is easy to check that a set X E E has the Baire property if and only if X can be represented in the form X = V A P where V is an open subset of E and P is a first category subset of E. The family of all subsets of a space E, having the Baire property (in E ) , is denoted by the symbol Ba(E). Obviously, B a ( E ) is the a-algebra of subsets of E generated by the family T ( E ) U Ii'(E), where T ( E ) is the topology of E (i.e, the family of all open subsets of E). Hence we have the inclusion B(E) E Ba(E). As a rule, this inclusion is proper. But there are some interesting examples of topological spaces E for which this inclusion becomes the equality. For instance, if E is a classical Luzin subset of the real line R, everywhere dense in R, then we have K(E) = [ E ] S ~ , Consequently, in this case, we obtain B ( E ) = Ba(E). Extensive information on Luzin subsets of the real line is contained in [88], [89], [I101 and [117]. We shall deal with Luzin sets in the subsequent sections of our book. At this moment, we wish only t o notice that the existence of Luzin subsets of R cannot be proved in the theory ZFC (on the other hand, the existence of such subsets of R follows easily from the Continuum Hypothesis). Another interesting example (in the theory ZFC)of a topological space E for which the equality B(E) = Ba(E) holds can be obtained if we take the set of all real numbers equipped with the so-called density topology (see, e.g., [117]). We shall consider below some elementary properties of the density topology.

20

CHAPTER

0

Let E be again an arbitrary topological space and let X be a subset of E. We say that X has the Baire property in the restricted sense if, for each subspace Y of El the set X n Y has the Baire property in the space Y. Clearly, the family of all subsets of the space E, having the Baire property in the restricted sense, is a a-algebra of subsets of E. We denote this a-algebra by B a r ( E ) . Obviously, we have the inclusion

It is also easy t o check that B ( E ) E B a r ( E ) . Moreover, it can be shown that A(E) C Bar(E), i.e, all analytic subsets of E have the Baire property in the restricted sense (see [89]). Let X and Y be any two topological spaces and let f be a mapping acting from X into Y. We say that f has the Baire property if, for each Borel subset B of Y, the set f - l ( B ) has the Baire property in X . Evidently, every Borel mapping acting from X into Y has the Baire property. The composition of two mappings, each of which has the Baire property, can be a mapping without the Baire property.

Exercise 6. Give an example of two functions

each of which has the Baire property, but their composition g o f does not possess this property. However, if X , Y, Z are three topological spaces, f : X -+ Y has the Baire property, g : Y -+ Z is a Borel mapping and h : X -+ Z is the composition o f f and g, then h has the Baire property, too. In a similar way we can define a mapping with the Baire property in the restricted sense. Namely, we say that f : X -+ Y has the Baire property in the restricted sense if, for each Borel subset B of Y, the set f - l ( B ) is a subset of X having the Baire property in the restricted sense. All Borel mappings have the Baire property in the restricted sense. Let X and Y be any two topological spaces and let

be a set-valued mapping. We say that F has closed graph if

21

INTRODUCTION

is a closed subset of the product space X x Y. It is clear that if the given set-valued mapping F has closed graph, then, for each element x E X I the set F ( x ) is a closed subset of the' space Y. The converse assertion is not true in general. Set-valued mappings with closed graphs are important in different domains of mathematics, especially, in those questions which concern the existence of fixed points of set-valued mappings (we recall that a point x E dom(F) is a fixed point for a set-valued mapping F if x E F ( x ) ) . Note that theorems on the existence of fixed points for set-valued mappings found many interesting applications (see, e.g., [38] and [153]). Let X and Y be again two topological spaces and let

be a set-valued mapping. We say that F is lower semicontinuous if (1) for each point x E X, the set F ( x ) is closed in Y; (2) for each open subset V of Y, the set

is open in X . There are certain relationships between set-valued mappings with closed graphs and lower semicontinuous set-valued mappings (cf., for example, Theorem 1 below).

Exercise 7. Let X be a topological space and let

be a real-valued function on X. We recall that f is lower (respectively, upper) semicontinuous if, for any t E R, the set {x E X : f(x) > t ) (respectively, the set {x E X : f ( x ) < t)) is open in X . Show that: a) f is lower semicontinuous if and only if - f is upper semicontinuous; b) f is lower semicontinuous if and only if the set

is closed in the product space X x R; c) f is lower semicontinuous if and only if, for each xo E X , we have the inequality liminf f (x) 1 f (go);

.+,,

d) f is continuous if and only if it is lower and upper semicontinuous.

CHAPTER

Exercise 8. Let

0

X be a set and let

be a real-valued function on

X .We introduce two set-valued mappings

by the following formulas:

Suppose now that X is a topological space. Show that the following two assertions are equivalent: a) f is lower semicontinuous (as an ordinary function); b) F 1 , j is lower semicontinuous (as a set-valued function). Show that the next two assertions are also equivalent: c) f is upper semicontinuous (as an ordinary function); d) F 2 , j is lower semicontinuous (as a set-valued function). Exercise 9. Let

X be a nonempty quasicompact topological space and

let

f : X+R be a ldwer semicontinuous function. Show that there exists a point go E X satisfying the relation f (xo) = i n S c ~ x(.I.f Formulate and prove an analogous result for upper semicontinuous realvalued functions defined on X. Exercise 10. Let

X and Y be any two topological spaces and let

be any two lower (respectively, upper) semicontinuous functions such that

Define a function h : XxY-,R

by the formula

23

INT R O D U C TIO N

Show that h is also lower (respectively, upper) semicontinuoue. Exercise 11. Let X be a completely regular topological space and let

be a lower semicontinuous function such that f (x) 2 0 for all x E X . Show ~ functions from X into R,satisfying the that there exists a family ( f i ) $ €of following conditions: 1) for each i E I, the function fi ie continuous; 2) for all i E I and for all x t! X , we have fi(x) 0; 3) f = supi€rfi; 4) card(I) 5 w(X) w where w(X) denotes the topological weight of X (i.e. w(X) is the smallest cardinality of a base of X ) . In particular, if X has a countable base, then f can be represented as a pointwise limit of an increasing sequence of positive continuoue real-valued functions defined on X .

>

+

Exercise 12. Let [a, b] be a closed subinterval of R and let

>

be two functions such that f g. Suppose, in addition, that f is lower semicontinuous and g is upper semicontinuoue. Demonstrate that there exists a continuoue function

satisfying the inequalities g 5 h S f This simple result admits a number of generalizations and, actually, is a direct consequence of the well-known Michael theorem on continuous selectors (see [105], [I061 or [122]). Exercise 13. Let X be a second category topological space and let {fi : i E I) be a family of real-valued lower semicontinuoue functions on X . Suppose, in addition, that, for each point x E X , the eet {fi(x) : i E I) is bounded from above. Show that there exists a nonempty open set V E X for which the set u{fi(V) : i E I) is bounded from above, too. Formulate and prove an analogous result for upper semicontinuous functions.

24

CHAPTER

0

Exercise 14. Let (GI+) be a second category topological group and let {fi : i E I} be a family of real-valued lower semicontinuous functions on G. Suppose that the following conditions hold: a) for each index i E I, the function fa is subadditive, i.e, we have

b) for each point x E G, the set {fi(x) : i E I ) is bounded from above. Show that the given family { f i : i E I ) is locally bounded from above. This means that, for any point x E G, there exists a neighbourhood V(x) for which the set u{fi(V(x)) : i E I ) is bounded from above. Formulate and prove an analogous statement for upper semicontinuous functions. Notice that the result presented in Exercise 14 easily implies the wellknown Banach-Steinhaus theorem (see [8] or [63]). For our further considerations, we need one auxiliary proposition on closed projections. This proposition is due to Kuratowski (see, for instance, [89]) and has numerous applications. Lemma 1. Let X be a topological space, let Y be a quasicompact space and let prl denote the canonical projection from X x Y into X , i.e. the mapping prl : X x Y - + X

is defined b y the formula

Then prl is a closed mapping, i.e. for each closed subset A of X x Y , the image prl(A) is closed in X . Proof. Take any point x E X such that U n prl(A) # 0 for all neighbourhoods U of x. We are going to show that x € prl(A). For this purpose, it is sufficient to establish that

Suppose otherwise, i.e. ({x} x Y ) f l A = 0. Then, for each point y E Y , there exists an open neighbourhood W((x, y)) of the point (x, y), satisfying the relation w ( ( x l Y)) n = 0.

e

INTRODUCTION

We may assume, without loss of generality, that

where U(x) is an open neighbourhood of x and V(y) is an open neighbourhood of y. Since the space {x} x Y is quasicompact, there exists a finite sequence ( x , Y ~ )(x,Yz), , .,* , (x, ~ t a ) of points from {x} x Y, such that {W((x, ys)) : 1 i n} is a finite covering of {x) x Y. Now, let us put

<
0, consider the set

< 1. For every

I(f (x + 6) - f (x))lsl In)I. It is not hard to check that a h , , is a closed subset of the space C[O, 11. Indeed, let us put

l(f (x -t 6) - f (2))/61 5 741. Then

Zh,n

is a closed subset of the product space C[O,11 x [0, 11 and

where pr1 : C[O, 11 x [0, 11 -+ C[O, 11

denotes the canonical projection onto C[O,11, Taking account of the compactness of the unit'segment [O, 11 and applying Lemma 1, we immediately obtain that the set Q h l n is closed in C[O,11. Simultaneously, Q h l n is nowhere dense in C[O, 11 (the latter fact is almost trivial from the geometrical point of view). Consequently, the set

is of first category in C[O, 11. Now, it is clear that any function belonging to the set C[O,11 \ D is nowhere differentiable on [ O , l ] . This completes the proof of Theorem 2. Exercise 18. Let f be a function acting from R into R and let x be a point of R.Recall that f possesses a symmetric derivative at the point x if there exists a (finite) limit

INTRODUCTION

29

In such a case, this limit is called the symmetric derivative off at x (denoted by the symbol f:(x)). Demonstrate that f can possess a symmetric derivative a t a point x being even discontinuous at this point. Show that if f is differentiable (in the usual sense) at a point x, then there exists a symmetric derivative fL(x) and the equality

is fulfilled. Show also that the converse assertion is not true in general. Does an analogue of Theorem 2 hold for the symmetric derivative (instead of the derivative in the usual sense)? The notion of a symmetric derivative of a function can be regarded as a simple example of the concept of a generalized derivative. In the subsequent sections of the book we shall discuss some other types of a generalized derivative. The notion of an approximate derivative (introduced by Khinchin in 1914) is of special interest and will be defined and discussed in Chapter 4. It is well known that this notion plays an important role in various questions of real analysis (for instance, in the theory of generalized integrals). The definition of an approximate derivative relies on the concept of a density point for a given Lebesgue measurable subset of R. Let X denote the standard Lebesgue measure on R and let X be an arbitrary X-measurable subset of R. We say that x E R is a density point for X if The classical theorem of real analysis, due to Lebesgue, states that almost all (with respect to A) points of X are its density points. In order to establish this fact, we need the concept of Vitali covering of a set lying in R , and the important result of Vitali concerning such coverings. For the sake of completeness, we formulate and prove this result. Let {Di : i E I) be a family of nondegenerate segments on R and let Z be a subset of R. We say that this family is a Vitali covering of Z if, for each point r E Z , we have

in f {X(Di) : i E I, r E Dd) = 0. The following fundamental result was obtained by Vitali (cf., e.g., [ l l l ] , [I171 or [127]).

Theorem 3. If Z is a subset of R and {Da : i E I ) is a Vitali covering of Z , then there exists a couniable set J C I such that ihe padial family {Dj : j E J) is disjoint and X ( z \ u { D j : j E J ) ) = 0.

CHAPTER 0

30

Proof. Without loss of generality we may assume that Z is bounded. Let U be an open bounded set in R containing Z . We may also assume that Di C U for each index i E I. Define by recursion a disjoint countable subfamily of segments

Take Dqo) arbitrarily. Suppose that Dqo), Di(l), been defined. Put

t ( k ) = sup{X(Di) : Di C U \ (Da(o)U

... , u u -

Dqlc) have already

U Di(k))).

Let Di(k+l) be a segment from {Di : i E I) such that

In this way we obtain the desired disjoint sequence {Di(k) : k E N ) . Note that

so we have, in particular,

We are going to show that

is the required subfamily. For this purpose, denote by Dl(k) the segment in R whose centre coincides with the centre of Dqk) and for which

Let us demonstrate that, for each natural number n, the inclusion Z \ u { D ~ ( ~: ) k E N ) 5 u{D:(~) : k E N , k

> n}

holds true. Indeed, let z be an arbitrary point from Z\U{Dd(k) : k E N}. Then, in particular,

Since {Da : i E I } is a Vitali covering of Z , there exists a segment Di for which E Di, Di fl (Di(0) U --.U Dd(n))= 0,

31

INTRODUCTION

Obviously, we have A(Di) > 0. At the same time, as mentioned above, limk-++ooA(Di(k)) = 0. So, for some natural numbers k, we must have

Let k be the smallest natural number with this property. Evidently, k Thus we get

> n.

In addition, V D i ) S 2X(Di(k+1))1 which immediately implies (in view of the definition of Dj(k+l)) the inclusion Di E D5(k+,). Consequently,

Finally, since, for each natural n , we have

we conclude that

A(Z \ U{Dj : j E J ) ) = 0,

and the theorem is proved. We shall present some standard applications of Theorem 3 in the subsequent sections of the book. Here we only recall how the above-mentioned Lebesgue result on density points of A-measurable sets can be easily derived from Theorem 3.

Theorem 4. Let X be an arbitrary A-measurable set on R and lei d(X) = {x E R : x is a density point of X}.

Then we have

A(X \ (X

n d ( x ) ) ) = 0.

Proof. We may assume, without loss of generality, that X is bounded. For any natural number n > 0, let us define

32

CHAPTER

0

Clearly, it suffices to show that A*(Xn) = 0 where A* denotes the outer measure associated with A. For this purpose, fix E > 0. Let U be an open subset of R such that

In virtue of the definition of X, , there exists a Vitali covering {Di : i E I } of X, such that (Vi E I)(A(Di n X)/X(Di)

< 1 - lln).

Obviously, we may suppose that Di 5 U for each index i E I. According to Theorem 3, there exists a disjoint countable subfamily {Dj : j E J) of this covering, for which we have A(X,

\ U{Dj

: jE

J}) = 0.

Then we can write

5 (1 - l/n)A(U) Since

E

< (1 - l/n)(X*(X,) + E ) .

> 0 was taken arbitrarily, we have

Finally, in view of the inequality A*(X,) 0, and the theorem is proved.

< +co,we conclude that A*(Xn) =

Exercise 19. For any A-measurable set X C R, show that the set d(X) is Bore1 in R. Exercise 20. Demonstrate that if z E R is a density point of two A-measurable sets X and Y , then z is a density point of the set Z = X n Y . This fact is important for introducing the so-called density topology on R which will be discussed in our further considerations.

1. Cantor and Peano type functions

It is well known that one of the first mathematical results of Cantor (which turned out to be rather surprising to him) was the discovery of the existence of a bijection between the set R of all real numbers and the corresponding product set R 2 = R x R (i.e, the Euclidean plane). For a time, Cantor did not believe that such a bijection exists and even wrote to Dedekind about his doubts in this connection. Of course, Cantor already knew of the existence of a bijection between the set N of all natural numbers and the product set N x N. A simple way to construct such a bijection is the following one. First, we observe that a function

defined by the formula

is a bijection between N and the set of all strictly positive natural numbers. Then, for each natural n > 0, we have a unique representation of n in the form n = 2k (21 + 1) where k and I are some natural numbers. Now, define a function

by the formula g(n) = (k,1 ) (n E N \ { O H . One can immediately check that g is a bijection, which also yields the corresponding bijection between N and N x N. By starting with the latter bijection, it is not hard to establish a one-toone correspondence between the real line and the Euclidean plane (respectively, between the unit segment [0, 11 and the unit square [O,112). Indeed, a simple argument (in ZF) shows that the sets

'

34

CHAPTER 1

are equivalent, i.e, there exists a bijective mapping from each of them to any other one. So we only have to check that the sets

are equivalent, too. But this is obvious since the product set 2N x 2N is equivalent with the set 2NxN and the latter set is equivalent with 2N because of the existence of a bijection between N and N x N . Keeping in mind these simple constructions, it is reasonable to introduce the following definition. We say that a mapping f acting from R into R 2 (respectively, from [O, 11 into [O, 112) is a Cantor type function if f is a bijection. As mentioned above, Cantor type functions do exist.

Remark 1. As pointed out earlier, one-to-one correspondences between N and N x N (respectively, between R and R x R or between [0, 11 and [O, 112) can be constructed effectively, i.e, without the aid of the Axiom of Choice. In this connection, let us recall that, for an arbitrary infinite set X , we also have a bijection between X and X x X , but the existence of such a bijection needs the whole power of the Axiom of Choice. More precisely, according to the classical result of Tarski (cf. [91]), the following two assertions are equivalent in the theory ZF: 1) the Axiom of Choice; 2) for any infinite set X , there exists a bijection from X onto X x X. Exercise 1. Let X be an arbitrary set. Show, in the theory Z F , that there exists a well ordered set Y such that there is no injection from Y into X . We may suppose, without loss of generality, that X n Y = 0. Demonstrate (in the same theory) that if card(X x Y) 5 card(X U Y), then there exists an injection from X into Y and, consequently, X can be well ordered. Show also (in ZF) that the relation

implies the inequality card(X x Y)

< card(X U Y).

CANTOR A N D PEANO T Y P E FUNCTIONS

35

Deduce from these results that, in ZF, the following two assertions are equivalent: 1) the Axiom of Choice; 2) for any infinite set X , the equality

is satisfied. Now, let f be an arbitrary Cantor type function acting, for example, from R onto R2. It is well known that, in such a case, f cannot be continuous. Indeed, suppose for a moment that f is continuous. Then we may write R2= U{f ([-n, n]) : n E N ) where each set f([-n,n]) (n E N ) is compact (hence closed) in R2.In accordance with the classical Baire theorem, at least one of these sets has a nonempty interior. Let k be a natural number such that

Then we have a bijective continuous mapping

which obviously is a homeomorphism between [-k, k] and f ([-k, k]). But this is impossible since [-k, k] is a one-dimensional. space and f([-k, k]) is a two-dimensional one. If we want to avoid an argument based on the notion of a dimension of a top.ologica1space (and it is reasonable to avoid here such an argument because we do not; discuss this important notion in our book), we can argue in the following manner. Consider the function

which also is a homeomorphism. Let L denote any circle contained in the set f ([-k, k]), i.e. let L be a subset of f ([-k, k]) isometric to

where r is some strictly positive real (the existence of L is evident since f([-k, k]) has a nonempty interior). We thus see that the function

36

CHAPTER 1

is injective and continuous. This immediately yields a contradiction since there is no injective continuous function acting from a circle into the real line (cf, the next exercise).

Exercise 2. Let L be a circle on the plane and let g : L --,R be a continuous mapping. By using the classical Cauchy theorem on intermediate values for continuous functions, prove that there exist two points z E L and z' E L satisfying the relations: a) g(z) = g(z'); b) z and z' are antipodal in L, i.e, the linear segment [z, z'] is a diameter of L. In particular, g cannot be an injection. This simple result admits an important generalization to the case of an n-dimensional sphere (instead of L) and of an n-dimensional Euclidean space (instead of R).The corresponding statement is known as the BorsukUlam theorem on antipodes and plays an essential role in algebraic topology (see, for example, [90]). In particular, this theorem shows that there are no injective continuous mappings from the sphere Sn into the space R n. The following statement is also of some interest in connection with Cantor type functions (see, e.g., [134]).

Theorem 1. Let f be a function from R 2 anto R continuous with respect to each of the variables x and y (se parately). Then f as not an injection. Proof. Suppose otherwise, i.e, that our f is injective. Denote

Then, according to the assumption of the theorem, tion from R into R.Let us put

+ is a continuous func-

Since f is injective, we have a # b. Consequently, either a < b or b < a. We may assume, without loss of generality, that a < b. The function $, being continuous on the segment [ O , l ] , takes all values from the segment

In particular, there exists at least one point xo E ]O,l[ such that

CANTOR AND PEANO T Y P E FUNCTIONS

Further, let us define

Then $ is a continuous function, too, and

Hence we have the inequalities

which imply the existence of a neighbourhood U(0) of the point 0, such that (VY E U(O))(a < +(Y)< b) or, equivalently, (VY E U(O))(a

< f(.o,Y) < b).

Of course, we may additionally suppose that U(0) is contained in ] - 1, I[. Thus, on the one hand, we have the inclusion

On the other hand,

so, for some reals yo

# 0 and

XI,

we get

which contradicts the injectivity of f . The contradiction obtained finishes the proof of Theorem 1. E x e r c i s e 3. Does there exist an injective mapping

such that f is continuous with respect t o one of the variables x and y? E x e r c i s e 4, Show that there exists a bijection

f

: [O, 11 -, [o, 112

CHAPTER

such that the function

prl o

1

f is continuous, where

denotes, as usual, the first canonical projection from [O, 112 onto [ O , l ] . We thus see that Cantor type functions cannot be continuous. In this connection, it is reasonable to ask whether there exist continuous surjections from R onto R 2 or from [0, 11 onto [O, 112. It turned out that such surjections do exist and the first example of the corresponding function acting from [O, 11 onto [O, 112 was constructed by Peano. Hence the following definition seems to be natural. Let

We shall say that f is a Peano type function iff is continuous and surjective. In order to demonstrate the existence of Peano type functions, we recall the classical Cantor construction of his famous discontinuum. Take the unit segment [O,1]on the real line R . The first step of Cantor's construction is to remove from this segment the open interval ]1/3,2/3[ whose centre coincides with the centre of [ O , l ] and whose length is equal to the one-third of the length of our segment. After this step we obtain the two segments without common points. Then we apply the same operation to each of these two segments, etc. After w-many steps we come to the subset C of [0, 11 which is called the Cantor discontinuum (or the Cantor space). The set C is closed (since we removed open intervals from [0, 11) and, in addition, C is perfect because the removed intervals are disjoint and pairwise have no common end-points. Moreover, since the sum of lengths of the removed intervals is equal to 1 (which can easily be checked), we infer that C is nowhere dense in R and its Lebesgue measure equals zero. Consequently, C is a small subset of R from the point of view of the Baire category and from the point of view of the standard Lebesgue measure A on R. The geometric construction of C described above and due to Cantor himself is rather visual but, sometimes, other constructions and characterizations of C are needed in order to formulate the corresponding results in a more general form. We present some of such constructions and characterizations of Cantor's discontinuum in the next two exercises.

Exercise 5. Take the two-element set 2 = {0,1) and equip this set with the discrete topology. Equip also the Cartesian product 2" with the product topology. Demonstrate that 2W is homeomorphic to the classical Cantor discontinuum C.

CANTOR A N D P E A N O T Y P E FUNCTIONS

39

Exercise 6. Let E be a topological space. Show that E is homeomorphic to C if and only if the conjunction of the following four relations holds: a) E is nonempty and compact; b) E has a countable base; c) there are no isolated points in E; d) E is zero-dimensional, i.e. for each point e E E and for any neighbourhood U(e) of el there exists a neighbourhood V(e) of e such that

where the symbol bd(V(e)) denotes the boundary of V(e). Actually, the last relation means that the family of all clopen subsets of E forms a base for E. The abstract characterization of the Cantor space, given in Exercise 6, implies many useful consequences. For instance, by using this characterization, it is not difficult to show that, for each natural number k 2 2, the product space k W is homeomorphic to the Cantor discontinuum (of course, here k is equipped with the discrete topology). Exercise 7. Demonstrate that k W is homeomorphic to C. Verify also that w W is homeomorphic to the space of all irrational numbers (where w is equipped with the discrete topology). Naturally, the Cantor discontinuum ha.s numerous applications in various branches of mathematics (especially, in topology and analysis). The next exercise presents a typical application of C in real analysis. Exercise 8. Construct a set C1on [O,1] such that: 1) C' is homeomorphic to C; 2) the Lebesgue measure of C1is strictly positive. Deduce from relations 1) and 2) that the measure A is not quasiinvariant with respect to the group of all homeomorphisms of R, i.e. this group does not preserve the u-ideal of all A-measure zero sets. It immediately follows from the construction of C that, for each clopen set X g C and for any E > 0, there exists a finite partition of X consisting of clopen subsets of X , each of which has diameter strictly less than E . It is also easy to check that every zero-dimensional compact metric space possesses an analogous property. At the same time, if E is an arbitrary

40

CHAPTER 1

compact metric space, then, for each closed set X E E and for any E > 0, there exists a finite covering of X consisting of closed subsets of X , each of which has diameter strictly less than E . These simple observations lead to the following important statement due to Alexandrov.

Theorem 2. Lei E be an arbitrary nonempty compact metric space. Then there exasts a continuous surjection from the Cantor space C onto E. Proof. Taking account of the preceding remarks, we can recursively define two sequences

satisfying the conditions: 1) for any n E w, the finite family

is a partition of C consisting of clopen subsets of C each of which has diameter strictly less than l / ( n 1); 2) for any n E w, the finite family

+

is a covering of E by nonempty closed sets each of which has diameter strictly less than l / ( n 1); 3) for any n E w, the family (Xn+l,k)llklm(n+l) (respectively, the family (Yn+l,k)lSklm(n+l)) is inscribed in the family (Xn,k)llk 0 was chosen arbitrarily, we conclude that A*(f (XI) qA*(X),


0, there exists a lower semicontinuous function f on [a, b], such that g 5 f and

Deduce from this fact that there exists a sequence { f n : n E N ) of Lebesgue integrable lower semicontinuous functions, such that: a ) fn+l S f n for any n E N; b) g 5 f n for any n E N ; C ) limn,+, fn(z) = g(z) for almost all points z E [a, b]. In particular, we may write

almost everywhere on [a, b]. Observe that

Putting F(x) =

l X ( f 0-

g)(t)dt

(x E la,b1)1

and applying Theorem 3 again, demonstrate that

for almost all z E [a, b]. Finally, prove the Lebesgue theorem stating t h a t if h is an arbitrary real-valued Lebesgue integrable function on [a, b], then

for almost all z E [a, b] T h e exercise presented above shows us that the classical Lebesgue theorem concerning the differentiation of a function H, where

66

CHAPTER

2

can be logically deduced from Theorem 3. However, this approach has a weak side because it does not yield the description of the set of those points z E [a, b] at which H'(x) = h(x). We now turn our attention t o the construction of a strictly increasing function whose derivative vanishes almost everywhere. Such a construction is essentially based on Theorem 3. Let us recall that the first step of the construction of the Cantor set on R is that we remove from the segment [0, 11 the open interval ]1/3,2/3[. Let us define f(x)=O (~50)l

Now, suppose that on the n-th step of the construction we have already defined the function f for all those points which belong t o the union of the removed (at this and earlier steps) intervals. Obviously, we obtain a finite family {[a,, bi] : 1 5 i 5 m) of pairwise disjoint segments on [O, 11. It is easy t o check that m = 2", but we do not need this fact for our further purposes. Pick any segment [ad, bi] from the above-mentioned family. Taking into account the inductive assumption, we may put

for all points x E](2ai+ bi)/3, (2bi +aa)/3[. So we have defined our function f for all points belonging t o the union of all intervals removed at the ( n + l ) t h step. Continuing the process in this way, we will be able t o construct f on the set R \ C, where C denotes the Cantor set. From the definition of f immediately follows that f is increasing and continuous on its domain. Moreover, it is easily seen that f can be uniquely extended t o an increasing continuous function f : R+[O,l]. Since f is constant on each removed interval, we obviously have fl(x) = 0

(x E R \ C),

i.e, the derivative of f vanishes almost everywhere on R. Thus, we have shown that there exists a non-constant increasing bounded continuous function f from R into R, whose derivative is zero almost everywhere. Now, let p and q be any two points of R such that p < q. Since f is not constant, there are some points x and y from R such that f (x) < f (y).

67

SINGULAR MONOTONE FUNCTIONS

Evidently, x < y and there exists a homothety (or translation) h of the plane R=,for which

Let f+ denote the function from R into R, whose graph coincides with t h e image of the graph o f f with respect t o h. Then we may assert that f * is also a n increasing bounded continuous function, whose derivative vanishes almost everywhere, and f * ( p ) < f*(q). In virtue of the remarks made above, we can formulate and prove the following classical result concerning the existence of strictly increasing continuous singular functions.

Theorem 4. There exasts a function g :

R- R

satisfying these three conditions: 1) g is continuous and strictly increasing; 8) (Vz E R)(O I g(z) _< 1); 3) the derivative of g as zero almost everywhere on R.

Proof. Let {(p,,q,) : n E N ) denote the countable family of all pairs of rational numbers, such that p, < q,. According t o the argument presented above, for each natural index n, there is a function

such that: a) g, is continuous and increasing; b) 0 g,(x) _< 1/2"'t1 for all x E R; c) the inequality g,(p,) < g, (9,) holds true; d) t h e derivative of g, vanishes almost everywhere. It follows from b) t h a t the series CnEN gn is uniformly convergent. So we may consider the function


S, then ( r - s)/(r2 - s 2 ) < 2/r;

2) if r > 1 and s > 1, then

CHAPTER

3

Proof. Indeed, we have

Thus 1 ) is true. Further, it can easily be checked that the inequality of 2) is equivalent t o the inequality

which, obviously, is true under our assumptions r completes the proof of Lemma 1.

Lemma 2. Let

> 1 and s > 1. This

4 be ihe function fiom R into R defined by

and let a and b be a n y two distinct real numbers. T h e n we h a v e

Proof. Without loss of generality we may assume that a < 6. Only three cases are possible. 1. 0 5 a < b. In this case, taking into account 1 ) of Lemma 1 , we can write ( l / ( b - a ) ) J b Q ( x ) d x = 2 ( ( 1 + b)li2 - ( 1 a

+ a ) ' 1 2 ) / ( ( l + b) - ( I + a ) )

2. a < b _< 0. This case can be reduced to the previous one, because of the evenness of our function q5. 3 , a < 0 < b. In this case, taking into account 2 ) of Lemma 1 , we can write ( l / ( b- a ) )

q5(x)dX = 2 ( ( 1 + b)'I2 a

+ ( 1 - a)'12 - 2 ) / ( ( 1 + 6 ) + ( 1 - a ) - 2 )

< 4 m i n ( 4 ( a ) ,4 ( b ) ) . This ends the proof of Lemma 2.

EVERYWHERE DIFFERENTIABLE NOWHERE MONOTONE FUNCTIONS

71

Lemma 3. Let n > 0 be a natural number and let $ : R -+ R be any function of the form

where $ is the function from Lemma 2,

are strictly positive real numbers, and

tl,

..., in are

real numbers. Then

for all de'stinct reals a and b.

Proof. The assertion follows immediately from Lemma 2, by taking into account the fact that

for any d > 0 and t E R.

Lemma 4. Let ( $ J , ) , be ~ ~a sequence offunctions as in Lemma 3. For any z E R and for each n 2 1, let U S define

and suppose that, for some a E R, the series Denote z $ . ( a ) = s < +m.

-

$,(a) is convergent.

tall

Then we have: I ) the seraes F ( z ) = - Q n ( z ) converges uniformly on every bounded subinterval of R; 2) the functa'on F is diflerentiable at a and

In particular, af

C a(4= f ( 4
0 be a natural numbep; let 11,..., In be pairwise disjoint nondegenemte segments on R and let tk denote the midpoint of Ik for each natural k E [l,n]. Fix any strictly positive real numbers

EVERYWHERE. DIFFERENTIABLE NOWHERE MONOTONE FUNCTIONS

73

Then there exists Q function $ as in Lemma 3, such that, for all natural k E [I, n], the following relations are fulfilled: 1) $(tk) > Y k ;

2) (Vz E I k ) ( $ b ) < Y k +&I; 3) (Vz € R \ (11 U U In))($(z)< 6)-

Proof. Let us denote

Then, for each natural k E [ I , n], define

where dk > 0 is chosen so large that

Finally, put

$=$I+

+$nu

Then, taking into account the fact that

it is easy to check that the function $ satisfies relations 1)

- 3).

Lemma 6. Let any two disjoint countable subsets

of R be ge'ven. Then there exasts

Q

function

such that: 1) F is dzffereniiable everywhere on R ; 2) 0 c FFl(z)5 1 for ail x E R; 3) F f ( t k )= 1 for each k E N \ ( 0 ) ; 4) F'(rk) c 1 for each k E N \ ( 0 ) .

Proof. We are going t o construct by recursion the sequence ($n)nl 1 of functions as in Lemma 3 with some additional properties. Namely, denoting

CHAPTER 3

74

we wish the following conditions t o be satisfied: ( 1 ) for any natural n 2 1 and for all natural k E [ I , n ] ,we have

(2) for any natural n 2 1 and for each x E R , we have

(3) for any natural n 2 1 and for all natural k E [ I , n],we have

Ili First choose a nondegenerate segment II with midpoint t l , such t h a t rl $ and apply Lemma 5 with

Evidently, we obtain and f l = such t h a t relations ( I ) , ( 2 ) and (3) are fulfilled for n = 1. Suppose now t h a t , for a natural n > 1 , we have already defined t h e functions $1, 1 $kt i4n-1

--

---

satisfying the corresponding analogues of (1) - (3) for n - 1. Pick disjoint nondegenerate segments I l l ... , In in such a way that: (a) tk is the midpoint of Ik for each natural k E [ I , n]; (b) Ik n { r l , ... , r n ) = 0 for each natural k E [ I , n]; (c) for any natural k E [ I , n] and for any x E I k , we have t h e inequality

where

+

6 = l/(n(n 1))

-

1/(2n - 2n).

We now can apply Lemma 5 with

and ylc = 1 - ( l / n )- fn-~(tlc) ( 1

I k 5 n).

Applying th above-mentioned lemma, we get the function $,. Clearly,

EVERYWHERE DIFFERENTIABLE NOWHERE MONOTONE FUNCTIONS

75

for all natural k E [I, n], so relation (3) holds true. Further, for any natural k E [I, n], we have

which shows that relation (1) holds true, too. Finally, in order to verify (2), fix any point x E R. If, for some natural k E: [I, n], the point x belongs to Ik,then we may write

1- l/n

+ l/(n(n + 1)) = 1 - l / ( n + 1)-

If r: does not belong to I1U

...

U In,then

Thus, relation (2) holds true, too. Proceeding in this manner, we are able to construct the required sequence ($n)nL1. Putting

and

we obtain a function F : R

-+

R.In view of Lemma 4, we also get

F1(x) = f (x)

(x E R).

Further, the definition of F immediately implies 0 < F1(x) _< 1

(x t. R),

Now, fix a natural k 2 1 and let n be a natural number strictly greater than k. Then Fi(rk) = fn-l(ri) $m(rk)
0)

> 0),

any two disjoint countable dense subsets of R. Using the result of t h e previous lemma, take any two everywhere differentiable functions

satisfying the relations: (a) 0 < F'(x) 1 and 0 < G1(x) 1 for all x E R; (b) F1(tfl) = 1 and F 1 (r n ) < 1 for each natural n 2 1; (c) G 1 (r n ) = 1 and G 1 (t n ) < 1 for each natural n 1. Now, define H=F-G.


0) and {r, : n E N , n > 0 ) are dense in R, we infer t h a t H cannot be monotone on any subinterval of R. Also, the relation 1 -1 < H (x) < 1 (x E R ) implies that H' is bounded, and the theorem has thus been proved. In fact, the preceding argument establishes the existence of many functions f : R+R which are everywhere differentiable, nowhere monotone and such t h a t f' is bounded. Let us mention some other interesting properties of any such a function f.

EVERYWHERE DIFFERENTIABLE NOWHERE MONOTONE FUNCTIONS

77

1. f has a point of a local maximum and a point of a local minimum in every nonempty open subinterval of R. Actually, for each nondegenerate segment [a, b] C R, we can find some points t l and t z satisfying the relations

Let us denote M = suPt,[t,,t,] f ( t ) . Then, for some T E [tl,tz],we must have f ( r ) = M , and it is clear that T must be in the interior of [tl,tz]. In other words, T is a point of a local maximum for f . Applying a similar argument, we can also find a point of a local minimum of f on the same nondegenerate segment [a, b].

.

2. Since f' is bounded, the function f satisfies the so-called Lipschitz condition, i.e, for some constant d 2 0, we have

Note that, in the latter relation, we may put

In particular, f is absolutely continuous. This also implies that f ' is Lebesgue integrable on each bounded subinterval of R.

3. The function f 1is not integrable in the Riemann sense on any nondegenerate segment [a, b] C R. To see this, suppose otherwise, i.e. suppose that f' is Riemann integrable on [a, b]. Then, according to a well-known theorem of mathematical analysis, f' must be continuous a t almost all (with respect t o the standard Lebesgue measure) points of [a, b] (see, e.g., [Ill]). Taking into account the fact that f ' changes its sign on each nonempty open subinterval of R , we infer that f ' must be zero at almost all points of [a, b]. Consequently, f must be constant on [a, b], which is impossible. The contradiction obtained yields the desired result. 4. Being a derivative, the function f' belongs to the first Baire class, i.e. it can be represented as a pointwise limit of a sequence of continuous functions. Hence, in virtue of the classical Baire theorem (see, e.g., [ l l I]), the set of all those points of R a t which f ' is continuous is residual (comeager), i.e, is the complement of a first category subset of R.

5 . Let us denote

CHAPTER 3

78

The sets X and Y are disjoint, Lebesgue measurable and have the property that, for each nondegenerate segment [a, b] C R, the relations X(X

n [a, b]) > 0,

X(Y

n [a, b]) > 0

are fulfilled (where X denotes, as usual, the Lebesgue measure on R ) . In order to demonstrate this fact, suppose, for example, that X(Y n [a, b]) = 0. Then we get f t ( t ) _> 0 for almost all points t E [a, b]. But this immediately implies that our function f , being of the form

is increasing on [a, b], which contradicts the definition o f f . E x e r c i s e 1. Give a direct construction of two disjoint Lebesgue measurable subsets X and Y of R, such that, for any nonempty open interval I E R,the inequalities

hold true. More generally, show that there exists a countable partition {X,, : n < w ) of R consisting of Lebesgue measurable sets and such that, for any nonempty open subinterval I of R and for any natural number n , the relation X(I n x,) > o is fulfilled. E x e r c i s e 2. Denote by E the family of all Lebesgue measurable subsets of the unit segment [0, I]. For any two sets X E E and Y E E, put

Identifying all those X and Y , for which d ( X ,Y ) ' = 0, we come t o the metric space ( E l d). Check that ( E , d ) is complete and separable, i.e, is a Polish space. Further, let E' be a subset of E consisting of all sets X E E such that X(X n I ) > 0, X(([O, 11 \ X ) n I ) > o

EVERYWHERE DIFFERENTIABLE NOWHERE MONOTONE FUNCTIONS

79

for each nondegenerate subinterval I of [O,l]. Show that E' is the complement of a first category subset of E . Hence, according to the Baire Category Theorem, E' # 0. Now, we are going t o present an essential generalization of Theorem 1 due t o Weil (see [165]). Namely, Weil gave a proof of the existence of everywhere differentiable nowhere monotone functions (with a bounded derivative) by using the Baire Category Theorem. We shall say that a function

is a derivative if there exists a t least one everywhere differentiable function

satisfying the relation (Vx E R)(F1(x) = f (x)). Let us consider the set D = {f : f i s a derivative a n d f i s bounded} Obviously, D is a vector space over d defined by the formula

R.We may equip this set with a metric

T h e latter metric produces the topology of uniform convergence. In view of the well-known theorem of analysis, a uniform limit of a sequence of bounded derivatives is a bounded derivative. This shows, in particular, that the pair ( D , d ) is a Banach space (it can easily be seen that it is nonseparable). Take any function f E D and consider the set f-l(O). We assert that this set is a Ga-subset of R. Indeed, we may write

where F : R -+ R is such that F' = f . This formula yields a t once the desired result. Let us put Do = {f E D : f-l(0) i s dense in R). We need the following simple fact.

80

C HAPT E R

3

L e m m a 7. The set Do is a closed vector svbspace of the space D. Consequenlly, Do as a Banach space, as well. P r o o f . First, we show t h a t D ois closed in D. Let { f k : k < w ) be a sequence of functions from Do,converging (in metric d) t o some function f E D. We put (k < w ) . z k = fi1(0) Then all the sets 21,are dense G6-subsets of R. Therefore, the set

is a dense Ga-subset of R, too. Obviously,

Thus we obtain f E Do. Now, let us demonstrate that Do is a vector subspace of D. Clearly, if f E Do and t E R, then t f E Do.Further, take any two functions g E Do and h E D oand consider the sets

Then the set Zg n Zh is a dense Ga-subset of R, and it is evident that

where

+

Zgth = ( g h)-'(o), This shows that Dois a vector space. Lemma 7 has thus been proved. Notice now that the space Dois nontrivial, i.e, contains nonzero functions. For instance, this fact follows directly from Theorem 1. But it can also easily be proved by another argument. E x e r c i s e 3. Give a direct proof (i.e, without the aid of Theorem 1) that Do contains nonzero elements.

Theorem 2. Let us denote

E = {f E Do

:

there i s a nondegenerate subinterval of R

on which f preserves its sign). Then the set E is offirst category in the space Do.

EVERYWHERE DIFFERENTIABLE NOWHERE MONOTONE FUNCTIONS

81

P r o o f . Let {I, : n E N ) be the family of all subintervals of R with rational end-points. For each n E N , put

Bn ={f € D O : ( V z E I n ) ( f ( x ) S O ) ) . Clearly, we have

= UnEN(An UBn), so it suffices t o demonstrate that each of the sets A, and B, is closed and nowhere dense. We shall establish this fact only for A, (for B,, the argument is analogous). T h e closedness of A, is trivial. In order t o prove that A, contains no ball in Do, take any f E Do and fix an arbitrary E > 0. Since f E D o , there exists a point x E In such that f(x) = 0. Now, by starting with the existence of a nonzero bounded derivative belonging t o D o , it is easy t o show that there is a function h E D o for which

Let us define g=f+h. Then the function g belongs t o the ball of Do with centre f and radius At the same time, g does not belong t o A, because

E.

This establishes that A, is nowhere dense in D O , and the theorem has thus been proved. In the next chapter, we shall consider one more proof of the existence of everywhere differentiable nowhere monotone functions, by applying some properties of the so-called density topology on R. E x e r c i s e 4. Let E be an arbitrary topological space. We recall that a family N of subsets of E is a net in E if each open subset of E can be represented as the union of some subfamily of N (the concept of a net, for topological spaces, was introduced by Archangelski; obviously, it generalizes the concept of a base of a topological space). We denote by the symbol nw(E) the smallest cardinality of a net in E . Let now f : E+R be a function. We put:

82

CHAPTER

3

lmaxv(f) = the set of all those t E R for which there exists a nonempty open subset U of E such that t = s u p ( f ( U ) )and, in addition, there is a point e E U such that f ( e ) = s u p ( f ( U ) ) ; lminv(f) = the set of all those t E R for which there exists a nonempty open subset V of E such that t = i n f ( f ( V ) )and, in addition, there is a point e E V such that f ( e ) = in f ( f ( V ) ) . Check that card(lmaxv(f)) nw(E) w,


f ( x ) for each x E U ( e ) ; slmin(f) = the set of all points e E E having the property that there exists a neighbourhood V ( e )such that f ( e ) < f ( x ) for each z E V ( e ) . Check that card(slmax(f)) nw(E)+ w,


o X(X n [x - h, x h])/2h = 1. According to the classical Lebesgue theorem (see, e.g., Chapter 0), almost all points of X are its density points.

+

Exercise 1. Let (tn)nEN be a sequence of strictly positive real numbers, such that

Let X be a Lebesgue measurable subset of R and let x E R. Prove that the following two assertions are equivalent: 1) z is a density point of X ; X(X n[X - t n lx tn])/2tn = 1. 2) limn,+,

+

Exercise 2. Let X be a Lebesgue measurable subset of R and let

x E R. Show that the following two assertions are equivalent: 1) x is a density point of X ; 2) limh,o+, k + ~ + X(X n [x- h , z k])/(h k) = 1.

+

+

T h e notion of a density point turned out to be rather deep and fruitful not only for real analysis but also for general topology, probability theory

84

C HAPT E R 4

and some other domains of mathematics. For example, by use of this notion the important concept of the density topology on R was introduced and investigated by several authors (Pauc, Goffman, Waterman, Nishiura, Neugebauer, Tall and others). This topology was studied, with its further generalizations, from different points of view (see, e.g., [51], [117], [I211 and [158]). We shall deal with the density topology (and with some of its natural analogues) in our considerations below. Now, let f : R + R be a function and let x E R. We recall that f is said to be approximately continuous at x if there exists a A-measurable set X such that: 1) x is a density point of X ; 2) the function f l(X U {x)) is continuous at x. The next two exercises show that Lebesgue measurable functions can be described in terms of approximate continuity. Exercise 3. Let g : R -+ R be a function, let E be a fixed strictly positive real number and suppose that, for any A-measurable set X with A(X) > 0, there exists a A-measurable set Y E X with A(Y) > 0, such that

Demonstrate that there exists a A-measurable function h : R which we have (Vx E R)(lg(x) - h(x)l < 6 ) .

4

R for

Infer from this fact that if the given function g satisfies the above condition for any E > 0, then g is measurable in the Lebesgue sense. Exercise 4. Let f : R -+ R be a function. By applying the result of Exercise 3 and utilizing the classical Luzin theorem on the structure of Lebesgue measurable functions (see, e.g., [ I l l ] ) , show that the following two assertions are equivalent: a) the function f is measurable in the Lebesgue sense; b) for almost all (with respect to A) points x E R, the function f is approximately continuous a t x. Exercise 5. Let f : R + R be a locally bounded Lebesgue measurable function and let PZ

Prove that, for any point x E R at which the function f is approximately continuous, we have F'(z) = f (x). Check that the local boundedness of f is essential here.

N O W HERE A PPR O XIM A TELY D IFF ERE N TI A BLE F U N C T I O N S

85

Let now f : R -t R be a function and let x E R. We say t h a t f is approximately differentiable a t x if there exist a Lebesgue measurable set Y C R, for which x is a density point, and a limit

This limit is denoted by fAp(x) and is called an approximate derivative of f a t the point x.

Exercise 6. Demonstrate that if a function f : R -t R is approximately differentiable a t x E R, then f is also approximately continuous a t x. Exercise 7. Check that an approximate derivative of a function

a t a point x E R is uniquely determined, i.e. it does not depend on the choice of a Lebesgue measurable set Y for which x is a density point and for which the corresponding limit exists. Check also that the family of all functions from R into R approximately differentiable a t x forms a vector space over R.

Exercise 8.' If a function f : R -t R is differentiable (in the usual sense) a t a point x E R, then f is approximately differentiable a t x and fAp(x) = f ' ( x ) . Give an example showing t h a t the converse assertion is not true. For our purposes below, we need two simple auxiliary propositions.

Lemma 1. Let f : R -t R be a function, let z be a point of R and suppose that f is approximately differentiable at Z . Then, for any real number M1 > f A p ( 2 ) ,we have

Similarly, for any real number Mz < f A p ( 2 ) , we have

Proof. Since the argument in both cases is comletely analogous, we shall consider only the case of MI. There exists a A-measurable set X such t h a t x is a density point of X and

CHAPTER 4

Fix e

> 0 for which

+

fAp(x) e < M I . Then there exists a real 5 > 0 such t h a t , for any strictly positive h have

< 6, we

But, if 6 > 0 is sufficiently small, then

for all strictly positive h

< 6.

So we obtain the relation

and the lemma is proved. Actually, in our further considerations we need only the following auxiliary assertion which is an immediate consequence of Lemma 1.

L e m m a 2 . Let f : R -t R be a function, let x be a point of R and suppose that, for every strictly positive real number M , the relation

holds true. Then f is not approxamately differentiable at x .

In particular, suppose that two sequences {hk : k E N ) ,

{Mk : k E N )

of real numbers are given, satisfying the following conditions: 1) hk > 0 and Mk > 0 for all natural k ; 2.) limk,+,hk = 0 and limk,+,Mk = +m; 3 ) the lower limit

is strictly positive. Then we can assert that our function f is not approximately differentiable a t the point x . After these simple preliminary remarks, we are able t o begin the construction of a nowhere approximately differentiable function.

NOWHERE APPROXIMATELY DIFFERENTIABLE FUNCTIONS

First of all, let us put

f1(4/9) = 2/3, fi(519) fi

= 113, fi(619) = 213, fi(71Q) = 313,

($19) = 213, fi(919) = 3/3

and extend (uniquely) this partial function to a continuous function

+

in such a way that f l becomes linear on each segment [k/9, (k 1)/9] where k = 0 , 1 , ..., 8. We shall start with this function f l . In our further construction, we also need an analogous function g acting from the segment [O, 91 into the segment [O, 31, Namely, we put

+

Obviously, g is continuous and linear on each segment [k, k 11 where k = 0 , 1 , ...,8. Also, another function similar t o g will be useful in our construction. Namely, we denote by g* the function from [0,9] into [0,3], whose graph is symmetric with the graph of g , with respect t o the straight line {(x,y) E R x R : y = 3/21, In other words, we put g*(x) = 3 - g(x) for all x E [O, 91. Suppose now that, for a natural number n 1, the function

>

has already been constructed, such that: (a) fn is continuous; (b) for each segment of the form [k/gn, (k k E {O, 1,

s.,

+ l)/gn], where

gn - I},

the function fn is linear on it and the image of this segment with respect t o fn is some segment of the form b/3", ( j 1)/3"], where

+

j E {O, 1, ..,,3n - 1).

Let us construct a function

88

CHAPTER

4

For this purpose, it suffices t o define f n t l for any segment [k/gn, ( k + l ) / g n ] where k E { O , l , ...,gn - 1). Here only two cases are possible. 1, fn is increasing on [k/gn, (k l)/gn]. In this case, let us consider the following two sets of points of the plane:

+

((0, O), (0,3), (9,3), (9, O H ,

Since we have here the vertices of two rectangles, there exists a unique affine transformation h : R~-+ R2 satisfying the conditions

h(993) = ((k

+ l)/gnJfn((k + l)/gn)),

+ 1)/9", fn(k/gn))t o the segment [k/gn, (k + l)/gn] h(9,O) = ((k

Let the graph of the restriction of f n t l coincide with the image of the graph of g with respect to h. 2, fn is decreasing on [k/gn, (k l)/gn], In this case, let us consider the following two sets of points of the plane:

+

Here we also have the vertices of two rectangles, so there exists a unique affine transformation h* : R2--, R2 satisfying the relations

+

Let the graph of the restriction of f n + l t o the segment [k/gn, (k l)/gn] coincide with the image of the graph of g* with respect'to h*. T h e function f n t l has thus been constructed. From the above construction immediately follows that the corresponding analogues of the conditions

N O W HER E A P P R O X IM A TEL Y D IFFE R E N TI A BLE F U N C T I O N S

89

(a) and (b) hold true for f n + l , too. In other words, f n + l is continuous and, for each segment of the form [k/9"+',(k + 1)/9"+l], where

the function f n t l is linear on it and the image of this segment with respect to f n t l is some segment of the form b/3"t1, ( j 1)/3"t1], where

+

Moreover, our construction shows that

In addition, let [u, v] = [k/gn, (k

+ 1)/9"]

be an arbitrary segment on which fn is linear. Then it is not hard to check that fn+l([u, ( 2 ~ v)/31) = fn([u, ( 2 ~ v)/31),

+

+

+

+ + fn+l([(u + 2 ~ ) / 3 , v I )= fn([(u + 2 ~ ) / 3 , v I ) .

f n + l ( [ ( 2 ~ v)/3, ( 2 +~ ~1131)= f n ( [ ( 2 ~ v)/3, ( 2 ~ ~ ) / 3 1 ) , Proceeding in this way, we come to the sequence of functions

uniformly convergent to some continuous function f also acting from [O,1] into [0, 11. We assert that f is nowhere approximately differentiable on the segment [O,l]. In order to demonstrate this fact, let us take an arbitrary point x E [O,1] and fix a natural number n 2 1. Clearly, there exists a number k E {O,1, ...,gn - 1) such that

Therefore, we have

+

f n ( ~ E) Ij/3n, ( j 1)/3"1 for some number j E {0,1, ...,3" - 1). For the sake of simplicity, denote

From the remarks made above it immediately follows that, for all natural numbers m > n, we have f m ( 4 E b,ql

90

CHAP T ER

4

and, consequently, f (x) E b, q], too. Further, we may assume without loss of generality that fn is increasing on [u, v] (the case when f n is decreasing on [u, v] can be considered completely analogously). Suppose first that f (x) (p q)/2 and put

< +

Then, for each point y E Dl, we may write

Hence, we get ( f (y)

- f(x))/(y - 2) L ((2q + PI13 - (P + q)/2)l(v - 21) = (1/6)(3").

Suppose now that f (x) 2 (p

+ q)/2 and denote

DZ = [u, (2u

+ v)/3].

In this case, for any point y E Dz, we may write

Hence, we get ( f (XI - f(y))/(x

- y) L ((P + q)/2 - (

+

2 ~ q)/3)/(v

- u) = (1/6)(3").

Thus, in the both cases, we have X({y E [x

- l/gn, x + l/gn] \ {x)

:

+ l/gn] \ {x)

:

or, equivalently, X({y E [x - l / g n , x

The latter relation immediately yields that our function f is not approximately differentiable a t x (see Lemma 2 and the comments after this lemma). Remark 1. The function f constructed above has a number of other interesting properties (for more information concerning f , see [loll and

P O 1 1.

NOWHERE APPROXIMATELY DIFFERENTIABLE FUNCTIONS

91

Now, starting with an arbitrary continuous nowhere approximately differentiable function acting from [O, 11 into [0, 11, we can easily obtain an analogous function for R. We thus come t o the following classical result (first obtained by Jarnik in 1934).

Theorem 1. There exist continuous bounded functions acting from R into R, which are nowhere approximately digerentaable. R e m a r k 2. Actually, Jarnik proved that almost all (in the sense of the Baire category) functions from the Banach space C[O, 11 are nowhere approximately differentiable. Clearly, this result generalizes the corresponding result of Banach and Mazurkiewicz for the usual differentiability, Further investigations showed that analogous statements hold true for many kinds of generalized derivatives. The main tool for obtaining such statements is the notion of porosity of a subset X of R a t a given point 2 L. R. this interesting topic is out of the scope of our book, So we only refer the reader t o the fundamental paper [19] where several category analogues of Theorem 1 for generalized derivatives are discussed from this point of view.

ow ever,

'

In Chapter 11 of our book we give an application of a nowhere approximately differentiable function t o the question concerning some relationships between the sup-measurability and weak sup-measurability of functions acting from R x R into R. Since the concept of an approximate derivative relies essentially on the notion of a density point, it is reasonable to introduce here the so-called density topology on R and t o consider briefly some elementary properties of this topology. E x e r c i s e 9. For any Lebesgue measurable subset X of R, let us denote

d ( X ) = {x E R : x is a density point for X ) . Further, denote by Td the family of all those Lebesgue measurable sets Y E R, for which Y E d ( Y ) . Show that: 1) Td is a topology on R strictly extending the standard Euclidean topology of R ; 2) the topological space ( R , T a ) is a Baire space and satisfies the Suslin condition (i.e, each disjoint family of nonempty open sets in ( R , T d ) is a t most countable); 3) every first category set in ( R , T d ) is nowhere dense and closed (in particular, the family of all subsets of ( R , T d ) having the Baire property coincides with the Bore1 o-algebra of ( R , Td));

92

CHAPTER

4

4) a set X C R is Lebesgue measurable if and only if X has the Baire property in ( R , Td); 5) a set X C R is of Lebesgue measure zero if and only if X is a first category subset of ( R , Td); 6) the space ( R , Td) is not separable. T h e above-mentioned topology Td is usually called the density topology of R. E x e r c i s e 10. Let f : R -t R be a function and let x E R. Prove that the following two assertions are equivalent: 1) f is approximately continuous a t x ; 2) f regarded as a mapping from ( R , T d ) into R is continuous a t x. E x e r c i s e 11. By starting with the result of the previous exercise, show t h a t the topological space (R, Td) is connected. For this purpose, suppose t o the contrary that there exists a partition {A, B) of R into two nonempty sets A E Td and B E Td. Then define a function

by putting f (x) = 1 for all x E A, and f (x) = -1 for all x E B. Obviously, f is a bounded continuous mapping from ( R , Td) into R and hence, according t o Exercise 10, f is approximately continuous a t each point of R. Further, define F(z) = f(t)dt (x E R ) .

1'

By applying Exercise 5 of this chapter, demonstrate t h a t the function F is differentiable everywhere on R and

for each x E R. This yields a contradiction with the Darboux property of any derivative. One of the most interesting facts concerning the density topology states t h a t ( R , T d ) is a completely regular topological space (see, for instance, [I171 and [158]). This property of T d implies some nontrivial consequences in real analysis. To illustrate, we shall sketch here a proof of the existence of everywhere differentiable nowhere monotone functions by applying the above-mentioned fact (note that this approach is due t o Goffman [50]). E x e r c i s e 12. Consider any two disjoint countable sets

NOWHERE APPROXIMATELY DIFFERENTIABLE FUNCTIONS

93

each of which is everywhere dense in R. Taking into account Exercise 10 and the fact that (R,Td) is completely regular, we can find, for any n E N , a n approximately continuous function

satisfying the relations

Analogously, for any n E N , there exists an approximately continuous function gn : R -t [O, 11 such that 0

< gn(x) < 1

(X E

R),

Now, define a function h : R-tR by the formula

Check that: (1) h is bounded and approximately continuous; (2) h(a) > 0 for all a E A; (3) h(b) < 0 for all b E B. Let H denote an indefinite integral of h. Show that: (i) H is everywhere differentiable on R and H1(x) = h(x) for each x E R; (ii) H is nowhere monotone. We thus see that with the aid of the density topology on R it is possible t o give another proof of the existence of everywhere differentiable nowhere monotone functions acting from R into R (cf. the proof presented in Chapter 3). R e m a r k 3. T h e density topology on R can be regarded as a very particular case of the so-called von Neumann topology. Let (ElS, p ) be a space with a complete probability measure (or, more generally, with a complete

94

CHAPTER 4

nonzero o-finite measure). Then, in conformity with a deep theorem of von Neumann and Maharam (see, e.g., [loo], [117], [121], [159]), there exists a topology T = T ( p ) on E such that: 1) ( E , T ) is a Baire space satisfying the Suslin condition; 2) the family of all subsets of ( E , T ) having the Baire property coincides with the o-algebra S ; 3) a set X E E is of p-measure zero if and only if X is of first category in ( E , T ) . We say that T = T ( p ) is the von Neumann topology associated with the given measure space ( E , S, p). Note that T, in general, is not unique. This fact is not so surprising, because the proof of the existence of T is essentially based on the Axiom of Choice. There are many nontrivial applications of a von Neumann topology in various branches of contemporary mathematics (for instance, some applications to the general theory of stochastic processes can be found in [121]). R e m a r k 4. For the real line R, an interesting analogue of the density topology, formulated in terms of category and the Baire property, was introduced and considered by Wilczyriski in [166]. Wilczyriski's topology was then investigated by several authors. An extensive survey devoted t o the properties of this topology and t o functions continuous with respect to it is contained in [30] (see also the list of references presented there). R e m a r k 5. There are some invariant extensions of the Lebesgue measure A , for which an analogue of the classical Lebesgue theorem on density points does not hold. For example, there exist a measure p on R and a p-measurable set X C R , such that: 1) p is an extension of A; 2) p is invariant under the group of all isometric transformations of R ; 3) there is only one p-density point for X , i.e. there exists a unique point x E R for which we have

A more detailed account of the measure p and its other extraordinary properties can be found in [78].

5 . Blurnberg's theorem and Sierpihski-Zygmund function

In various questions of analysis, we need to consider some restrictions of a given function (e.g, acting from R into R ) , having nice additional properties which do not hold, in general, for the original function. In order to illustrate this, let us recall two widely known statements from t h e theory of real functions. T h e first of them is the classical theorem of Luzin concerning the structure of an arbitrary Lebesgue measurable function acting from R into R. Undoubtedly, this theorem plays the most fundamental role in the theory of real functions. Let X denote the standard Lebesgue measure on the real line. Let

be a function measurable in the Lebesgue sense. Then, according t o the Luzin theorem (see, e.g., [ I l l ] ) , there exists a sequence {D, : n E N ) of closed subsets of R, such that

and, for each n E N , the restricted function

is continuous. It immediately follows from this important statement t h a t , for any Lebesgue measurable function f : R + R, there exists a continuous function g : R 4 R such that

Indeed, it suffices to take a set Dn with X(D,) > 0 and then t o extend the function f 1 D, to a continuous function g acting from R into R (obviously, we are dealing here with a very special case of the classical Tietze-Urysohn

96

C HA P TER

5

theorem on the existence of a continuous extension of a continuous realvalued function defined on a closed subset of a normal topological space). In particular, we have the equality

where c denotes, as usual, the cardinality of t h e continuum. Also, we may formulate the corresponding analogue of the Luzin theorem for real-valued functions possessing the Baire property. This analogue is essentially due to Baire. Let

be a function having the Baire property. Then there exists a subset D of R such that: 1) the set R \ D is of first category; 2) the function f ID is continuous. In particular, since card(D) = c and cl(D) = R, we conclude that the restriction o f f t o some everywhere dense subset of R having the cardinality of the continuum turns out to be continuous. I t can easily be observed that the Luzin theorem and its analogue for t h e Baire property hold true in much more general situations. T h e following two exercises show this.

Exercise 1. Let E be a Hausdorff topological space, let p be a finite Radon measure on E and let pi denote the usual completion of p. Prove that, for any p'-measurable function

and for each E > 0, there exists a compact set Ii' E E for which these two relations are fulfilled: 1) p ( E \ I 0 such that (Vh E R)((hl < E

* ( X + h ) n X # 0).

In other words, aset X C R has the Steinhaus property if the corresponding difference set X - X = {x' - x" : x' E X, x" E X ) is a neighbourhood of point 0. It turns out that, as a rule, all good subsets of R are either of Lebesgue measure zero, or of first category, or have the Steinhaus property. In this connection, it is reasonable to mention here that Steinhaus himself observed that all Lebesgue measurable sets on R with strictly positive measure have this property (see [154]). Some years later, it was also established that an analogous result is true for second category subsets of R having the Baire property. Let X denote, as usual, the standard Lebesgue measure on R. We now formulate and prove the following classical result.

Theorem 1. Let X be a subset of R satasfying at least one of these t w o assumpta'ons: 1) X E dom(X) and X(X) > 0; 2) X E Ba(R) \ K ( R ) . T h e n X possesses the Steinhaus property.

Proof. Suppose first that assumption 1) holds. Let x be a density point of X and let ]a, b[ be an open interval containing x for which we have

Obviously, there exists a strictly positive (Vh E R)((hl < E

E

such that

+ X(]a + h, b + h[ U ]a, b[) 5 4(b - a)/3).

Take an arbitrary h E R with J h (< E . We assert that

LEBESGUE NONMEASURABLE FUNCTIONS

Indeed, if ( X A(]a

+ h) n X = 0, then we must have

+ h, b + h[ U la,b[) 1 X(((Xn la, b [ ) + h) U ( x n ]a,b[))=

which is impossible. Thus X possesses the Steinhaus property. Suppose now that assumption 2) holds. Then X can be represented in the form X = unx, where U is a nonempty open set in R and X I is a first category subset of R.Evidently, there exists a real E > 0 such that (Vh E R ) ( ( h J< E

+ (U + h) n U # 0).

Let us fix any h E R with Ih( < E . It is easy t o check the inclusion

+

Taking account of the fact that (U h) fl U is a nonempty open subset of R and (XI h) U XIis a first category subset of R , we infer that

+

Consequently, (X+h)nx#0, and this finishes the proof of Theorem 1. The following statement is an easy consequence of Theorem 1 but, sometimes, is much more useful in practice.

Theorem 2. Let X and Y be subsets of R such that at least one of these two conditions holds: 1) {X, Y) c dom(X), X(X) > 0, X(Y) > 0; 2) { X , Y ) C B a ( R ) \ li'(R). Then the vector s u m

has a nonempty anteraor. Proof. Clearly, under assumption I), there exists an element t E R such that X((X t t ) n Y ) > 0.

116

CHAPTER

6

Actually, this relation follows from the metrical transitivity of the measure A (also, from the Lebesgue theorem on density points). Similarly, under assumption 2), there exists an element r E R such that

In fact, here we have the metrical transitivity for the Baire property. Let us put

z=(x+t)nr in the first case, and

Z=(X+r)nY in the second one. It suffices to show that the set Z + Z has a nonempty interior. If 2 is symmetric with respect to zero, then we may directly apply Theorem 1. Otherwise, we can find an element t E R such that

in the first case, and

in the second one. Finally, define

+

The set 2' is symmetric with respect to zero and Z' 2 Z z / 2 . Moreover, X(2') > 0 in the first case, and 2' E Ba(R) \ IC(R) in the second one. Applying Theorem 1 to 2' and taking account of the relation

we come to the required result, The following exercises show that Theorems 1 and 2 have analogues in much more general situations.

Exercise 1. Let E be an arbitrary topological space and let

be a family of first category open subsets of E. Prove, by applying the Zorn Lemma, that the set U=LJ{Ui : i E I )

LE BE S G U E N O N M E A S U R A BL E F UN C T I O N S

117

is of first category, too. This classical result is due to Banach (see, e.g., [89], [I101 or [117]). It is of some interest because the set of indices I may be uncountable here. Obviously, the assumption that all sets Ui are open in E is very essential for the validity of this result. Exercise 2. Let (GI .) be an arbitrary topological group and let X be a subset of G such that

Using the previous exercise, show that the set

is a neighbourhood of the neutral element of G. Deduce from this result that if A and B are any two subsets of G satisfying the relation

then the set A - B = { a e b : ~ E A ~, E B ) has a nonempty interior. Exercise 3. Let (G, .) be a a-compact locally compact topological group with the neutral element e and let p be the left invariant Haar measure on G. We denote by p1 the usual completion of p. Let X be an arbitrary p'-measurable subset of G. Starting with the fact that p is a Radon measure, prove that

In particular, if p'(X) > 0, then there exists a neighbourhood U ( e ) of e such that ('Jg E U ( ~ ) ) ( P / ( (-SX )n X ) > 0) and, consequently,

Conclude from this fact that if A and B are any two pl-measurable subsets of G with $(A) > 0 and pl(B) > 0, then the set A . B has a nonempty interior.

Now, we are ready to present the first classical construction of a subset of the real line, nonmeasurable in the Lebesgue sense and without the Baire property. As mentioned earlier, this construction is due to Vitali (see [162]). First, let us consider a binary relation Rv(x, y) on R defined by the formula X E R & ~ E R & X - ~ E Q where Q denotes, as usual, the set of all rational numbers. Since Q is a subgroup of the additive group of R, we infer that' Rv(x, y) is an equivalence relation on R.Consequently, we obtain the partition of R canonically associated with R v ( x , y). This partition will be called the Vitali partition of R and will be denoted by R / Q . Any selector of the Vitali partition will be called a Vitali subset of R. Theorem 3. There exist Vitali subsets o f R . I f X is an arbitrary Vitadi subset of R , then X is Lebesgue nonmeasurable and does not possess the Baire property. Proof. The existence of Vitali sets follows directly from the Axiom of Choice applied t o the Vitali partition. Let now X be a Vitali set and suppose for a moment that X is either Lebesgue measurable or possesses the Baire property. Then, taking account of the relation

we infer that X must be of strictly positive measure (respectively, of second category). But this immediately yields a contradiction. Indeed, for each rational number q # 0, we have

because X is a selector of R / Q . Here q may be arbitrarily small. In other words, we see that our X does not have the Steinhaus property. This contradicts Theorem 1. We thus obtained that Vitali sets are very bad from the points of view of the Lebesgue measure and the Baire property. However, these sets may be rather good for other nonzero a-finite invariant measures given on R . Exercise 4. Prove that there exists a measure p on the real line, satisfying the following conditions: a) p is a nonzero complete a-finite measure invariant under the group of all isometric transformations of the real line; b) dom(A) C dom(p) where A denotes the standard Lebesgue measure on R ;

LEBESOUE NONMEASURABLE FUNCTIONS

c) (VY E dom(A))(A(Y) = 0 + p(Y) = 0); d) (VY E dom(A))(A(Y)> 0 p(Y) = too); e) there is a Vitali set X such that X E dom(p). Moreover, since p is complete and a-finite, we can consider a von Neumann topology T(p) associated with p. Let R* denote the set of all real numbers, equipped with T(p). Then the a-ideal li'(R*) and the a-algebra Ba(R*) are invariant under the group of all translations of R* and the Vitali set X mentioned in e) belongs to Ba(R*), i.e. possesses the Baire property with respect t o T(p).

+

The next exercise shows that any Vitali set remains nonmeasurable with respect to each invariant extension of the Lebesgue measure. Exercise 5. Prove that, for any measure v on R invariant under the group Q and extending the Lebesgue measure A , no Vitali subset of R is v-measurable.

It is not hard to see that the argument used in the Vitali construction heavily relies on the assumption of the invariance of the Lebesgue measure with respect to translations of R . This argument does not work for those nonzero a-finite complete measures p on R which are only quasiinvariant (i.e. p is defined on a a-algebra of subsets of R , invariant under translations, and the a-ideal of all p-measure zero sets is preserved by translations, too). So the following question arises: how to prove the existence of nonmeasurable sets with respect to such a measure p . We shall consider this question in the next chapter of the book. Namely, we shall show there that a more general algebraic construction is possible yielding the existence of nonmeasurable sets with respect to p . The main role in that construction will be played by the so-called Hamel bases of R. Now, we want to turn our attention to another classical construction of a Lebesgue nonmeasurable set (of a set without the Baire property). As pointed out earlier, this construction is due to Bernstein (see [9]). First, let us introduce one useful notion closely related to the Bernstein construction. Let E be a topological space and let X be a subset of E . We say that X is totally imperfect in E if X contains no nonempty perfect subset of E. We say that X is a Bernstein subset of E if X and E \ X are totally imperfect in E. Equivalently, X is a Bernstein subset of E if, for each nonempty perfect set P C E, we have

It immediately follows from this definition that X C E is a Bernstein set if and only if E \ X is Bernstein. Clearly, each subset of the real line, having cardinality strictly less than the cardinality of the continuum, is totally imperfect. The question concerning the existence of totally imperfect subsets of the real line, having the cardinality of the continuum, turns out to be rather nontrivial. For its solution, we need uncountable forms of the Axiom of Choice (cf. the next exercise). Exercise 6. Prove, in the theory ZF & DC, that if there exists a totally imperfect subset of R of cardinality c, then there exists a Lebesgue nonmeasurable subset of R. Prove also an analogous fact for the Baire property. Exercise 7. Let n be a natural number greater than or equal to 2, and let X be a totally imperfect subset of the n-dimensional Euclidean space Rn.Show that the set R n \ X is connected (in the usual topological sense). Infer from this fact that any Bernstein subset of Rn is connected. There are many examples of totally imperfect subsets of the Euclidean space R n . A wide class of such sets was introduced and investigated by Marczewski (see [156]). Let E be a Polish topological space and let X be a subset of E. We say that X is a Marczewski subset of E if, for each nonempty perfect set P C_ E, there exists a nonempty perfect set P' C_ E such that

It immediately follows from this definition that every Marczewski set is totally imperfect in E, and that any subset of a Marczewski set is a Marczewski set, too. Also, it can easily be observed that any set Y E with card(Y) < c is a Marczewski set. Indeed, let us take an arbitrary nonempty perfect set P 5 E. Then, as we know (see Chapter I), there exists a disjoint family {Pa : i E I) consisting of nonempty perfect sets and satisfying the relations card(I)=c, (vi~I)(PicP).

c

Now, since card(Y)

< card(I), it is clear that there exists at least one index

io E I such that Pi, n Y = 0, and thus Y is a Marczewski set.

Let us recall the classical result of Alexandrov and Hausdorff stating that every uncountable Bore1 set in a Polish topological space contains a subset homeomorphic to the Cantor discontinuum (hence contains a nonempty perfect subset). Taking this result into account, we can give

12 1

LEBESGUE NONMEASURABLE FUNCTIONS

another equivalent definition of Marczewski sets. Namely, we may say that a set X lying in a Polish space E is a Marczewski set if, for each uncountable Borel subset B of E l there exists an uncountable Borel set B' C E satisfying the relations

B'EB,

B1nX=O.

In some situations, the second definition is more convenient. For instance, let El and E2 be two Polish spaces and let f : El

-+

E2

be a Borel isomorphism between them. Then, for a set X 5 El, the following two assertions are equivalent: 1) X is a Marczewski set in E l ; 2) f ( X ) is a Marczewski set in E2. In other words, the Borel isomorphism f yields a one-to-one correspondence between Marczewski sets in spaces El and E2. This fact is rather useful. For instance, suppose that we need to construct a Marczewski subset of El having some additional properties which are invariant under Borel isomorphisms. Sometimes, it turns out that such a set can much more easier be constructed in E2. Let us denote it by X'. Then we apply the Rorel isomorphism f-I to X' and obtain the required Marczewski set f q l ( X ' ) in the space E l . Later, we shall demonstrate the usefulness of this method. Namely, we shall show that there exist Marczewski subsets of R nonmeasurable in the Lebesgue sense (respectively, without the Baire property). One simple fact concerning Marczewski sets is presented in the next statement.

Lemma 1. Let { X k : k < w ) be a countable family of Marczewski subsets of a Polish space E. T h e n U { X k : k < w ) is a Marczewski set, too. I n particular, if the space E is uncountable, then the family of all Marczewski subseis of E forms a proper a-ideal i n the Boolean algebra of all subsets of E. Proof. Fix a nonempty perfect set P E E. Since Xo is a Marczewski set, there exists a nonempty perfect set Po E such that Poc P,

Pofl Xo = 0,

diam(Po) < 1.

Further, since X I is also a Marczewski set, there exist nonempty perfect sets Poo E E and Pol C E such that

CHAPTER 6

Proceeding in this manner, we will be able to define a dyadic system

{Pji...j b : jl = 0, j2

€ {O, I},

...,jk

€ {0, I}, 1 5 k

< W}

of nonempty perfect sets in E whose diameters converge t o zero, and

for each natural number k

> 1. Now, putting

we obtain a nonempty perfect set D E E satisfying the relation

This shows that U{Xk : k

< w} is a Marczewski subset

of E.

We thus see that, in an uncountable Polish topological space El the family of all Marczewski subsets of E forms a a-ideal. It is usually called the Marczewski u-ideal in E and plays an essential role in classical point set theory (cf. [17]). As mentioned above, Marczewski subsets of E can be regarded as a certain type of small sets in E . In our further considerations, we shall also deal with some other types of small sets which generate proper a-ideals in the basic set E. For instance, we shall deal with the a-ideal generated by all Luzin subsets of R (respectively, by all Sierpiliski subsets of R). In addition, we shall consider the a-ideals of the so-called universal measure zero subsets of R and of strongly measure zero subsets of R. Various properties of these subsets will be discussed in subsequent chapters of the book (note that valuable information about different kinds of small sets can be found in [17], [89] and [107]). Let us return to Bernstein sets. We now formulate and prove the classical Bernstein result on the,existence of such sets.

Theorem 4 . There exists a Bernstein subset of the real line. Ail such subsets are Lebesgue nonmeasurable and do not possess the Baare property. Proof. Let a denote the least ordinal number for which c a r d ( a ) = c. We know that the family of all nonempty perfect subsets of R is of

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LEBESGUE NONMEASURABLE FUNCTIONS

cardinality c . So we may denote this family by {PC : E < a ) . Moreover, we may assume without loss of generality that each of the partial families

{PE : E < a, t is an even ordinal), {PC : E < a, ( is an odd ordinal) also consists of all nonempty perfect subsets of R. Now, applying the method of transfinite recursion, we define an a-sequence of points

satisfying the following two conditions: 1) if t < C < a, then xc # zc; 2) for each E < a , we have xe E PC. Suppose that, for P < a, the partial P-sequence already been defined. Take the set Pa.Obviously,

{XC : E < P ) has

Hence we can write

Pp \ {"E

:

E < PI # 0.

Choose an arbitrary element z from the last nonempty set and put x p = x. Continuing in this manner, we will be able to construct the a-sequence {xE : [ < a) of points of R, satisfying conditions 1) and 2). Further, we put X = {zF : ( < a, [ i s an even ordinal). It immediately follows from our construction that X is a Bernstein subset of R (because X and R \ X are totally imperfect in R). It remains t o demonstrate that X is not Lebesgue measurable and does not possess the Baire property. Suppose first that X is measurable in the Lebesgue sense. Then the set R \ X is Lebesgue measurable, too, and at least one of these two sets is of strictly positive measure. We may assume without loss of generality that X(X) > 0. Then a well-known property of X implies that there exists a closed set F C R contained in X and having a strictly positive .measure. Since X is a diffused (continuous) measure, we must have card(F) > w and hence card(F) = c. Denote by Fo the set of all condensation points of F . Obviously, Fo is a nonempty perfect subset of R contained in X . But this contradicts the fact that X is a Bernstein set in R.

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6

Suppose now that X possesses the Baire property. Then the set R \ X possesses the Raire property, too, and a t least one of these two sets is not of first category. We may assume without loss of generality that X is of second category. Consequently, we have a representation of X in the form

where V is a nonempty open subset of R and Y is a first category subset of R . Applying the classical Baire theorem, we,see that the set V \ Y contains an uncountable Gs-subset of R . This immediately implies that X contains also a nonempty perfect subset of R, which contradicts the fact that X is a Bernstein set in R . A result much more general than Theorem 4 is presented in the following exercise. Exercise 8. Let E be an infinite set and let {Xj : j E J) be a family of subsets of E, such that: 1) card(J) card(E); 2) (Vj E J)(card(Xj) = card(E)). Prove, by applying the method of transfinite recursion, that there exists a family {Yj : j E J) of subsets of E, satisfying the relations: a) ( V j E J)(Vjl E J ) ( j # j1 Yjn Yjl = 0); b) ( V j E J)(VjlE J)(card(Xj f l Y j l ) = card(E)).


0, we have

In other words, X is A2-thick in R~ if and only if the equality

is satisfied, where the symbol (A2), denotes the inner measure associated with X2. Let us point out that if a subset X of R 2 is Az-measurable and Azmassive simultaneously, then it is of full A2-measure, i.e.

Thus if we already know that a set X C R2 is not of full X2-measure but is Xz-thick, then we can immediately conclude that X is not X2-measurable. The next statement shows us that there are functions acting from R into R whose graphs are X2-thick subsets of the plane. T h e o r e m 5. There esists a jknction

whose graph is a Az-thick subset of R 2. Consequently, the followang two a.ssertions are true:

LEBESGUE NONMEASURABLE FUNCTIONS

1) the graph off as noi a A2-measur~blesubset of R ~ ; 2) f is not a A-me~surablefunction. Proof. Let a be the least ordinal number of cardinality continuum. Consider the family {Be : ( < a) consisting of all Bore1 subsets of R~ having strictly positive A2-measure. We are going to construct, by transfinite recursion, a family of points

satisfying these two conditions: (1) if < C < a, then xe # X C ; (2) for each I < a, the point (xe, ye) belongs to Be. Suppose that, for an ordinal P < a, the partial family

has already been defined. Let us take the set Bp. Applying the classical Fubini theorem, we see that the set

is A-measurable and has a strictly positive measure. Consequently, this set is of cardinality of the continuum, and there exists a point x belonging to it and distinct from all the points xe (( < P ) . We put xp = x. Then we choose an arbitrary point y from the set Bp(xp) and put yp = y. Proceeding in this manner, we will be able to construct the required family {(xe, ye) : ( < a). Now, it easily follows from condition (1) that the set

can be regarded as the graph of a partial function acting from R into R. We extend arbitrarily this partial function to a function acting from R into R and denote the latter function by f . Then condition (2) implies that the graph o f f is Az-thick in R 2 . Since there are continuum many pairwise disjoint translates of this graph, we conclude that the graph is not of full Az-measure and hence it is not a Az-measurable subset of R 2 . Finally, the function f is not A-measurable. Indeed, otherwise the graph of f will be a Az-measure zero subset of the plane, which is impossible. This ends the proof of the theorem. Exercise 15. By applying the Kuratowski-Ulam theorem which is a topological analogue of the classical Fubini theorem (see, e.g., [89] or [117]),

prove a statement for the Baire property, analogous to Theorem 5. Namely, show that there exists a function

such that its graph is thick in the sense of the Baire property, i.e, the graph intersects each subset of R 2 having the Baire property and not belonging to the a-ideal of all first category subsets of R2, Deduce from this fact that the graph of f does not have the Baire property in R a and f does not have the Baire property as a function acting from R into R. Exercise 16. Theorem 5 with the previous exercise show us that there exist functions from R into R whose graphs are thick subsets of the plane (in particular, those graphs are nonmeasurable in the Lebesgue sense or do not have the Baire property). On the other hand, prove that there exists a measure p on R 2 satisfying the following conditions: a) p is an extension of the Lebesgue measure X 2 ; b) p is invariant under the group of all translations of R2 and under the central symmetry of R~ with respect to (0,O); c) the graph of any function acting from R into R belongs to dom(p) and, for any such graph I?, we have p ( r ) = 0. We thus see that the graphs of all functions acting from R into R are small with respect to the above-mentioned measure p. This is a common property of all functions from R into R. Another interesting common feature of all functions from R into R was described by Blumberg's theorem (see Chapter 5 of this book). If we deal with some class of subsets of R which are small in a certain sense, then, as a rule, it is not easy to establish the existence of a set belonging to this class and nonmeasurable in the Lebesgue sense (or without the Baire property). Theorem 5 yields us that there exist functions from R into R whose graphs are nonmeasurable with respect to A 2 . At the same time, as mentioned above, the graphs of such functions may be regarded as small subsets of R 2 with respect to the measure p of Exercise 16. More generally, suppose that a a-ideal I of subsets of R is given. Then the following natural question can be posed: does there exist a set X E I nonmeasurable in the Lebesgue sense or without the Baire property? Obviously, the answer to this question depends on the structure of I and simple examples show that the answer can be negative. Let us consider the two classical a-ideals: I(X) = the a-ideal of all A-measure zero subsets of R;

L EBE SG U E N O N ME A S U R A BL E F U N C TI O N S

129

K ( R ) = the a-ideal of all first category subsets of R. These two a-ideals are orthogonal, i.e. there exists a partition { A , B ) of R such that A E I(A), B E II'(R). The reader can easily check this simple fact which immediately implies the existence of a Lebesgue nonmeasurable set belonging to K ( R ) and the existence of a Lebesgue measure zero set without the Baire property. Indeed, let X be a Bernstein subset of R. We put

Then it is easy to check that: 1) Xo E I(A) and Xo does not possess the Baire property; 2) X I E K ( R ) and XIis not measurable in the Lebesgue sense. A more general result is presented in the next exercise.

Exercise 17. Let X be a Bernstein subset of R. Let Y be a Ameasurable set with A(Y) > 0 and let Z be a subset of R having the Baire property but not belonging to I 0, we infer that B is uncountable and hence card(B) = c. This yields again a contradiction with the fact that f is a Sierpiriski-Zygmund function.

Exercise 20. Let Ii' be a compact subset of R2.Obviously, the set p r l ( I < ) is compact in R.Show that there exists a Borel mapping

such that the graph of 4 is contained in Ii'. In addition, give an example of a compact subset P of R 2 for which there exists no continuous mapping

such that the graph of $ is contained in P. This simple result is a very particular case of much more general statements about the existence of measurable selectors for set-valued mappings measurable in various senses. For example, suppose that a Borel subset B of the Euclidean plane R~is given satisfying the relation

Can one assert that there exists a Borel function

such that its graph is contained in B? Luzin and Novikov (see, e.g., [97]) showed that, in general, the answer to this question is negative, i.e, there

are Borel subsets B of R 2 with prl(B) = R which do not admit a Borel uniformization la. On the other hand, suppose that we have an analytic subset A of R 2 and consider its first projection prl(A) which is an analytic subset of R.Then, according to the classical theorem of Luzin, Jankov and von Neumann (see, for instance, [65]), there exists a function

such that: 1) the graph of g is contained in A; 2) g is measurable with respect to the a-algebra generated by the family of all analytic subsets of R. In particular, one may assert that the above-mentioned function g has the Baire property in the restricted sense and is measurable with respect to the completion of any a-finite Borel measure given on R. This important theorem has numerous applications in modern analysis and probability theory (some such applications are presented in [65]). Exercise 21. By using the result of the previous exercise, show that the graph of any Sierpiriski-Zygmund function is a Marczewski subset of the Euclidean plane. Exercise 22. Construct, by using the method of transfinite recursion, a Sierpiriski-Zygmund function whose graph is a &-thick subset of the Euclidean plane. Applying a similar method, construct a Sierpiriski-Zygmund function whose graph is a thick subset of the plane in the category sense. Deduce from these results that there are Marczewski subsets of the plane, nonmeasurable in the Lebesgue sense (respectively, without the Baire property). On the other hand, show that there exists a Sierpiriski-Zygmund function whose graph is a A2-measure zero subset of the plane. Analogously, show that there exists a Sierpiriski-Zygmund function whose graph is a first category subset of the plane. Concluding this chapter, we wish to make some remarks about logical aspects of the question concerning the existence of a Lebesgue nonmeasurable subset of the real line (or of a subset of the same line without the Baire property). Namely, in 1970, Solovay published his famous paper [I511 where he pointed out a model of ZF & DC in which all subsets of the real line were Lebesgue measurable and possessed the Baire property. However, the existence of such a model was based on the assumption of the existence of an uncountable strongly inaccessible cardinal number and this seemed to be

L EBE S G U E N O N M E A S U R A B LE F U N C T I O N S

135

a weak side of the above-mentioned result. But, later, Shelah showed in his

remarkable work [I291 that large cardinals appeared here not accidentally. More precisely, he established that: 1) there are models of ZF & D C in which all subsets of R possess the Baire property; 2) the existence of a model of ZF & D C in which all subsets of R are Lebesgue measurable implies the existence of some large cardinal. Solovay constructed also another model of set theory in which all projective subsets of R are Lebesgue measurable and possess the Baire property (see [151]). In this connection, it is reasonable to recall that in the Constructible Universe of Gijdel there are projective subsets of R (even belonging to the class P r 3 ( R ) ) which are not Lebesgue measurable and do not have the Baire property (for more details, see e.g. [55] and 1561). From among many other results connected with the existence of sets nonmeasurable in the Lebesgue sense (respectively, of sets without the Baire property), we want to point out the following ones: 1. Kolmogorov showed in [83] that the existence in the theory ZF & D C of a universal operation of integration for all Lebesgue measurable functions acting from [ O , l ] into R implies the existence of a Lebesgue nonmeasurable function acting from R into R. A similar result is true (in the same theory.) for a universal operation of differentiation. We thus conclude that the two fundamental operations of mathematical analysis - integration and differentiation - lead directly to real-valued functions which are nonmeasurable in the Lebesgue sense. This result seems to be interesting and important from the point of view of foundations of real analysis. In Chapter 13, we shall present some statements concerning generalized derivatives, which are closely related to the above-mentioned Kolmogorov result. 2. Sierpiriski proved that the existence of a nontrivial ultrafilter in the Boolean algebra of all subsets of w implies (within the theory ZF & D C ) the existence of a subset of R nonmeasurable in the Lebesgue sense and without the Baire property.

The proof of this result can be found, e.g., in [26]. 3. Shelah and Raisonnier (cf. 11201) established that the implication wl

holds in

0 and we can find an uncountable set Z c Y of Lebesgue measure zero. But then the set X n Z is uncountable, so we get a contradiction with the definition of the Sierpiliski set X. The contradiction obtained ends the proof of Theorem 5.

170

CHAPTER 8

Exercise 14. Let X be a Sierpiriski subset of R considered as a topological space with the induced topology. Show that any Borel subset of X is simultaneously an F,-set and a G6-set in X . In particular, each countable subset of X is a G6-set in X . Exercise 15. Let X be a Sierpiliski subset of R.As mentioned above, all countable subsets of X are Ga-sets in X . Applying this fact, demonstrate that, for any nonempty perfect set P E R , the set X n P is of first category in P. This result strengthenes the corresponding part of Theorem 5. Exercise 16. Let X be a Sierpiliski set on the real line R . Equip X with the topology induced by the density topology of R . Prove that the topological space X is nonseparable and hereditarily Lindelof (the latter means that each subspace of X is Lindelof, i.e, any open covering of a subspace contains a countable subcovering). Exercise 17. Assume that the Continuum Hypothesis holds. Let X be a Sierpiriski set on the real line R.Equip X with the topology induced by the Euclidean topology of R.Prove that

where A(X) denotes the class of all analytic subsets of X and B ( X ) denotes the class of all Borel subsets of X . Exercise 18. Let J1 and J2 be two orthogonal a-ideals of subsets of R, each of which is invariant with respect to the group of all translations of R. Let A1 and A2 be two subsets of R satisfying the relations

Demonstrate that: 1) there exists a set B1 E J1 for which we have

B1 - A2 = U { B 1- a : a E A 2 ) = R; 2) there exists a set B2 E J2 for which we have

B2 - A1 = U { B 2- a : a E A 1 ) = R. Further, put: J1 = the a-ideal of all first category subsets of R ; Jz = the a-ideal of all Lebesgue measure zero subsets of R. Deduce from relations 1) and 2) that if X is an arbitrary Luzin set on R and Y is an arbitrary Sierpiriski set on R, then the equalities

are fulfilled. We thus conclude that the simultaneous existence in R of Luzin and Sierpiriski sets immediately implies that the cardinality of these sets is as minimal as possible (i.e, is equal to the smallest uncountable cardinal). This result wae obtained by Rothberger (see [124]). It is easy to observe that if we replace the Continuum Hypothesis by Martin's Axiom (which is a much weaker assertion than CH), then we can prove the existence of some analogues of Luzin and Sierpiriski sets. Namely, if Martin's Axiom holds, then there exists a set X C R such that: 1) ccard(X) = c; 2) for each set Y E I'(R),we have

A set X with the above property is usually called a generalized Luzin subset of the real line. Similarly, if Martin's Axiom holds, then there exists a set X c R such that: (1) card(X) = c; (2) for each set Z E I ( X ) , we have

A set X with the above property is usually called a generalized Sierpiriski subset of the real line. Let us remark that, for the existence of generalized Luzin sets or generalized Sierpiriski sets, we do not need the full power of Martin's Axiom. In fact, the existence of generalized Luzin and Sierpiliski sets is implied by some additional set-theoretical assumptions which are much weaker than Martin's Axiom. The next exercise contains the corresponding result for an abstract a-ideal of sets.

>

Exercise 19. Let E be a set with card(E) w and let J be a proper a-ideal of subsets of E, containing in itself all one-element subsets of E. We denote: cov(J) = the smallest cardinality of a covering of E by sets belonging to J; cof(J) = the smallest cardinality of a base of J. Prove that if the equalities cov(J) = cof (J) = ccard(E) are fulfilled, then there exists a subset D of E such that ccard(D) = card(E)

CHAPTER

8

and, for any set Z E J, we have

In particular, if our basic set E coincides with the real line R and J is the a-ideal of all first category subsets of R (respectively, the a-ideal of all Lebesgue measure zero subsets of R), then we obtain, under Martin's Axiom, the existence of a generalized Luzin subset of R (respectively, the existence of a generalized Sierpiliski subset of R). Some facts about generalized Luzin sets and generalized Sierpiriski sets are presented in the next three exercises. Exercise 20. Assume that the Continuum Hypothesis holds. Prove that there exists a set X c R satisfying the following conditions: a) X is a vector space over the field Q; b) X is an everywhere dense Luzin subset of R. Show also that there exists a set Y c R satisfying the following conditions: (a) Y is a vector space over the field Q ; (b) Y is an everywhere dense Sierpiliski subset of R . Moreover, by assuming Martin's Axiom, formulate and prove analogous results for generalized Luzin sets and for generalized Sierpiriski sets. In addition, infer from these results, by assuming Martin's Axiom again, that there exist an isomorphism f of the additive group of R onto itself and a generalized Luzin set X in R such that f (X) is a generalized Sierpiriski set in R. Exercise 21. Suppose that Martin's Axiom holds. Prove that any generalized Luzin subset of R is universal measure zero. In addition, by using a generalized Luzin set on R, show that there exists a a-algebra S of subsets of R, such that: a) for each point x E R, we have {x} E S; b) S is a countably generated a-algebra, i.e. there exists a countable subfamily of S which generates S; c) there is no nonzero a-finite diffused measure defined on S. In particular, we see that Martin's Axiom implies that the cardinal c is not real-valued measurable.

A result similar to the one presented in Exercise 21 can be proved in the theory ZFC if we replace R by a certain uncountable subspace E of R. Namely, it suffices to take as E a universal measure zero subset of R with cardinality equal to w l . In particular, we immediately obtain from this result that w l is not real-valued measurable (cf. [161]).

Exercise 22. Assume Martin's Axiom. By applying a generalized Luzin set, prove that there exist two a-algebras S1and S2of subsets of the real line R, satisfying the following conditions: 1) B(R) E Sl ~ S Z ; 2) both S1 and S2are countably generated a-algebras; 3) there exists a measure p1 on S1extending the standard Borel measure on R ; 4) there exists a measure p2 on Sz extending the standard Borel measure on R ; 5) there is no nonzero a-finite diffused measure defined on the a-algebra of sets, generated by S1 U Sz. We wish to present here one application of a generalized Luzin set to the construction of a function which is extremely bad from the point of view of measure theory. First, we must give the corresponding definition. Let E be a set (in particular, a topological space) and let f be a function acting from E into R. We shall say that f is absolutely nonmeasurable if, for any nonzero a-finite diffused measure p on E, our f is nonmeasurable with respect to p. Let us stress that, in this definition, the domain of p is not a fixed aalgebra of subsets of E (actually, d o r n ( ~ )may be an arbitrary a-algebra of subsets of E , containing all singletones). T h e o r e m 6. Suppose that Marian's Axiom holds. Then there exists an injective functaon f : R-+R

whach samultlaneously is absolulely nonmeasurable. Proof. We know that Martin's Axiom implies the existence of a generalized Luzin subset of R . Let X be such a subset. Since we have

there exists a bijection

f : R-+X. Obviously, we can consider f as an injection from R into itself. Let us verify that f is the required function. Suppose, for a moment, that our f is not absolutely nonmeasurable. Then there exists a nonzero a-finite diffused measure p on R such that f is p-measurable, i.e. for any Borel subset B of R, the relation f - l ( ~ E) d o m b )

is satisfied. Equivalently, for any Borel subset B' of X , we have f-'(8') E dom(p). Clearly, without loss of generality, we may assume that p is a probability measure. Now, for each Borel subset B1 of X , we put

In this way we obtain a Borel diffused probability measure v on X , which is impossible since X is a universal measure zero space (see Exercise 21 above). This contradiction ends the proof of Theorem 6.

Remark 3. The preceding theorem was formulated and proved under the assumption that Martin's Axiom is valid. In this connection, it is reasonable to point out here that we cannot establish the existence of an absolutely nonmeasurable function within the theory ZFC.Indeed, if the cardinality of the continuum is real-valued measurable, then such functions do not exist. At the same time, one can easily demonstrate (in ZFC) that there exists an absolutely nonmeasurable function In order to show this, it suffices to pick a universal measure zero subspace X of R with card(X) = w l and then to take as f any bijection acting from w l onto X . In our further considerations, we shall meet some other applications of Luzin sets and Sierpifiski sets (respectively, of generalized Luzin sets and generalized Sierpiriski sets). But now we shall use once more Martin's Axiom and construct a generalized Sierpiriski set with the Baire property in the restricted sense.

Theorem 7. Suppose that Madin's Axiom holds. Then there exists a set X c R such that: 1) for every nonempty perfect set P E R, the intersection X n P is a first category set in P; 2) for each Lebesgue measurable set Y E R with X(Y) > 0, the inter'section X n Y is nonempty; 3) X is a generalized Sierpin'ski subset of R.

Proof. Let, as usual, c denote the cardinality of the continuum. Obviously, we may identify c with the smallest ordinal number a whose cardinality is equal to c. Let (Z()( 0, then obviously there exists a nonzero a-finite diffused Borel measure p on Y. Putting

we obtain a nonzero a-finite diffused Borel measure v on X . But this is impossible because X is a universal measure zero space. Implication 1) =+ 2) has thus been proved.

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C H A PTER

8

Let now X satisfy relation 2 ) . We are going to demonstrate that relation 1) holds for X , too. Of course, without loss of generality, we may assume that our X is a subset of the unit segment [O,l]. Suppose for a moment that X is not a universal measure zero space. Then there exists a Borel diffused probability measure p on [O,1]such that

We may also assume that p does not vanish on any nonempty open subinterval of [O,1] (replacing, if necessary, p by (p X)/2). Now, define a function f : [O,ll -+[O,lI

+

by the formula

f ( 4 = P([0>4 )

(a: E [0,11).

Evidently, f is an increasing homeomorphism from [O,l] onto itself. Let us put 4 2 ) = ~ (- l (fz ) ) (2 E B([O, 11)). In this way we get a Borel probability measure v on [0, 11 such that

Furthermore, it turns out that v coincides with the standard Borel measure on [0, 11. Indeed, for each interval [a, b] C_ [0, 11, we may write

where ~ ( [ 0cl) , = a,

cl([O,dl) = b .

Then we have .([a, bl) = P ( f -l([a, bl)) = P([c,

4)=

= cl([0,4)- ~ ( [ 0cl) 1 = b - aConsequently, the measures v and X are identical on the family of all subintervals of [0, 11 and hence these two measures coincide on the whole Borel a-algebra of [ O , l ] . Thus

which contradicts relation 2 ) . The contradiction obtained establishes implication 2 ) 3 1) and ends the proof of Theorem 8.

Exercise 26. Let X be an uncountable topological space such that all one-element subsets of X are Borel in X . We say that X is a Sierpiliski space if X contains no universal measure zero subspace with cardinality equal to card(X). Show that: a) any generalized Sierpiliski subset of R is a Sierpiriski space; b) if X is a Sierpiliski space of cardinality w l , then A(X) = B(X), where A(X) denotes the class of all analytic subsets of X (i.e, the class of all those sets which can be obtained by applying the (A)-operation to various (A)-systems consisting of Borel subsets of X ) and B(X) denotes, as usual, the class of all Borel subsets of X ; c) if X1 and X2 are two Sierpiliski spaces and X is their topological sum, then X is a Sierpiriski space, too; d) if X is a Sierpiriski space, Y is a topological space such that card(Y) = car$(X), all one-element subsets of Y are Borel in Y , and there exists a Borel surjection from X onto Y, then Y is a Sierpiliski space, too. Consequently, if X is a Sierpiriski subset of R and

is a Borel mapping such that card(f(X)) = card(X), then f ( X ) is a Sierpiliski subspace of R . Exercise 27. By assuming Martin's Axiom and applying the method of transfinite recursion, construct a generalized Sierpiriski subset X of R such that X+X=R. Infer from this equality that there exists a continuous surjection from the product space X x X onto R. Further, by starting with this property of X and taking into account assertion d) of the preceding exercise, show that the product X x X is not a Sierpiriski space. Conclude from this fact that the topological product of two Sierpiliski spaces is not, in general, a Sierpiliski space. Exercise 28. Let H be a Hilbert space (over the field R ) whose Hilbert dimension is equal to c (in particular, the cardinality of H equals c, too). Assuming that c is not a real-valued measurable cardinal, demonstrate that there exists a subset X of H satisfying the following conditions: 1) car$(X) = c; 2) X is everywhere dense in H (in particular, X is nonseparable); 3) X is a universal measure zero subspace of H.

Suppose now that c is not cofinal with w l , i.e, c cannot be represented in the form c= IC(

C

€<w1

where all cardinal numbers I E ~(< < wl) are strictly less than c . By starting with the fact that there exists an wl-sequence of nowhere dense subsets of H covering H, show that there is no generalized Luzin subset of H.In other words, show that there is no subset Y of H satisfying the next two relations: a) ccard(Y) = c; b) for each first category set Z C H ,the inequality

is fulfilled. Rich additional information about Luzin sets and Sierpiliski sets (also, about other small subsets of the real line) can be found in [89] and [107].

9. Egorov type theorems

It is well known that one of the earliest important results in real analysis and Lebesgue measure theory was obtained by Egorov [37] who discovered close relationships between the uniform convergence and the convergence almost everywhere of a sequence of real-valued Lebesgue measurable functions. This classical result (known now as the Egorov theorem) has numerous consequences and applications in analysis. For example, it suffices to mention that another classical result in real analysis - the so-called Luzin theorem on the structure of Lebesgue measurable functions - can easily be deduced by starting with the Egorov theorem. Here we wish to discuss some aspects of the Egorov theorem and, in addition, to show that, for a sequence of arbitrary real-valued functions there is no hope to get a reasonable analogue of this theorem. In other words, we are going to demonstrate in our further considerations that there are some sequences of rather strange real-valued functions, for which even very weak analogues of the Egorov type theorems fail to be true. First of all, we want to present the Egorov theorem in a form slightly more general than those in which this theorem is usually formulated in various standard courses of real analysis and classical measure theory. In order to do this, we need some auxiliary notions and facts. Let E be a nonempty set and let S be some class of subsets of E, satisfying the following conditions: 1) 0 E S, E E S; 2) S is closed under countable unions and countable intersections. Suppose also that a functional

is given, such that: a) for every increasing (with respect to inclusion) sequence of sets

we have U(u{xn :

72

< w)) < ~ u P { v ( X ~: )71 < w ) ;

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b) for every decreasing (with respect to inclusion) sequence of sets

we have v(fl{Yn : n

< w)) 2 inf{v(Yn)

: n

< w).

In this case, we say that S is an admissible class of subsets of E and v is an admissible functional on S. Note that, in analysis, there are many natural examples of admissible functionals. This can be confirmed by the following example. E x a m p l e 1. Let E be a nonempty set and let S be some a-algebra of subsets of E . Then it is obvious that S is an admissible class. Suppose, in addition, that v is a finite measure on S. Then it can easily be observed that v is an admissible functional on E . Exercise 1. Give an example of an admissible functional on E, which is not a measure on E. Similarly to the concept of measurability of real-valued functions with respect to ordinary measures, the concept of measurability of real-valued functions with respect to admissible functionals may be introduced and investigated. Namely, we say that a function

is measurable with respect to an admissible functional v on E (or, simply, f is Y-measurable) if, for each open interval It, t l [ c R, we have

Obviously, the same definition can be introduced for partial functions acting from E into R. The properties of functions (partial functions) measurable with respect to admissible functionals turn out to be rather similar to the properties of functions (partial functions) measurable in the usual sense. The next two exercises vividly illustrate this fact. Exercise 2. Let v be an admissible functional on ,E and let

+

be any two v-measurable functions. Show that the functions f g and f - g are v-measurable, too. In addition, show that if g(x) # 0 for all x E E, then the function f/g is also v-measurable.

E G O R O V T Y P E T HE O R E M S

183

Let X be an arbitrary set from d o m ( v ) . Check that the restriction f (X is a v-measurable partial function from E into R.

Exercise 3. Let v be an admissible functional given on a set E and let {fn : n < w ) be a sequence of v-measurable functions, pointwise convergent on E . Let us denote

Demonstrate that the function f is measurable with respect to v . Now, we are able to formulate and prove a direct analogue of the Egorov theorem for sequences of real-valued functions measurable with respect to an admissible functional.

Theorem 1. Let E be a basic set and let v be an admissible functional on E . Suppose, in addition, that a sequence {fn : n < w ) of v-measurable functions as given, pointwise convergent on E l and let f denote the corresponding limit function. Then, for each E > 0, there exasts a set X E d o m ( v ) satisfying these two relations: 1 ) v ( E ) - v(X) _< E ; 2) the sequence of functions { f n ( X : n < w ) converges uniformly to the function f J X . Proof. For any natural number m , let us denote

It is easy to check that the set E o l mbelongs to d o m ( v ) ,and

Consequently, there exists a natural index mo such that

Let us put Xo = Eo,m, and, for each m < w , consider the set

Evidently, E l , , belongs to d o m ( v ) , and

CHAPTER

9

Consequently, there exists a natural index m l such that

Let us put X1 = El,,, . Continuing in this manner, we will be able to define a certain sequence { X I , : k < w ) of sets belonging to d o m ( v ) and satisfying the relations: a) Xo 2 X I 2 ... 2 X k ,.,; b) for any k < w , we have v(E) u ( X l c )< E ; c) for any k < w , there is a natural number mr, for which we have

>

Finally, we put

-

x =nixk :

k <w).

Then, in virtue of the definition of an admissible functional, we get

and it can easily be verified that the sequence of the restricted functions { f n l X : n < w ) converges uniformly to the restricted function f ( X . This completes the proof of Theorem 1. Obviously, the theorem presented above immediately implies the classical Egorov theorem (see [37]). It suffices to take as v an arbitrary finite measure on E . In this connection, let us recall that, for a-finite measures, a direct analogue of the Egorov theorem is not true in general. Exercise 4. Let X denote, as usual, the standard Lebesgue measure on the real line R. Give an example of a sequence { f , : n < w ) of real-valued uniformly bounded A-measurable (even continuous) functions on R , which is convergent everywhere on R but there exists no unbounded subset X of R such that the sequence { f n l X : n < w) is uniformly convergent on X . Exercise 5. Let E be a normal topological space and let p be a finite inner regular Borel measure on E , i.e. for each Borel subset Y of E, we have the equality p(Y) = sup{p(F) : F

Y & F is closed in E ) .

We denote by p' the usual completion of p . Let

E G O R O V T Y PE THE O REM S

185

be an arbitrary pl-measurable function. By applying the Tietze-Urysohn theorem on the existence of a continuous extension of a continuous realvalued function defined on any closed subset of E , show that, for each E > 0, there exists a continuous function

satisfying the relation

Deduce from this fact that, for any pl-measurable function

there exists a sequence (4, : n < w ) of continuous real-valued functions on E, convergent to 4 almost everywhere (with respect to pl). Exercise 6. Let E be again a normal topological space, let p be a finite inner regular Borel measure on E and let p1 denote the completion of p. By starting with the Egorov theorem and applying the result of Exercise 5, prove the following Luzin type theorem: for any pl-measurable function

and for each real

E

> 0, there exists a continuous function

such that PI({. E E : f ( x ) # g(z))) < E The latter relation just expresses that the given function f has the so-called (C)-property of Luzin. It is frequently said that all pl-measurable functions possess this property (and the converse assertion is true, too). Let us now return to the Egorov theorem. In conformity with it, any convergent sequence of measurable real-valued functions converges uniformly on some "large" measurable subset of E ("large" means here that the measure of the complement of this subset may be taken arbitrarily small). In particular, if a given finite Borel measure on E is nonzero diffused and inner regular, then we immediately obtain that every convergent sequence of measurable real-valued functions on E converges uniformly on an uncountable closed subset of E. Hence, if E is an uncountable Polish topological space equipped with a nonzero finite diffused Borel measure, then, for any

186

CHAPTER

9

convergent sequence of measurable real-valued functions on E , there exists a nonempty compact perfect subset of E (actually, a subset homeomorphic to the Cantor discontinuum) on which the sequence converges uniformly. In connection with these observations, it makes sense to consider the following more general situation. Let E be an arbitrary uncountable complete metric space without isolated points and let { f , : n < w ) be a sequence of real-valued Borel functions on E , such that, for some constant d 2 0, we have

in other words, our sequence of functions is uniformly bounded. Do there exist a nonempty perfect compact subset P of E and an infinite subset K of w , for which the partial sequence of functions { f n l P : n E I 112,for which the sequence of functions { f n l X : n E I 0, we obviously obtain the relation

and, consequently,

card(X) = c because X is Borel in P'. It is clear now that X contains a nonempty perfect compact subset P for which the sequence of functions { f n l P : n E Ii'} converges uniformly, too.

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E G O R O V T Y PE THE O REM S

Mazurkiewicz was the first mathematician to prove that, for any uniformly bounded sequence of real-valued Borel functions given on an uncountable Polish space E, there exists a subset of E homeomorphic to the Cantor discontinuum, on which some subsequence of the sequence converges uniformly. In order to present a detailed proof of this interesting result, we need some auxiliary notions and simple facts. Let E be an uncountable Polish space and let be a family of realvalued functions defined on E . We shall say that the family is semicompact if, for each sequence { : n < w) E and for each nonempty perfect set P E, there exist an infinite subset IC of w and a nonempty perfect set PI contained in P , such that the partial sequence of functions (4, : n E IC) converges pointwise on PI. We shall say that a family S consisting of some Borel subsets of E is semicompact if the corresponding family of characteristic functions

is semicompact in the sense of the definition above. The following auxiliary statement yields a much more vivid description of semicompact families of Borel sets in E.

Lemma 1. Let S be a family of Borel subsets of an uncountable Polish space E. Then these two asseriions are equivalent: 1) the family S is semicompaci; 2) for any sequence {X, : n < w ) of sets from S and for each nonempty perfect subset P of E , there exists an infinite set I< E w such that

Proof. Note first that implication 2) for example, we have

a 1) is almost

trivial because if,

for some infinite subset K of w, then the set

n{Xn

: n E IC) fl P

contains a noneinpty perfect subset PI, and the sequence of characteristic functions {xx, : n E Ii') converges pointwise to the function xp, on the set PI (since all xx, (n E Ii') are identically equal to 1 on P').

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9

Now, let us establish implication 1) j 2). Suppose that relation 1) is fulfilled. Let { X , : n < w ) be an arbitrar y sequence of sets from S and let P be a nonempty perfect subset of E. We may assume, without loss of generality, that P = E. Then, according to I ) , there exists an infinite subset Ii' of w such that the corresponding sequence of characteristic functions { x x , : n E Ii') is convergent on an uncountable Borel subset Y of E. Let us denote

Obviously, x is a Borel function on Y, and

Therefore, at least one of the sets

is uncountatile. Suppose, for example, that card(Yl) > w . Then, by taking account of the formula Yl = Yl r l limsup{Xn : n E IC) = Yl r l l i m i nf { X , : n E I f ) , it can easily be checked that, for some infinite subset

of Ii', the inequality

is satisfied. If card(Yo) > w , then an analogous argument applied to the sequence of characteristic functions

yields the existence of an infinite subset KO of IC for which the inequality

is fulfilled. This establishes implication 1) Lemma 1.

j

2) and finishes the proof of

The next two auxiliary propositions also are not hard t o prove. Lemma 2. Let be a semicompact family of real-valued functions defined on an uncountable Polish space E . Then, for any real number d 2 0, the famaly offundaons

189

E G O R O V T Y PE THE O R EM S

is semicompact, too. Lemma 3. Let a1 and be any two semicompact families of realvalued functions defined on an uncountable Polish space E . Then the family of functions {41+42 :

41 € @ I , 4 2 € @ 2 )

is semicompact, too. Exercise 7. Give the detailed proofs of Lemma 2 and Lemma 3. It immediately follows from these lemmas (by using the method of induction) that if d 1 0 and alla2,.,., akare semicompact families of functions on an uncountable Polish space E , then the family of all those functions which can be represented in the form

where It11

41

5 d, It21 5 d, .,. , ltkl 5 d, 42

E

. a -

4 k E ahl

is also semicompact.

Lemma 4. Let be a semicompact family of bounded real-va/ued functaons on an uncountable Polish space E , and let a* denote the family of all those functions which are uniform limits of sequences of funclaons belonging to (in other words, a* is the closure of with respect to the topology of una'form convergence on E). Then the family a* is semicompact, too. Proof. Let (4: : n < w ) be an arbitrary sequence of functions from the family a*.In virtue of the definition of a*,for every natural number n , there exists a function 4, E such that

Let us consider the family of functions (4, : n < w ) . According to our assumption, is semicompact. Hence, for each nonempty perfect set P 2 E l there exist an infinite subset Ii' of w and a nonempty perfect subset P' of P, such that the partial sequence of functions

{dn1P1 : n E I nk for which all the sets

19 1

E G O R O V T Y PE T H E O R EM S

are uncountable. In this case, we may put nk+l = n and, for any a E 2<W with Ih(a) = k, we can construct two nonempty perfect sets POoand Pal satisfying the relations

So we see that, in this case, the process of our construction can be continued. 2 . For each natural number n > nk, there exists a u from 2<W with Ih(a) = k, such that card(P, n X,) 5 w . Since the family {Po : Ih(u) = k) is finite, we can find an infinite subset M of w and a a / E 2<W with lh(al)= k, such that

card(P,~n X,) 5 w for all numbers n belonging to M . From the latter relation we obtain

which immediately gives the desired result. Thus we may restrict our considerations only to case 1. As mentioned above, in this case, the construction described can be continued and, after w many steps, yields a dyadic family

of nonempty perfect subsets of E. Now, putting

P' =

(U{P, : Ih(u) = k)),

we get a nonempty perfect set P' such that

Hence, we may write

and the proof of Lemma 5 is complete. Now, taking into account the preceding lemmas, we are able to formulate and prove the following result of Mazurkiewicz.

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C HAPTER 9

T h e o r e m 2. Let be an arbatrary uniformly bounded famaly of realvalued Borel functions on an uncountable Polish space E. Then @ is semicompact. Proof. Since our @ is uniformly bounded, there exists a real number d 2 0 such that (Vd E @)(IldJIIL 4. Let us denote by \E' the family of all those functions II, which satisfy the following two relations: 1) IIII,II L d ; 2 ) II, is a linear combination of characteristic functions of some Borel subsets of E . Then, according to Lemmas 2, 3 and 5, the family \E' is semicompact. Also, it is clear that the original family @ is contained in the closure of \E' (with respect to the topology of uniform convergence on E). Hence, in view of Lemma 4, the family is semicompact, too. This ends the proof of Mazurkiewicz's theorem. Let us observe that if E = R, then Theorem 2 can be extended to an arbitrary family of uniformly bounded real-valued Lebesgue measurable functions on E and to an arbitrary family of uniformly bounded real-valued functions on E having the Baire property. Indeed, in order to obtain the corresponding results, it suffices to apply the following well-known fact: for any Lebesgue measurable (respectively, having the Baire property) realvalued function on R, there exists a real-valued Borel function II, on R such that the set 1. E R : dJ(4 # II,(x))

+

is of Lebesgue measure zero (respectively, of first category)< Exercise 8. Let E be an uncountable Polish topological space. Prove an analogue of Theorem 2 for real-valued functions on E possessing the Baire property and for real-valued functions on E measurable with respect to the completion of a fixed nonzero a-finite diffused Borel measure on E. In addition, give an example of a sequence {f, : n < w ) of uniformly bounded real-valued Borel functions on R, such that, for each infinite subset M of w , the corresponding partial sequence {f, : n E M ) is convergent only on a first category subset of R being simultaneously of Lebesgue measure zero. As demonstrated above, for any uniformly bounded sequence of realvalued functions possessing good descriptive properties, we have the pointwise convergence (and even the uniform convergence) of an appropriate

193

E G O R O V T Y PE TH E O REM S

subsequence on some nonempty perfect set, hence, on some set of cardinality continuum. However, various uniformly bounded sequences of realvalued functions are possible, which are extremely bad from the point of view of convergence pointwise. The following statement (essentially due to Sierpiriski) shows that the existence of such sequences can be directly deduced from the existence of a Luzin set Z in an uncountable Polish space E with card(Z) = c . In this connection, it is reasonable to recall here that the existence of a Luzin set of cardinality continuum is easily implied by the Continuum Hypothesis (see, e.g., Chapter 8 of the book).

Theorem 3. Let Z be a Luzin subset of the Cantor discontinuurn 2", satisfying the equality card(Z) = c . Then there exists a sequence {X, : n < w ) of subsets of 2") such that, for each infinite subset I< of w, the corresponding partial sequence of characteristic functions {xx, : n E I 0, there exists ) f(y) a Borel set Y c X such that p(Y) > 1 - E and l i m t _ , ~ F ( y , t = uniformly with respect to y E Y . This result is due to Tolstov.

10. Sierpiliski's partition of the Euclidean plane

In this chapter, we discuss several results and statements which are tightly connected with the classical Sierpidski partition of the Euclidean plane R~ = R x R. It turns out that these results and statements can be successfully applied in various fields of mathematics. Especially, they can be applied to certain questions and problems from mathematical analysis, measure theory and general topology.

,

Let w denote, as usual, the least infinite ordinal number and let w l denote the least uncountable ordinal number. It is a well-known fact that Sierpidski was the first mathematician who considered, in his classical paper [142],a partition {A, B) of the product set w l x w l , defined as follows:

B = { ( € t o :Wl > E > 0. He observed that, for any E < w l and 5 < w l , the inequalities

are fulfilled, where

= {E : (€56) E A), BE = (6.: ( € , < IE B ) . In other words, each of the sets A and B can be represented as the union of a countable family of "curves" lying in the product set w l x w l . This property of the partition {A, B) implies many interesting and important consequences. For instance, it immediately :follows from the existence of {A, B ) that if the Continuum Hypothesis

holds, then there exists a partition {A', B') of the Euclidean plane R 2 , satisfying the relations:

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C HAPTER 10

1) for each straight line L in R2 parallel to the line R x {O), the inequality card(A1n L) w


w)

206

CHAPTER

10

We also recall that a partial function f acting from R into R satisfies the Lipschitz condition if there exists a real L 2 0 such that

for all u and v belonging to dom(f). In this case, the constant L is usually called a Lipschite constant for f . It is not hard to prove the following auxiliary statement.

L e m m a 1. Let (X, d) be a metric space, let Y be a subset of X and let f be a function acting from Y into R and satisfying the Lipschitz condition with a Lipschitz consiant Lip(f) = L 2 0. Then f can be extended to a function

g : X+R which fulfils thas condition, too, with the same Lipschitz constant L. Proof. Assume that Y

# 0 and, for any point x E X , define

g(x) = in f {f (y)

+ Ld(x, Y) : y E Y).

In this way we obtain a mapping g from X into R . Let us check that g is the required extension o f f . Fix an arbitrary x E Y. Obviously, for each y E Y, we have g(x) 5 f (?4) Ld(8J ?4).

+

In particular, putting y = x, we get

On the other hand, the relation

implies that

f (x) 5 f (?4)+ Ld(x, ?4)

(?4 E Y)

and, hence, f ( x ) 5 g(x). Consequently, g(x) = f (x) and g is an extension of f . Now, let X I and xz be two arbitrary points from X and let exist points yl E Y and yz E Y such that

E

> 0. There

Then we may write

and, finally, 19(xi) - 9(~2)15 L d ( z l 9 ~ 2 ) Since c is an arbitrary strictly positive number, we conclude that

which finishes the proof of Lemma 1. Actually, we need only a very special case of this lemma when X = R. In this case, the proof can be done directly.

.

Lemma 2. Let f be a partial function acting from R into R, and suppose that f is differentiable at all points of dom(f), i.e. for each point t E dorn(f), there exists a derivative fi,,(f)(t) relative t o the set dom(f). Then the domain o f f can be represented an the form

where card(I) _< w and all sets Pi have the property that f lPi satisfies the Lipschitz condition.

Proof. First, let us denote

and, for any natural number n

> 0, define the set

D, = {t E D : (Vt' E D)(ltt - t (

5 l/n

D, by

=+ (f(t') - f (t) 1 < nlt' - to}.

Then, by taking account of the assumption that f is differentiable relative to D , it is not hard to check the equality

Further, for each natural n

> 0, we may write

CHAPTER 10

where

(Vk

< w ) ( d i a m ( D n k )5 l / n ) .

Now, it immediately follows from the definition of D , that all the restrictions f)Dnk (O 0 and q, let us denote by P ( E ,q) the set of all those elements ( a , t o , YO)E Ca(R x R ) x R x R

ORDINARY DIFFERENTIAL EQUATIONS WITH B A D RIGHT-HAND SIDES

241

for which there exist at least two real-valued continuous functions qil and 4z such that:

dom(41)= dom(qi2)= R,

rni(x) = 42(x) =

1: 1:

+

ro

(X

+

yo

(3 E R),

ect,m1ct))dt O(t,42(t))dt

E R),

It is not difficult to establish that P(E,q) is a closed subset of the product space Ca(R x R) x R x R. Indeed, suppose that a sequence

of elements of P(E,q) converges to some element

(@,xo, yo) E Ca(R x R) x R x R. Then we obviously have

and the sequence of functions

converges uniformly to the function O. We may assume without loss of generality that

(Vn E ~ ) ( l l @ ( ~ ) l l 5

11@11+

1).

and denote two real-valued conFor every natural number n, let tinuous functions satisfying the following relations:

&)(x) =

1'

xf'

d n ) ( t &)(t))dt ,

+

yp)

(x E R),

242

CHAPTER

l$ln)(q)

12

- 4P)(q)l L & -

Then it is not hard to verify that all functions from the family

are equicontinuous. More precisely, for each function $ from this family and for any two points x' E R and x" E R , we have the inequality

So, applying the classical Ascoli-Arzeld theorem (see, e.g., [119]), we can easily derive that there exists an infinite subset Ii' of N for which the partial sequences of functions

converge uniformly (on each bounded subinterval of R ) to some functions $1 and $2, respectively. Also, it can easily be checked that, for $1 and $ z , we have the analogous relations

Thus we see that ( @ I 80,YO )

E P ( & ,Q),

and hence P ( s ,q) is closed in the product space C b ( R x R ) x R x R . Now, let us put

P = u { P ( E , ~ ):

E

> 0,

C E Q , q E Q).

Then it is clear that a function E Cb(R x R ) does not belong to the set U if and only if there exist a rational number E > 0, a rational number q and some points xo E R and yo E R , such that (XQ, t o , yo) belongs to the set P ( E ,q). In other words, we may write

ORDINARY DIFFERENTIAL EQUATIONS WITH BAD RIGHT-HAND SIDES

243

where prl

:

Cb(R x R) x R x R + Cb(R x R)

denotes the canonical projection. It immediately follows from the definition of the set P that P is an Fa-subset of the product space Cb(Rx R) x R x R. In addition, the plane R x R is a a-compact space. So, applying Lemma 1, we conclude that prl(P) is an Fa-subset of Cb(R x R) and, consequently, U is a Ga-subset of Cb(R x R). This finishes the proof of Theorem 1.

Remark 1. Evidently, the Banach space Cb(Rx R) is not separable. Let E denote the subset of this space, consisting of all those functions which are constant at infinity. In other words, @ E E if and only if there exists a constant M = M ( @ )E R such that, for any for which we have

> 0, a

positive real number a = a(@,&)can be found

Notice that E is a closed vector subspace of Cb(R x R) and hence .E is a Banach space, as well. Moreover, one can easily verify that E is separable. Clearly, a direct analogue of Theorem 1 holds true for E. Actually, in [115] Orlicz deals with the space E. A number of analogues of Theorem 1, for other spaces similar to Ca(Rx R) or E, are discussed in [I].

Remark 2. Unfortunately, the set U considered above has a bad algebraic structure. In particular, U is not a subgroup of the additive group of Cb(Rx R) and, consequently, U is not a vector subspace of Cb(Rx R). Indeed, suppose for a while that U is a subgroup of Cb(Rx R), Then U must be a proper subgroup of Cb(Rx R). Let us take a function

Obviously,

U n ({s)+ U ) = 0.

+

But each of the sets U and {Q) U is the complement of a first category subset of Cb(Rx R). Therefore their intersection U n ({Q}+ U) must be the complement of a first category subset of Cb(R x R), too, and hence

We have thus obtained a contradiction which yields that U cannot be a subgroup of Cb(R x R).

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For some other properties of U interesting from the set-theoretical and algebraic points of view, see e.g. [I]. Theorem 1 proved above shows us that, for many functions from the space Ca(Rx R ) , we have the existence and uniqueness of a solution of the Cauchy problem. In fact, this is one of the most important results in the theory of ordinary differential equations. Naturally, we may consider a more general class of functions

not necessarily continuous or Lebesgue measurable and investigate for such functions the corresponding Cauchy problem from the point of view of the existence and uniqueness of a solution. For this purpose, let us recall that, as shown in the previous chapter of our book, there exists a subset Z of the plane R x R , satisfying the following relations: 1) no three distinct points of Z belong to a straight line; 2) Z is the graph of some partial function acting from R into R; 3) Z is a Xz-thick subset of the plane R x R, where X z denotes the standard two-dimensional Lebesgue measure on R x R; 4) for any Bore1 mapping

the intersection of Z with the graph of d, has cardinality strictly less than the cardinality of the continuum. We denote by @ the characteristic function of the above-mentioned set 2 . Then, obviously, O is a Lebesgue nonmeasurable function and, furthermore, if [R]" C dom(X), then O is sup-measurable as well. Now, starting with the function O described above, we wish to consider an ordinary differential equation

with the Lebesgue nonmeasurable right-hand side 9, and we are going to show that, in some situations, it is possible to obtain the existence and uniqueness of a solution of this equation (for any initial condition). First of all, we need to determine the class of functions to which a solution must belong. It is natural to take the class ACl(R) consisting of all

ORDINARY DIFFERENTIAL EQUATIONS WITH B A D RIGHT- HAND SIDES

245

locally absolutely continuous real-valued functions on R. In other words, II, E ACr(R) if and only if, for each point x E R, there exists a neighbourhood V(x) such that the restriction $lV(x) is absolutely continuous. Another characterization of locally absolutely continuous functions on R is the following one: a function $ belongs to ACl(R) if and only if there exists a Lebesgue measurable function

such that f is locally integrable and

for any point x E R. Let ly be a mapping from R x R into R and let (30,yo) E R x R. We say that the corresponding Cauchy problem

has a unique solution (in the class ACl(R)) if there exists a unique function $ E AC,(R) satisfying the relations: a) $'(x) = ly(x, $(x)) for almost all (with respect to the Lebesgue measure A) points x E R; b) $(go) = Yo. For example, if our mapping ly is bounded, Lebesgue measurable with respect to x and satisfies locally the Lipschitz condition with respect to y, then, for each (xo,YO)E R x R, the corresponding Cauchy problem has a unique solution. The reader can easily verify this fact by using the standard argument. Notice that, in this example, ly is necessarily Lebesgue measurable and sup-measurable (cf. Exercise 2 from Chapter 11). Notice also that an analogue of Theorem 1 holds true for a certain class of Banach spaces consisting of mappings (acting from R x R into R ) which are Lebesgue measurable with respect to x and continuous with respect to y.

Exercise 2. Prove that an analogue of Theorem 1 remains true for any Banach space E of bounded mappings acting from R x R into R, for which there exists an everywhere dense set D 2 E such that each function from D is Lebesgue measurable with respect to x and satisfies locally the Lipschitz condition with respect to y. The next statement shows that the existence and uniqueness of a solution can be fulfilled even for some ordinary differential equations whose

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12

right-hand sides are extremely bad, e.g. nonmeasurable in the Lebesgue sense.

Theorem 2. There is a Lebesgue nonmeasurable mapping

such that the Cauchy problem

has a unique solution for any point (xD,yo) E R x R. Proof. Let Z be a subset of the plane, constructed in the previous chapter (see Theorem 2 therein). Denote again by the characteristic function of Z and fix a real number t . Further, put

We assert that ly is the required mapping. Indeed, is Lebesgue nonmeasurable because @ is Lebesgue nonmeasurable. Let now (30,yo) be an arbitrary point of the plane R x R. Consider a function

defined by the formula

The graph of this function is a straight line, so it has at most two common points with the set 2. Consequently, the function

is equal to t for almost all (with respect to the Lebesgue measure A) points from R. We also have $'(x) = t for all x E R. In other words, $ is a solution of the Cauchy problem

It remains to show that $ is a unique solution from the class ACl(R). For this purpose, let us take an arbitrary solution 4 of the same Cauchy problem, belonging to ACl(R). Then, for almost all points x E R, we have the equality 4'(.) = @ ( X I 4(.>) t.

+

ORDINARY DIFFERENTIAL EQUATIONS WITH BAD RIGHT-HAND SIDES

247

It immediately follows from this equality that the function @+ is measurable in the Lebesgue sense. But, as we know,

So we obtain that @,p is equivalent to zero and hence

for almost all x E R. Therefore we can conclude that

This completes the proof of Theorem 2.

Remark 3. The preceding theorem was proved in the theory ZFC.In this connection, let us stress once more that the function ly of Theorem 2 is Lebesgue nonmeasurable and, under a certain set-theoretical hypothesis, is also sup-measurable (hence weakly sup-measurable). At the same time, we do not know whether it is possible to establish within the theory ZFC the existence of a sup-measurable mapping which is not measurable in the Lebesgue sense. Exercise 3. Let n be a natural number and let

be a polynomial of degree n. Show that there exists a mapping

satisfying the following relations: a) ly is nonmeasurable in the Lebesgue sense; b) for any initial condition (xD,yo) E R x R, the differential equation y' = ly(x, y) has a unique solution $J with $J(xo)= yo; c) all solutions $J of the above-mentioned differential equation are of the form $(x) = aoxn alxn-I ... an,lx a (x E R ) ,

+

+ +

+

where a E R. (constructed in the previous Now, starting with the same function chapter of the book), we shall show that, under the set-theoretical assumpt ion [R]" c cdom(X),

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12

Theorem 1 of Orlicz can be generalized to Banach spaces of mappings acting from R XR into R, essentially larger than the classical Banach space C ~ ( R X R) (notice that all spaces of real-valued bounded mappings, considered in this chapter, are assumed to be equipped with the norm of uniform convergence). More precisely, we can formulate and prove the next result.

Theorem 3. Suppose that [R] 0.

In other words, Bo can be regarded as a topological direct sum of the two Banach spaces Cb(R x R) and {tao : t E R). Consequently, we may identify Bo with the product space Cb(R x R) x R. Let now 91 be an arbitrary function from Cb(Rx R) such that the corresponding ordinary differential equation

ORDINARY DIFFERENTIAL EQUATIONS WITH BAD RIGHT-HAND SIDES

249

has a unique solution for any initial condition y(xo) = yo. Then it is not difficult t o check (by using the properties of our function Qo) that, for each real number t l , the ordinary differential equation

has also a unique solution for any initial condition y(xo) = yo. Conversely, if a function ly = ly, +tl@,, from the space Bo (where q1 E C b ( R x R ) ) is such that the ordinary differential equation Y' = Q(x1 possesses a unique solution for every initial condition, then the ordinary differential equation Y' = *1(x,y) possesses a unique solution for every initial condition, too. Let us recall that the symbol U denotes (in this chapter) the family of all functions l y l from Cb(R x R ) such that the differential equation y' = g l ( x , y) has a unique solution for any initial condition. Denote now by V an analogous family for the space Bo, i.e. let V be the family of all functions ly from Bo such that the differential equation y' = ly(x, y) has a unique solution for any initial condition. Then, taking account of the preceding argument, we can assert that

V = U + {tQ0 : t E R ) . Since, according t o Theorem 1, U is a dense Gs-subset of the Banach space C b ( R x R ) , we easily conclude that V is a dense Gs-subset of the Banach space Bo . Theorem 3 has thus been proved. Exercise 4. By assuming the same hypothesis

give an example of a Banach space B1 of functions acting from R x R into R, satisfying the following relations: 1) Cb(R x R ) C B1; 2) there are discontinuous Lebesgue measurable sup-measurable functions belonging to B1; 3) there are Lebesgue nonmeasurable sup-measurable functions belonging to B1;

250 4) an analogue of Theorem 1 holds true for

C HA PTER

12

B1.

Remark 4. Let B be a Banach space of bounded sup-measurable mappings, for which an analogue of Theorem 1 is valid, i.e, the family of all Q E B such that the differential equation

has a unique solution for every initial condition y(xo) = yo, is a dense Gasubset of B. It is not difficult to see that the class of all Banach spaces B is rather wide. In particular, it follows from Theorem 3 that the situation is possible where a space of this class contains a Lebesgue nonmeasurable mapping. In this connection, it would be interesting to obtain a characterization (description) of the above-mentioned class of Banach spaces. Finally, let us point out that some logical and set-theoretical aspects of the classical Cauchy-Peano theorem on the existence of solutions of ordinary differential equations are discussed in the paper by Simpson [145].

13. Nondifferentiable functions from the point of view of category and measure

Earlier we were concerned with various nondifferentiable functions acting from R into R. In this chapter, we wish to discuss one general approach to such functions from the viewpoint of category and measure. Roughly speaking, our goal is to demonstrate that, for a given generalized notion of derivative (introduced within the theory ZF & DC), the set of nondifferentiable functions (with respect to this notion) turns out t o be sufficiently large. We begin with an approach based on the concept of Baire category. More precisely, it is based on the important theorem of Kuratowski and Ulam from general topology (for the formulation and proof of this theorem see, e.g., [89] or [117]). Note that the Kuratowski-Ulam theorem can be interpreted as a purely topological analogue of the classical Fubini theorem from measure theory. It is widely known that the Fubini theorem is fundamental for all of measure theory. Moreover, this theorem has many applications in analysis, probability theory and other domains of mathematics. Also, it is well known that the Kuratowski-Ulam theorem possesses a number of nontrivial applications in general topology and in modern mathematical analysis (some of them are presented in the books 1891 and [117]). In our further considerations, the main role is played by the following statement.

Theorem 1. Let El and Ez be a n y t w o topological spaces with countable bases ( or, more g e n e ~ a l l y ,with countable P - bases ) and let E3 be a topological space. Let Z be a subset of the product space El x E z . Suppose that a certain mapping @:Z-+E3 i s given, and that this mapping satisfies the conditions: 1) the partial function @ acting f r o m the topological space El x E2 i n t o the topological space E3 has the Baire property, i.e., f o r a n y open set V

from E3, the preimage W1(V) has the Baire property an El x Ez; 2) for almost all (in the sense of category) points x E El, the domaan of the ~artialmapping @(x,.) given b y

is a first category set in the space Ep. Then the following two relations hold: (a) Z is a first category subset of the product space El x E2; (b) for almost all (in the sense of category) points y E E2, the set

as of first category in the space El; roughly speaking, almost each point y E El is almost singular with respect to the partial mapping @(.,y). The proof of this general statement is very simple. Indeed, according to the Kuratowski-Ulam theorem, relation (a) implies relation (b). Therefore it is sufficient to establish relation (a) only. Since, in virtue of condition I), the partial function @ has the Baire property, the set

has the Baire property in the product space El x Ep. Using condition 2) and the Kuratowski-Ulam theorem once more, we get the required result. In connection with Theorem 1, a natural question arises: how can condition 2) be checked for the given partial mapping @? The following situation can be frequently met in analysis and it will be the most interesting for us in the sequel. Suppose that Ez is a Polish topological vector space, E3 is a topological vector space with a countable base and our partial mapping @ satisfies condition 1) and the next condition: 2') for almost each (in the sense of category) point x E E l , the partial mapping @(x,.) is linear and discontinuous on its domain. Then it can be shown that @ satisfies condition 2), as well. Indeed, for almost all points x E El, the function @(x,.) has the Baire property and is linear and discontinuous on the vector space

Let us prove that, for the points x mentioned above, the set Z(x) is of first category in the space Ep. Suppose otherwise, i.e, suppose that Z(x) is a second category set with the Baire property. Then we may apply to Z(x) the well-known Banach-Kuratowski-Pettis theorem from the theory of

253

N O N DIFFE R E N T I A BLE F U N C TI O N S

topological groups (see, for example, [66] or [89]). This theorem is a topological analogue of the classical Steinhaus property of Lebesgue measurable sets with a strictly positive measure. Namely, according to this theorem, the set Z(x) Z(x) = { y - 2 : y E Z(x), 2 E Z(x))

-

contains a nonempty open subset of the topological vector space Ez (more precisely, the set Z(x) - Z(x) is a neighbourhood of zero of Ez). But since the set Z(x) is a vector space, too, we come to the equality

and, finally, we obtain Z(X) = Ez. Hence the function @(x,.) is defined on the whole Polish topological vector space Ez and is linear on this space. Now, by taking account of the fact that the function @(x,.) has the Baire property, it is not difficult to prove (by using the same Banach-Kuratowski-Pettis theorem) that @(x,.) is a continuous mapping. But this contradicts the choice of the point x. The contradiction obtained shows us that the set Z(x) must be of first category in the space E2,Therefore condition 2) is satisfied for our partial mapping

a. Remark 1. Theorem 1 may be considered as one of possible formalizations of a well-known principle of mathematical analysis which is frequently called "the principle of condensation of singularities". Among various works devoted to this principle, the most famous is the classical paper of Banach and Steinhaus [8]. It is easy to see that the Banach-Steinhaus principle of condensation of singularities is closely connected with Theorem 1 and can also be obtained as a consequence of the Kuratowski-Ulam theorem. Indeed, let us take El = N where the set N of all natural numbers is equipped with the discrete topology, and let E2 be an arbitrary Banach space. Suppose that E g is another Banach space and a double sequence of continuous linear operators

Lm,, : E2 -+ E3

( m ,n E N)

is given, such that, for any m E N , we have

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13

Let us define a partial mapping from the product space El x Ez into the space E3 by the following formula:

is clear that this partial mapping has the Baire property and, for each E N , the partial mapping @(m,.) is defined on a first category subset the space E z . Hence the domain of the partial mapping @ is also a first category set in the product space El x E z . Now, we may apply the Kuratowski-Ulam theorem and, evidently, we obtain that, for almost all elements x E El, the set { m : (m, x ) E dam(@))

is empty. But, actually, this is the Banach-Steinhaus principle of condensation of singularities.

Remark 2. The general scheme of applications of Theorem 1 is as follows. First of all, we must check that a given partial mapping @ has the Baire property. Obviously, @ has this property if it is a Bore1 mapping or, more generally, if it is a measurable mapping with respect t o the aalgebra generated by a family of analytic sets (such situations are typical in modern analysis). Now, suppose that our partial mapping @ of two variables has the Baire property. Then the second step is to check that the corresponding partial mappings of one variable are defined on the first category sets. This will be valid if Ez and E3 are Polish topological vector spaces and if, for almost all elements x E E l , the corresponding mappings @(x,.) are linear and discontinuous on their domains (notice that if the given space Eg is a normed vector space, then we need to check the linearity and the unboundedness of the corresponding partial mappings). Finally, we can apply Theorem 1. Now, we wish to present an application of Theorem 1 in a concrete situation. Namely, we will be interested in a certain type of generalized derivative. Let co denote the separable Banach space consisting of all real-valued sequences converging to zero. Let R denote the real line and let [O,l] be the closed unit interval in R. Suppose that a mapping

is given. Evidently, we may write

NONDIFFERENTIABLE FUNCTIONS

where 4n:[0,1]-,R

EN).

Let us assume that the mapping 4 satisfies the following condition: for each point x E [O,11 and for each index n E N, the value &(x) is not equal to zero. Moreover, let us assume (without loss of generality) that

for all natural numbers n. If f is a real-valued function defined on the segment [0, 11 and a point x belongs to this segment, then the real number

is called the +derivative of f at x (if this limit exists, of course). In our further considerations, we denote the limit mentioned above by the symbol f$(x>. Let us put El = [O, 11, Ez = C[O, 11, E3 = R and consider a partial mapping acting from the product space El x E2 into the space E3 and defined by the formula

Suppose that the original function 4 has the Baire property. We assert that, in such a case, the partial mapping has the Baire property, too. Indeed, it suffices to observe that, for every natural number n, the mappings

have the Baire property. For the second mapping, this is obvious since the function 4% has the Baire property. Further, the mapping

is continuous and the mapping

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13

can be represented as the following superposition:

In this superposition the first mapping has the Baire property and the two other mappings are continuous. Therefore we conclude that the superposition also has the Baire property. Let us notice, by the way, that the same result can be established in a different manner. Namely, if in the function of two variables (31f ) f(x dln(x)) f (XI

+

+

-

we fix a point x , then we obtain a continuous function of one variable, and if in the same function of two variables we fix a second variable f , then we obtain a function of one variable having the Baire property. So, we see that, for our function of two variables, the conditions similar to the classical Caratheodory conditions (i.e, the measurability with respect to one of the variables and the continuity with respect t o another one) are fulfilled. From this fact it immediately follows that our function of two variables has the Baire property (in this connection, see also [lo21 where a general problem concerning the measurability of functions of two or more variables is investigated in detail). Taking the above remarks into account, we conclude that the partial mapping (2, f fk<x> has the Baire property. Moreover, it is easy to see that if a point x is fixed, then this partial mapping yields a linear discontinuous function of one variable f. Consequently, we can apply Theorem 1 and formulate the following statement.

>

-

Theorem 2. If a mapping

has the Baire property, then almost each function from the Banach space C[O, 11 does not possess a 4-derivative almost everywhere on the segment

LO, 11. We want to point out that the basic operations used in classical mathematical analysis are, as a rule, of the projective type, i.e, these operations are described completely by some projective sets lying in certain Polish topological spaces. In many natural situations, it can happen that the graph of our partial mapping from Theorem 1 is a projective subset of the corresponding Polish product space. Then, according to the important

N O N D I FFE R E N T I A B LE F U N C TI O N S

257

results of Solovay, Martin and others, we must apply some additional settheoretical axioms for the validity of the corresponding version of Theorem 1. For example, suppose that satisfies only condition 2) of Theorem 1, the graph of lies in a Polish product space El x E2 x E3 and this graph is a continuous image of the complement of an analytic subset of a Polish topological space. Then if we wish to preserve the assertion of Theorem 1 for a, we need the existence of a two-valued measurable cardinal or Martin's Axiom with the negation of the Continuum Hypothesis. Analogously, if the graph of our partial mapping is a projective subset of a Polish product space, belonging to a higher projective class, then we need the Axiom of Projective Determinacy or a similar set-theoretical axiom (for more details, see [55] and [56]). Actually, suppose that we work in the following theory: ZF & D C & (each subset of R has the Baire property). Then the assertion of Theorem 1 will be true for all Polish topological spaces El, Ez, Es and for all partial mappings acting from El x E2 into E3 and satisfying condition 2) of this theorem. See, e.g., [67] where the theory mentioned above is applied to some questions connected with the existence of generalized derivatives of various types. In particular, it is established in 1671 that if we work in the above-mentioned theory, then almost each function from the space C[O, 11 does not possess a generalized derivative almost everywhere on the segment [O, 11. Obviously, such an approach can also be applied to special types of generalized derivatives, for instance, to the so-called path derivatives (for the definition and basic properties of path derivatives, see, e.g., [21]). In addition, let us stress that the direct analogue of the classical BanachMazurkiewicz theorem (which was considered in Chapter 0) cannot be established for all generalized derivatives, since there is (in the theory ZF & D C ) a certain notion of a generalized derivative having the property that, for any continuous function there exists at least one point x from the segment [O, 11, such that f is differentiable at x in the sense of this generalized derivative (cf. [67]). Further, the following natural question arises: does there exist an an* logue of the above-mentioned result in terms of measure theory'? In other words, does there exist a Bore1 diffused probability measure p on the space C[O, 11 such that, for any generalized derivative introduced in the theory ZF & D C , almost all (with respect to p ) functions from C[O, 11 are not differentiable, in the sense of this derivative, at almost all (with respect to A) points of [0, l]?

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13

At the present time, this question remains open. Here we give a construction of the classical Wiener measure p, on C[O, 11 and demonstrate that, for the derivative in the usual sense, p , yields a positive answer to this question. We recall that historically the Wiener measure appeared as a certain interpretation (mathematical model) of the Brownian motion (for an interesting survey of this phenomenon, see e.g. [14] and, especially, [96]). Note that the construction of the Wiener measure is not easy and needs a number of auxiliary facts and statements. To begin, we first of all wish to recall some simple notions from probability theory and the famous Kolmogorov theorem on mutually consistent finite-dimensional probability distributions. Let E be a set, let S be a a-algebra of subsets of E and let p be a probability measure on S . So we are dealing with the basic probability space (ElS , ~ 1 ) In our further considerations we assume, as a rule, that p is a complete measure. Let f be a partial function acting from E into R. We say that f is a random variable i f f is measurable with respect to the a-algebra S (i.e, for any open set U C_ R, the preimage f"(U) belongs to S ) and

-

For any random variable f , we may define the Bore1 probability measure p j on R, putting

The measure p j is usually called the distribution of a random variable f . Actually, the measure p j is defined in such a way that it becomes the homomorphic image of the measure p under the homomorphism f , so we may write p j = p o f - l . Obviously, p~ is uniquely determined by the function Fj : R [O,11 such that

F j ( x ) = p ( { e ~ E: f ( e ) < x ) ) (xER). This function is also called the distribution o f f . It is increasing and satisfies the relations: (a) limt,-o,Fj(t) = 0,

NONDIFFERENTIABLE FUNCTIONS

(b) limt,+, Ff (t) = 1, (c) (Vx E R)(limt,$- Ff (t) = Ff (x)).

Exercise 1. Let F be an increasing function acting from R into R and satisfying the relations analogous to (a), (b) and (c). Show that there exist a probability space (E,S, p) and a random variable

such that F = Fj.

Exercise 2. We recall that a probability measure p is separable if the topological weight of the metric space canonically associated with p is less than or equal to w (in other words, the above-mentioned metric space is separable). For instance, the classical Lebesgue measure on the unit segment [0, 11 is separable. Check that this fact is a trivial consequence of the following statement: the completion of any probability measure given on a countably generated a-algebra of sets is separable. Show the validity of this statement. Check that any homomorphic image of a separable measure is separable, too. Give an example of a topological space T and of a Borel probability measure on T which is not separable. Remark 3. In connection with the result of Exercise 2, let us note that there exist nonseparable measures on the segment [0, 11 extending the classical Lebesgue measure on [O,l]. Moreover, there are nonseparable extensions of the standard Lebesgue measure on the unit circle, which are invariant under the group of all rotations of this circle around its centre (for more information, see e.g. [57], [69] and references therein). Exercise 3. Let (E,S,,u) be a basic probability space and let T be a topological space (equipped with its Borel a-algebra B(T)).Any pmeasurable partial mapping

satisfying the condition

is usually called a T-valued random variable on E. The Borel probability measure p j on T defined by the formula

is called the distribution of f in T, and we write p j = ,u o f - l . Show that there exist a probability space ( E , S , p ) , a topological space T and a Borel probability measure v on T, such that there is no T-valued random variable f on E for which pf = v . Let ( E l S , p ) be again a basic probability space and let

be a random variable. We recall that JE f(e)dp(e) is the mathematical expectation of f (of course, under the assumption that this integral exists). We also recall the simple formula

More generally, for any Borel function

we have

under the assumption that the corresponding integrals exist. Exercise 4. Prove the formula presented above. Deduce, in particular, that, for each natural number n, the equality

holds true (if these integrals exist). In many cases, it may happen that the distribution p j of a random variable f can be defined with the aid of its density. We recall that a Lebesgue measurable function

is a density of p j (of F j ) if, for each Borel set X C_ R, we have

NONDIFFERENTIABLE FUNCTIONS

26 1

This means that the measure p j is absolutely continuous with respect to the Lebesgue measure X on R. Evidently, any two densities of p j are equivalent with respect to A. In addition, if p j exists, then we can write

and, more generally,

for every Bore1 function q5 :

R-+R

such that the corresponding integrals exist. The classical example of a probability distribution is the normal (or Gaussian) distribution. For the real line, the density of the so-called centered normal distribution is given by the formula

where a

> 0 is a fixed constant.

It can easily be checked in this case that

Taking a derivative (with respect to a ) in the last equality, we obtain

where c is some strictly positive constant whose precise value is not interesting for us. We now wish to recall the Kolmogorov theorem on the existence of a probability measure with given finite-dimensional distributions (see, e.g., [14], [36], [113], [121]). This theorem plays the fundamental role in the contemporary theory of stochastic processes. Let T be an arbitrary set of indices. Consider a family {& : t E T ) where, for each index t E T, the set & coincides with R. Suppose that, for any finite set T = { t l , ..., tn) T,

a Borel probability measure p, on the space

is given in such a way that the whole family

{pr :

7

E

[WW)

of probability measures is consistent, i.e, for any two finite subsets T' of T such that T E r', we have

T

and

where

p r T ~ , :, RTj + RT denotes the canonical projection from R,, onto R,. Further, consider the product space

with the a-algebra S generated by the family of mappings

where, for each index t E T , the mapping

coincides with the canonical projection from RT onto Rt. In other words, we may say that S is the smallest a-algebra of subsets of RT,such that all mappings prt (t E T) are measurable with respect to S ( S is also frequently called the cylindrical a-algebra of the space RT). Exercise 5. Show that the cylindrical a-algebra of the topological R coincides with its Borel a-algebra if and only if product space '

Exercise 6. Let X be a set and let { f i : i E I) be a family of realvalued functions defined on X. We say that this family separates the points of X if, for any two distinct points x and y from X , there exists an index i E I such that

fa(x> 2 fa(!/).

N O N DIFFERE N TI A BLE F U N C TI O N S

263

Let now X be a Polish topological space and let {fd : i E I ) be a countable family of Borel real-valued functions on X , separating the points of X . Denote by S({fi : i E I ) ) the smallest a-algebra of subsets of X , for which all functions fr (i E I ) become measurable. Consider a mapping

defined by the formula

Note that, since carcd(I) 5 w , the space RI is isomorphic to one of the spaces R W , R n (n E N). Check that: a) f is injective and Borel; b) S({fi : i E I)) = {f-l(Z) : Z E B(R')). By using the classical theorem from descriptive set theory, stating that the image of a Borel subset of a Polish space under an injective Borel mapping into a Polish space is also Borel, infer from a) and b) the equality S({fi : i E I ) ) = B(X). In particular, consider the separable Banach space C[O, 11 of all continuous real-valued functions on the segment [0, 11 and take as I a countable subset of [0, 11 everywhere dense in [0, I]. For each i E I, let

be the mapping defined by

Conclude from the result presented above that S({fi : i E I ) ) = B(C[O, 11). Give also a direct proof of this equality, without the aid of the mentioned result. The Kolmogorov theorem states that there exists a unique probability measure p~ defined on the cylindrical a-algebra of RT and satisfying the relations (r E Clr = PT 0 PTT,;

CHAPTER 1 3

where, for each finite set r 5 T, the mapping

is the canonical projection from RT onto R,. The original measures p, are usually called the finite-dimensional distributions of p ~ . The proof of the Kolmogorov theorem is not difficult. Indeed, using the consistency conditions, we first define the functional p~ on the cylindrical algebra (consisting of all finite unions of elementary subsets of R ~ in) such a way that the equalities Pr = pT P~T,: will be fulfilled for all finite sets r C T . Then we have to show that this functional is countably additive on the above-mentioned algebra. This is not hard because all finite-dimensional spaces R, are Radon, i.e, for any Bore1 set X C RT and for each E > 0, there exists a compact set K 2 X such that p r ( X \ K) < E . Finally, utilizing the classical Carathkodory theorem, we can extend our functional onto the whole cylindrical a-algebra S (for details, see e.g. [14], [I131 or [121]). Exercise 7. With the previous notation, show that, in the formulation of the Kolmogorov theorem, it suffices to assume only the consistency conditions of the form -1 PT = PT' 0 Prrl,r, where r and r' are any finite subsets of T for which T

c r',

carcd(rl \ r ) = 1.

R e m a r k 4. There are various generalizations of the Kolmogorov theorem. For example, this theorem may be regarded as a particular case of the statement asserting the existence of a projective limit of a given projective system of Radon probability measures. Furthermore, there are some abstract versions of the Kolmogorov theorem in terms of the so-called compact classes of sets. For more details, see e.g. [14], [113] or [121]. For our further purposes, we need only that case of the Kolmogorov theorem when T = [O,l].

NONDIFFERENTIABLE FUNCTIONS

Let us fix a finite set

= { t l l *..,tn) c [011]\ ( 0 ) . Clearly, we may suppose that 0 < t l measure p, on R, by the formula

< ... < t n .

Define a Borel probability

where the density p, satisfies the relation

e~p((-1/2)(x:/tl for all points

+(

-

~ 2~

l ) ~ / (-t tal ) ( ~ 1 s1s

s , ~ n )

+ ... + ( x n - ~ n - ~ ) ~ /-( ttnn- I ) ) )

E

If T is a finite subset of [0,11 whose minimal element coincides with 0 , then we put P7 = Po x C17\{0) where po is the Borel probability measure on Ro concentrated a t the origin of Ro (i.e. the so-called Dirac measure). It is not difficult to check the consistency of the family of probability measures

{p, :

T

i s a f i n i t e subset of [ O l l ] ) .

Exercise 8. By starting with the equality

ItW ex p (- ax /2)dx = ( 2 r / a ) ' f 2 2

( a > 0),,

show that

= ( 2 n c d / ( c+ $))'la - exp((-1/2)((a - b)'/(c + d ) ) ) , where a , b, c, d are strictly positive real numbers. Exercise 9. By using the result of Exercise 8, demonstrate the consistency of the above-mentioned family of measures

{p, :

T

i s a finite subset o f [ O , l ] ) .

266

C H A P T ER 13

Applying the Kolmogorov theorem to this family of measures, we get the probability measure p, on the product space We shall demonstrate below that the latter measure canonically induces the required Wiener measure on the space C[O, l ] C (in this connection, note that the initial measure p, also is called the Wiener measure on the product space ~['l']). In order to obtain the main result of this section, we need some simple but important notions from the general theory of stochastic processes. Let (E,S, p ) be a space endowed with a probability measure and let T be a set of indices (parameters). We shall say that a partial function of two variables H : ExT-tR ~ [ ' l l ] .

~ [ ' l l ]

is a stochastic process if, for each t E T, the partial function is a random variable on the basic probability space ( E , S, p ) . In this case, for any fixed e E E, the partial function H(e,.) : T - R is called the trajectory of a given process H , corresponding to e. Suppose that T is equipped with a a-algebra S' of its subsets, i.e, the pair (T, S') turns out to be a measurable space. We say that a stochastic process H is measurable if it (regarded as a partial function on E x T) is measurable with respect to the product a-algebra of S and S'. Exercise 10. Let us put E = T = [O, 11 and equip [O,1] with the standard Lebesgue measure A . Give an example of a nonmeasurable stochastic process H such that dom(H) = E x T and all trajectories H(e, .) (e E E ) and all random variables H(., t) (t E T ) belong to the first Baire class. Suppose that some two stochastic processes H and G are given on E x T. We say that they are stochastically equivalent if, for each t E T, the random variables H(., t ) and G(., t) are equivalent (i.e, coincide almost everywhere with respect to p ) . Stochastically equivalent processes have very similar properties and, as a rule, are identified. However, in certain problems of probability theory (e.g. in those where special features of trajectories of a given process play an essential role) such an identification cannot be done. Assume now that a set T of parameters is a topological space. We say that a stochastic process H : ExT-tR

267

N O N D I F F ER E N T I A B L E F U N C T I O N S

is stochastically continuous at a point to E T if, for each

E

> 0, we have

Further, we say that a process H is stochastically continuous if H is stochastically continuous at all points t E T . Note that if H I and H z are any two stochastically equivalent processes, then H I is stochastically continuous if and only if H z is stochastically continuous. Exercise 11. Suppose that the unit segment [0, 11 is equipped with the Lebesgue measure A . Give an example of a measurable stochastic process H with d o m ( H ) = [O,11 x [0, I], which is stochastically continuous but almost all its trajectories are discontinuous.

Lemma 1. Let T = [O,l] wath the usual topology and let

be a stochastic process. Then the following two conditions are equivalent: I ) H is stochastacally continuous; 2) for any E > 0, we have

Proof. Suppose that condition 1) is fulfilled. Fix E > 0 and 6 > 0. For each t E T, there exists an open neighbourhood V ( t ) o f t such that

The family { V ( t ) : t E T ) forms an open covering of T = [0,1]. Since [O, 11 is compact, there exists a Lebesgue number d > 0 for this covering, i.e, d has the property that any subinterval of [O,I] with diameter 2d is contained in one of the sets of the covering. Consequently, if

then t 1 E It

- d , t + d[ and, for some r E T , we get

Thus, for almost all e E E, we may write

268

CHAPTER 13

{e : IH(e,tl)

- H ( e , r ) J> &/2) U {e

:

J H ( e , t )- H(e,y)I > e/2)

and, taking into account the definition of V ( r ) ,we obtain

This establishes implication 1) j 2). The converse implication 2) trivial, and the lemma has thus been proved.

1) is

Exercise 12. Show that Lemma 1 holds true in a more general situation when T is an arbitrary compact metric space. Exercise 13. Let T = [O, 11 and let

be a stochastic process. Suppose also that, for some real number cu there exists a function

4

>

0,

: [O, 11 -t [O, +oo[

satisfying the following two conditions: 1) limd,o+d(d) = 0; 2) for all t and t' from [O,l], we have

Show that the process H is stochastically continuous. The simple result presented in Exercise 13 can directly be applied to the Wiener measure pw introduced above. Indeed, we have the basic probability space (R[OJl, S,pw) and the stochastic process W :

~

[

~

l

'

x [0, 11 -t R ]

canonically associated with p,, which is defined by the formula

In particular, we see that dom(W) = R[OI'] t l and tz from [O,1] such that

x [O,l]. Choose any two points

269

N O N DIF F ERE N T I A BL E F U N C T I O N S

According to the definition of p,, the two-dimensional distribution of the random vector (W(.,tl), W ( ~ l t 2 ) ) is given by the corresponding density

where, for all (xl, x2) E R 2 , we have Pt1,ta( X I , X Z= ) (1/2n)(tl(t2

- t1))-'/~ezp((-1/2)(x~/tl + (22 - ~ 1 ) ~ / (-t 2tl))).

Consider the random variable W(., t l ) - W(*,t2)e It is easy to see that the density p : R-tR

.

of this variable is defined by the formula p(x) = (2n(t2 - tl))'1/2exp(-x2/2(t2

- tl))

(x E R ) .

Indeed, this immediately follows from the general fact stating that if (fl, f2) is a random vector whose density of distribution is q(f1 , f 2 ) : R 2

then the density of distribution of fi -

+

f2

R,

is

where qfi-,a(x) = JR P(fl,h)(X+ I.I)d8

(x E R ) .

Exercise 14. Prove the fact mentioned above. Now, if t and t' are any two points from [O,l], we may write p,({e

-

E R['I~]: (W(e,t) W(e,t1)l > E ) )


0, we can write

> 0, there exists a finite family of reals

belonging to Q and satisfying the conditions: (1) the length of each segment [t;,tytl] is less than l l n ; (2) if t and t' belong to some segment [t?, t7+"+], then

Moreover, we may choose the above-mentioned families

in such a way that the following conditions will be fulfilled, too: (3) for any n E N \ {O), the set Q, is contained in the set Qntl; (4) Q = U{Q, : n E N , n > 0).

Now, let us put

E' = n{dom(H(., t))

:

t E Q).

Obviously, we have p(E1) = 1. Further, for each natural n function G,: ~'x[O,l]-tR

> 0, define a

by the equalities

Gn(e,1) = H(e, 1). Evidently, the partial function Gn is measurable with respect to the product a-algebra of S and B([O, I]). Furthermore, the series

is convergent for any point t E [O,l]. Hence, for each t E [0, 11, we get

almost everywhere in E (with respect to p, of course). Let us put

for all those pairs (e, t ) E El x T for which the above-mentioned limsup exists. In this way, we obtain a partial mapping

The definition of G implies a t once that G is a measurable stochastic process stochastically equivalent to H and, for any point t E Q, we have

Let now t be an arbitrary point from [O,11 \ Q. Then there exists an increasing sequence

such that

NONDIFFERENTIABLE FUNCTIONS

In virtue of the definition of G, we easily obtain

for any point e E E 1 n d o m ( G ( . t, ) ) . This completes the proof of the lemma.

Remark 5. The process G of Lemma 2 is usually called a separable modification of the original process H. Note that the existence of a separable modification of a given process can be established in a much more general situation than in that described by Lemma 2. For our further purposes this lemma is completely sufficient. More deep results may be found in [I131 and [121]. It is interesting to mention here that the general theorem concerning the existence of a separable modification of a stochastic process essentially relies on the notion of a von Neumann topology (multiplicative lifting). For details, see e.g. [I211 where such an approach is developed. Lemma 3. Let and (/?n)nEN be two sequences of st~ictly positive real numbers, such that

and let { f n : n E N) be a sequence of random variables on ( E , S , p ) satasfying the relataons ~ ( { Ee E : Ifn(e)l > a n } ) < Pn Then there exists a p-measure zero set A

c

(n E N ) .

E such that, for any point

s's convergent.

Proof. For each n E N , let us denote

Then, according to our assumption,

Let us put

A = n n c ~ ( U r n Ern>nAm). ~,

CHAPTER

13

Then we obviously have p(A) = 0.

Take any point e from E \ A. There exists a natural number k for which

This means that, for each natural m

is fulfilled. Hence the series

> k, the inequality

CnEN Ifn(e)l is convergent.

Lemma 4. Let H be a stochastic process such that d o m ( H ) = E x [0, 11 and

+

for all t E [0, 11 and t r E [0, 11, where d > 0 is some fixed constant. Then there exists a stochastic process G satisfying the relations: 1) G and H are stochastically equivalent; 2) G is measurable; 8) G is separable with a set of separability

4)

for any point t E Q , we have

5) almost all (wath respect to p) trajectories o f G are continuous realvalued functions defined on the whole segment [0,1]. Proof. First of all, we may write

+

for any t E [O, 11 and t r E [0, 11. This immediately implies that H is stochastically continuous. Applying Lemma 2, we can find a process

satisfying relations 1) - 4). Indeed, relations 1) - 3) are satisfied in virtue of Lemma 2, and relation 4) is valid since d o m ( H ) = E x [O,l]. Let us denote

275

NONDIFFERENTIABLE FUNCTIONS

where k and m are assumed to be natural numbers. Obviously, random variable. Furthermore, we have

a,

is a

In view of Lemma 3, the series

is convergent almost everywhere in E , i.e, there exists a p-measure zero set A such that @m(e) < +m rn€N

C

for all elements e E E \ A. Now, we fix n E N and easily observe that if t E [O,l], t' E [O,1] and It - t'I < 2-", then, for some natural k, the number k/2" is less than or equal to 1 and

Evidently, IG(., t)

- G(*,t')l < (G(s,t) - G(., k/2")( + IG(., t') - G(., k/2")(.

But if, in addition, t E Q and t' E Q , then it can directly be checked that

which yields the relation

Utilizing the separability of G, we infer that there exists a p-measure zero set B having the following property: if e is an arbitrary element from E \ (A U B) and t and t' are any two points such that

then

But we know that, for e E E\(AUB), the series C, N cPrn(e)is convergent. Thus, we conclude that the trajectory G(e, .) is uniformly continuous. This immediately implies that G(e, .) is a restriction of a continuous real-valued function defined on 10, I]. So we may extend G to a new process in such a way that all trajectories of this process, corresponding to the elements from E \ ( A U B), turn out to be continuous on [0,1]. It can easily be seen that the new process (denoted by the same symbol G) is separable and measurable as well. Indeed, the separability of G holds trivially and the measurability of G follows from the fact that G is measurable with respect to e E E and is continuous with respect to t E [O, 11. Lemma 4 has thus been proved. We now are ready to establish the following result.

Theorem 3 . The W i e n e r measure pw induces a Borel probability naeasure p on the space C[O, 11, with propertaes analogous t o the corresponding properties of pw.

Proof. Indeed, we have the probability measure space

and the standard Wiener process W = ( p ~ . t ) ~ for ~ [ this ~ , ~space. ] In view of the preceding lemma, there exists a process G for the same space, such that: 1) W and G are stochastically equivalent; 2) G is measurable; 3) G is separable with a set of separability

4) for any point t E Q, we have W(.,t) = G(.,t); 5) almost all trajectories of G are continuous real-valued functions on [O, 11.

277

NONDIFFERENTIABLE FUNCTIONS

Let El denote the set of all those elements e E E = for which the trajectory G(e, .) is continuous on [O,l]. Obviously, pw(E1)= 1. Define a mapping (b : E1+CIO,l] ~

[

~

l

l

]

by the formula (e E E'). 4(e) = G(e, Observe that (b is measurable with respect to pw (this fact easily follows from the result of Exercise 6). So we can put s)

Since p is a homomorphic image of p,, we have P(X) = P W ( { E~ E : G(e, .) E X ) ) for each Borel subset X of C[O, 11. In particular, if a any two points of [0,1], then ~ ( { fE C[O, 11 : If(t)-f(t')l

= p, ({e E E

< a)) = pw({e E E

:

> 0 and t and t' are

IG(e,t)-G(e,tl))

< a))

1

: W(e, t) - W(e, tl)l < a)).

In a certain sense, we may identify p and p,. So it will be convenient to preserve the same notation pw for the obtained measure p. Irother words, we consider p, as a Borel probability measure on the space C[O, 11. At last, we are able to return to the question of the differentiability of continuous real-valued functions on [O,1] (from the point of view of p,). Namely, the following statement is true.

Theorem 4. Almost all (with respect to p,) finctions from C[O, 11 are nondafferentaable almost everywhere on [O, 11 (with respect to A). Proof. Let us introduce the set

D = {(f,t) E C[O, 11 x [O,1] : f is differentiable at t). It can easily be checked that the set D is (p, x A)-measurable in the product space C[O, 11 x [ O , l ] . So, taking into account the F'ubini theorem, it suffices to show that, for each t E [O, I], the set

is of pw-measure zero. In order to do this, we first observe that the inclusion

CHAPTER 13

278

is satisfied. Hence, it suffices to prove, for each n E N , that

where Dt,n

= {f E C[O,lI :

/imsupl,l,~+

If

(t

+ r ) - f(t)J/lrl < n ) .

Further, one can easily verify that

and Dt,n,a,r =

{f E C[O, 11

:

If (t + r) - f (t)I/IrI < n).

Thus, it remains to demonstrate that But, for any r satisfying 0 < lrI < 6 , we may write

(&r)''/'

/nli'l/'

e ~ p ( - ~ ~ / 2=) 0d (~1 ~ 1 ~ / ~ ) .

-nlrll/a

This immediately implies the desired result, since lrI arbitrarily small.

> 0 can be chosen

Remark 6. A more general result obtained by Wiener and LBvy holds true; namely, they proved that almost all (with respect to p w ) functions from C[O, 11 are nowhere differentiable on [O,l]. Briefly speaking, almost all trajectories of the modificated Wiener process are nowhere' differentiable on [ O , l ] . For extensive information concerning the relationships between stochastic processes and Brownian motion, we refer the reader to the fundamental monograph by Levy [96]. Remark 7. As mentioned earlier, the standard Wiener process is a very particular case of a Gaussian process. Gaussian processes form a natural class of stochastic processes which have many interesting properties (see, e.g., [36], [113], [121], [146], [23]) and are important from the point of view of numerous applications.

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Index

absolutely continuous function, 62 absolutely nonmeasurable function, 173 admissible functional, 182 almost symmetric function, 155 analytic set, 16 analytic space, 16 antichain, 7 (A)-operation, 16 approximate derivative, 29 approximately continuous function, 84 approximately differentiable function, 85 Axiom of Choice, 1 Axiom of Dependent Choices, 6 Axiom of Projective Determinacy, 257 axioms of set theory, 1

Baire property, 5 Baire property in the restricted sense, 20 Banach-Kuratowski matrix, 203 Banach condition, 205 Banach-Tarski paradox, 136 Bernstein set, 119 bijection, 12 binary relation, 5 Blumberg space, 100 Borel isomorphism, 17 Borel mapping, 17 Borel measure, 16 Borel set, 16

INDEX

Bore1 a-algebra, 16

Cantor discontinuurn, 38 canonical Baire space, 47 Cantor space, 38 Cantor type function, 34 Caratht5odory conditions, 220 cardinality of the continuum, 4 cardinal number, 3 Cartesian product, 3 Cauchy functional equation, 137 Cauchy problem, 237 chain, 7 class of sets, 2 closed graph, 20 C-measurable mapping, 232 compact topological space, 15 continuous mapping, 17 Continuum Hypothesis, 4 convex function, 149 countable chain condition, 8 countable form of the Axiom of Choice, 15 countable set, 3 C-set, 232 cylindrical a-algebra, 262

Darboux property, 57 decreasing function, 55 density point, 29 density topology, 19 derived number, 58 difference of sets, 2 dihedral angle, 158 disjoint family of sets, 13 domain of a partial function, 11 duality between two vector spaces, 51

INDEX

element of a set, 1 empty set, 2 equivalence relation, 6 Euclidean space, 5 extension of a partial function, 11

family of sets, 2 finite set, 3 first category set, 19 function, 10 functional graph, 10

Gaussian distribution, 261 generalized Cantor discontinuum, 43 Generalized Continuum Hypothesis, 4 generalized derivative, 29 generalized integral, 29 generalized Luzin set, 171 generalized Sierpiriski set, 171

Hamel basis, 5 Hausdorff metric, 46 Hilbert cube, 16 Hilbert dimension, 179 Hilbert space, 179

increasing function, 55 infinite set, 3 initial condition, 237 injection, 11 intersection of sets, 2 iterated integrals, 199

INDEX

Jensen inequality, 149

largest element, 9 Lebesgue measure, 5 limit ordinal, 3 linearly ordered set, 9 Lipschitz condition, 77 local maximum, 77 local minimum, 77 lower semicontinuous function, 21 lower semicontinuous set-valued mapping, 21 Luzin set, 19

mapping, 10 Marczewski set, 120 Martin's Axiom, 8 mathematical expectation, 260 maximal element, 9 measurable stochastic process, 266 membership relation, 1 minimal element, 9

normal distribution, 261 normal topological space, 101 nowhere dense set, 19 nowhere approximately differentiable function, 83 nowhere differentiable function, 27

one-to-one correspondence, 12 ordered pair, 2 ordinal number, 3 ordinary differential equation, 217 oscillation of a function, 106

INDEX

partial function, 10 partial mapping, 10 partial order, 7 partially ordered set, 7 partition associated with an equivalence relation, 7 Peano type function, 38 perfect set, 27 Polish space, 16 polyhedron, 157 principle of condensation of singularities, 253 probability distribution, 258 probability space, 258 projective set, 18

quasicompact topological space, 14

Radon measure, 17 Radon space, 17 random variable, 258 range of a partial function, 11 real line, 4 regular cardinal, 3 regular ordinal, 3 relation, 1 restriction of a partial function, 11

selector, 14 semicompact family of functions, 187 separable measure, 259 separable stochastic process, 270 set, 1 set-valued mapping, 12 Sierpidski-Erdos Duality Principle, 45

INDEX

Sierpiriski set, 159 Sierpiriski's partition of the plane, 197 Sierpiriski-Zygmund function, 95 simple discontinuity point, 56 singular cardinal, 3 singular monotone function, 55 singular ordinal, 3 smallest element, 9 Steinhaus property, 114 stochastic process, 266 stochastically continuous process, 266 stochastically equivalent processes, 266 strictly decreasing function, 55 strictly increasing function, 55 strongly measure zero set, 166 subset, 1 sup-continuous mapping, 217 sup-measurable mapping, 217 surjection, 12 Suslin space, 16 symmetric derivative, 28 symmetric difference of sets, 2 symmetric group, 12

topological group, 24 topological structure, 15 topological weight of a space, 23 totally imperfect set, 119 transformation of a set, 12

Ulam transfinite matrix, 142 uncountable set, 3 union of sets, 2 universal measure zero space, 164 universal object, 54 upper $emicontinuous function, 21

INDEX

vector space, 5 Vitali covering, 29 Vitali partition, 118 Vitali set, 118 von Neumann topology, 93

weakly sup-measurable mapping, 217 well ordered set, 9 Wiener measure, 258 Wilczyriski's topology, 94

Zermelo-Fkaenkel set theory, 1 Zorn Lemma, 9