Measure-Theoretic Notions of Prevalence

Measure-theoretic notions of prevalence

Hongjian Shi

B. Sc., Henan Normal University, China, 1984 M. Sc., Simon Fraser University, Canada, 1995

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@ Hongjian Shi 1997 SIMON FRASER UNIVEXSITY October 1997

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Haar null sets (or shy sets) play an important role in studying properties of function spaces. In this thesis the concepts of transverse measures, left shy sets and right shy sets are studied in general Polish groups, and the basic theory for the left shy sets and right shy sets is established. In Banach spaces, cornparisons of shy sets with other notions of srnall sets (e-g., Aronszajn ndsets and Gaussian null sets in Phelps sense) are made, and examples of non-implications are given. Their thickness and preservations of various kinds of null sets under isomorphisms are investigated. A new description of shy sets is given and is used to study shy sets in finite dimensional spaces. The theory is applied to a number of specific function spaces. Some known typical properties are shown to be also prevaient. Findy, the notions of left shy sets and right shy sets are applied to study properties of the space of homeornorphisms on [O, 11 that leave O and 1 fixed. Several interest ing examples are given of non-shy sets which are null in the sense of some other notions. Examples of left-and-right shy sets are found which can be decomposed into continuum many disjoint, non-shy sets in %[O, 11. This shows that the O-ideal of shy sets of the non- Abelian, non-locdy compact space 'H[O, 11 does not satisfy the countable chah condition. Dedicat ion

To m y Parents and Guina, Tianyu and Jeanne Acknowledgments

1 wish to express my deepest gratitude and appreciation to my senior supervisor Dr. B&n S. Thomson for his encouragement, guidance and patience in the preparation of this thesis. His many constructive suggestions and ideas are very helpful in the studying of topics in this thesis. Without his help in many ways it would not be possible for this thesis to be in its present form. 1 would like to thank Dr. Andrew M. Bruckner and Dr. Jonathan M. Borwein for careful reading of earlier drafts of the thesis. and many suggestions for improvements of this thesis. Thanks aIso go to Dr. Dave Renfro for information about the Besicovitch set and related material. Finally, 1 would tike to thank Dr. Brian S. Thomson again, Department of Math- ematics & Statistics and Simon Fraser University for their financial support during my study in Simon Fraser University. Contents

Title ...... 1

* .. Abstract ...... 111 Dedication ...... iv Acknowkdgrnents ...... v ... List of Figures ...... vrii Contents ...... ix 1 Introduction ...... 1 2 Shysets ...... 6 2.1 Introduction ...... 6 2.2 The transverse notion ...... 7 2.3 Nul1 sets in a separable Banach space ...... 9 2.4 Gaussian null sets ...... 13 2.5 Sets of Wiener measure in Co[O,11 ...... 15 2.6 Non-measure-theoretic variants ...... 19 2.7 Basictheory ...... 21 2.8 The completeness assumption ...... 24 2.9 Haar null sets in Polish groups ...... 24 2.10 Preservations of nul1 sets under isomorphisms ...... 33 2.1 1 Measurability issues ...... 36 2.12 Extension to non-separable Abelian groups ...... 38 2.13 Extension to semigroups ...... 41 2.14 Why not measure zero sets of a single measure ...... 44 2.15 Classification of non-shy sets ...... 48 2.16 Prevalent versus typical ...... 55 2.17 Fubini's theorem ...... 58 2.18 Countable chah condition ...... 59 2.19 Thickness of non-shy sets ...... 61 3 Probes ...... 63 3.1 Introduction ...... 63 3.2 Finite dimensional probes ...... 64 3.3 Elementary linear arguments ...... 66 3.4 Compact sets ...... 70 3.5 b1easure.theoreticarguments...... 73 4 Prevdent properties in some function spaces ...... 77 4.1 Introduction ...... 77 4.2 Continuous, nowhere differentiable functions ...... 78 4.3 Prevalent properties in C[a,b] ...... 80 4.4 Prevalent properties in bd, bVB1, bB1 ...... 83 4.5 Prevalent properties in D[a,b] ...... 85 4.6 Prevalent properties in BSC[a,b] ...... 87 4.7 MultiplicativelyshysetsinC[O,l] ...... 93 5 Space of automorphisms ...... 99 5.1 Introduction ...... 99 5.2 Space of automorphisms ...... 101 5.3 Sorne examples of prevalent properties in %[O, 11 ...... 104 5.4 Right transverse does not imply left transverse ...... 110

vii 5.5 A set to which the measure p is left-and-right transverse ... 114 5.6 A compact set argument ...... 117 5.7 Examples of non-left shy. non-right shy sets ...... 120 5.8 Examples of left-and-right shy sets ...... 124 5.9 Non-shy sets that are of the first category ...... 127 5.10 Non-prevalent properties that are typical ...... 130 5.11 Corresponding results at x = 1 ...... 132 5.12 A non-shy set that is left-and-right shy ...... 139 5.13 %[O, 11 does not satisfy the countable chain condition ..... 147 5.14 Functions with infinitely many fixed points ...... 145 5.15 Concluding remarks ...... 150 Bibliography ...... 152 Glossary ...... 158 List of Figures

2.1 The relations of non-shy sets in general ...... 51 2.2 The relations of non-shy sets in Rn ...... 55

5.1 The construction of the automorphisms g and k ...... 142 Chapter 1

Introduction

A glossary is made at the end of this thesis. The words are mainly from Red Andysis, Measure Theory, Topology and Functional Analysis. Some words (e-g., left shy and probe) are either new or in the literature and their definitions are given in the context of the thesis. Some terms (e-g., Banach space and first category) without explanations can be found in the text books [Il] and [48]. Since 1899 when R. Baire [35, pp. 481 introduced the definitions of first category, second category of sets, and established the Baire category thearem, the Baire category t heory has been widely used to characterize properties of sets, especially typical properties, and to show the existence of some pathological functions that are usuaily difficult to construct. Here a typical property is a property which holds for all points in a cornplete metric space except for a set of the first category. There are some restrictions in applications of the Baire category theory. For example, when we discuss a probabilistic result on the likelihood of a given property on a function space, the Baire category theory just provides us the topological structures; Lebesgue measure theory is not subsumed by the Baire category theory. Further in a Banach space it cm be shown that there are Lipschitz functions which are not differentiabte on a dense Ga set, thus the topological category theory is not well suited for differentiability theory of Lipschitz functions, Frequently we need a notion of measure zero in a general setting, that can be used much as sets of the first category have been used to describe properties as typical. In 1972 Christensen [12] first generalized the concept of sets of Haar measure zero on a locally compact space to an Abelian Polish group which is not necessarily locally compact. Such zero sets have been called Haar zero sets, Haar nul1 sets, Christensen nul1 sets or shy sets [12], [56], [SI, [27]. We state it in the foilowing definition. A uniuersally measurable set S in an Abelian Polish group G is said to be a Haar zero set if there is a Borel probability measure p on G such that p(S + x) = O for al1 XEG. Christensen [14] showed that this concept of Haar zero sets is very successful in many respects, for example in extending Rademacher's classical theorem that a locally Lipschitz mapping £rom Rn to Rm is differentiable almost everywhere to mappings between general Banach spaces. In 1980 T. Tops~eand J. Hoffman-Jmgensen [56] extended the notion of Haar zero by Christensen to general topological semigroups. In 1994 Brian R. Hunt, T. Sauer and J. A. Yorke [27] rediscovered the concept of Haar zero sets in Banach spaces. They called such sets shy sets. (In 1271 ail subsets of a Borel shy set are also termed shy.) In [27] the complement of a shy set is called prevalent, and a property in a metric space is said to be prevalent if it holds for al1 points except for a shy set, or it is said that almost every function satisfies such property. In [27] the notion of shy sets is used to study properties of certain classes of functions in function spaces. For exarnple, it is easy to show that almost every sequence (ai}& in ér has the property that Czl ai diverges. An attractive result by Hunt [28] is that aimost every function in the space of continuous functions is nowhere differentiable. It has been popular to study typicd properties of function spaces since Banach (31 and Mazurkiewicz [40] applied the Baire category theorern to CHAPTER 1. INTROD UCTiûN give existence proofs of nowhere differentiable, continuous functions independently. The study of prevalent properties of function spaces offers an interesting contrast to the typical properties. Usually, it is more difficult to show a property is prevalent than to show a property is typical- A typical property may not be a prevalent property and vice versa. This thesis is devoted to systematically studying shy sets, left shy sets and right shy sets in non-separable Banach spaces and general Polish groups, and investigating whether a typical property is prevalent, and showing that some known sets characterized by other notions are shy or non-shy. In Chapter 2 we introduce various transverse notions, and the concepts of left shy

sets, right shy sets in general Polish groups. The basic theory for the left shy sets and right shy sets is established, showing that, in a general Polish group, the couutable union of left shy sets is again left shy, and open sets are neither left shy nor right shy. There are several extensions of the concept of shy sets to non-separable Banach spaces (see Christensen [14], Topsae and Hoffman-Jargenson [56], Hunt et al. [27], Borwein and Moors [a]). The latter three are equivalent. We compare shy sets, in a Banach space, with s-null sets (see Section 2.3), Aronszajn null sets [2], Preiss-Tiier null sets [46] and Gaussian nul1 sets in Phelps sense [45], giving examples of non-implications. Also we discuss their thickness of various kinds of nul1 sets, and their preservations under isomorphisms. In [18] Dougherty has classified non-shy sets into eight types, and mentioned some examples of non-implications, while we give examples in details for al1 non-implications, and exact characterizations for some non-shy sets. In [27] Hunt, Sauer and Yorke introduced the definition of a probe. A finite dimensional subspace P of a Banach space is cded a probe for a set T or its complement if Lebesgue measure supported on P is transverse to a universaliy measurable set which contains the complement of T. To show a set is shy we often try to find some probe of this set. In Chapter 3 we use probes and elementary linear arguments to study some simple prevalent properties in function spaces. Also we use the dimensions of CHAPTER 1. INTRODUCTION probes to give a new description of shy sets as foliows. A shy set S G X is said to be m-dimensional if S has a m-dimensionai probe but no n-dimensional probe for n < ml and S is said to be infinite-dimensional if S has no finite dimensional probes. We find that, in finite dimensional spaces, this new concept for shy sets is related to Kakeya problems and Besicovitch sets (see [21] and [22]),and so some results are obtained. In [12], [56] and [27] it is shown independently that compact sets are shy in Abelian Polish groups, Abelian sernigroups and Banach spaces. A method using results from functionai analysis is given to show that certain sets including compact sets are shy in Banach spaces. In Chapter 4 several hown typical properties are shown to be also prevalent. In 1101 Bruckner and Petruska showed that, in the spaces F = bd, bDB1, bB1 of bounded approximately continuous functions, bounded Darboux Baire 1 functions and bounded Baire 1 functions equipped with the supremum norm, for any a-finite Borel rneasure p on [O, 11, the typical functions are discontinuous p almost everywhere on [O, 11. We show that, in the spaces F = bd, bZ)B1, bBL, for any a-finite Borel measure p on [O, 11, the prevalent functions are also discontinuous p almost everywhere on [O, 11. In [50] it was shown that, in the space BSC[a, b] of bounded symmetrically continuous functions on [a, b] equipped with supremum norm, the typical functions have c-dense sets of points of discontinuity. Here we show that the prevalent functions of BSC[a,b] also have c-dense sets of points of discontinuity. On the space C[O,11 of continuous functions we can aiso impose the operation of multiplication of functions so that C[O,11 becomes a semigroup. In Chapter 4 we also study the multiplicatively shy sets in the space C[O,11 of continuous functions, and discuss the relations of multiplicatively shy sets and additively shy sets. For example, we show that the set of continuous functions on [O, lj with at least one zero is not additively shy, but is multiplicatively shy in C[O,11. We use R[O, 11 to denote the space of homeomorphisms on [O, 11 that leave O and CHAPTER 1. INTRODUCTION

I hed. In Chapter 5 we show that, in H[O, 11, there exists (2) a Borel probability measure p that is both left transverse and right transverse to a Borel set X, but is not traasverse to X. (2) a Borel probability measure p that is left transverse to a Borel set X, but is not right transverse to X. (3) a Borel probability measure p that is right transverse to a Borel set X, but is not left transverse to X. We also find examples of left-and-right shy sets which can be decomposed into continuum many disjoint, non-shy sets in X[O, 11. This answers the problem (Po)posed by Jan Mycielski in [41]:Does the existence of a Borel probabili ty measure left transverse to a set Y imply that the set Y is shy in a non-locally compact, cornpletely metrizable group? From this we conclude immediately that the 0-ideai of shy sets in %[O, I] does not satisfy the countable chah condition. In [25] and [26] Graf, Mauldin and Williams defined a Borel probability measure Pa on Rio, 1) £rom a probabilistic point of view, and studied whether some interesting sets are nuli under this measure P,. In Chapter 5 we use Our notions of shy sets, Ieft shy sets and right shy sets to study some of the sets discussed by Graf, Maddin and Williams. For example, Graf, Mauldin and Williams showed that, for any m E (O, 11 and 1 E (1, +cm),the sets {h E H[O,l] : h(x) 2 mz) and {h E 'H[O,l] : h(x) 5 fa) are null under the measure Pa.We show that these two sets axe neither shy nor prevalent. Cornparisons of prevdence results are made with some typical results, and some known results in [25] and [26]. Chapter 2

Shy sets

2.1 Introduction

Christensen [12], Topsae and Hoban-Jargensen [56], and Hunt et al. [27] studied shy sets in Abelian Polish groups, topological semigroups and Banach spaces independently. In this section we study left shy sets and right shy sets in general Polish groups, establishing the basic theory for left shy sets and right shy sets. We compare shy sets with other notions of nul1 sets in Banach spaces. For example, in the Banach space of continuous functions on [O, 11 that are zero at x = 0, the set of continuous, nowhere Hdder continuous functions with exponent a, O < a! < 1/2, bas Wiener measure zero, but this set is prevalent (see Hunt [28]). Christensen [12] asked whether any collection of disjoint universally measurable non-shy sets must be countable. (This property of a a-ideal is cailed the countable chain condition; see page 59.) Dougherty [la] answered this question in the negative by giving an example. Solecki [52] showed that a Polish group admitting an invariant metric satisfies the countable chain condition iff this group is locally compact. CHAPTER 2. SHY SETS

The transverse notion

In al1 of our discussions in this section we suppose that X is a linear topological space and we assume we have been given a set S C X that is a universdy rneasurable set in X. Some of the tenninology applies as well to an Abelian topological group since, for some of the terminology, we use only the additive group structure to define the concepts. Most of oui. discussion in the sequel, however, will be in the setting of a Banach space, and, occasionally, in a non-Abelian Polish group. Our goal is to define a measure-theoretic notion of smallness analogous to the topological notions of first category and a direct generalization of the notion of a set of Lebesgue measure zero in finite dimensional spaces. Al1 measures in the sequel are assumed to be defined on the Borel subsets of the space and can be extended to aU universdy measurable subsets.

Definition 2.2.1 We say that a probability measure p on X is transverse to a set S if @+Y) = 0 for all y E X. Thus p assigns zero measure to S and to every translate of S.

Note that there is no interest in measures here that are not diffuse. If ~((xo})> O for some point xo E X then p cannot be transverse to any non-empty set. In fact, if p is transverse to some non-ernpty set A t hen p(A +y) = O for al1 y E X. C hoose yo E A then xo E A - y0 + xo. So O < p({xO)) 5 p(A - y0 + xo) = O. This is impossible.

Some authors (e-g., [SI) cdthe measure p a test-measure for S if C( is transverse to S. We could also adapt language from [27] and cdp a probe in the sense that it is used to test or prove or "proben the measure-theoretic nature of the set S. In many applications the construction of a transverse measure for a set S (if there is one) can be doue by a simpler device. Often a measure p can be fomd that CHAPTER 2. SHY SETS is supported on a finite dimensional subspace or on some simple compact set. For example if every line in the direction x for some x E X mets S in a set of one dimensiona1 linear measure zero then a probability measure p transverse to S cm be constructed by writing

p(E) = Xl({t E [O, 11 : tx E El)

where Al denotes one-dimensional Lebesgue measure.

Lemma 2.2.2 A set S has the property that every line in the direction x (x E X) meefs S in a set of linear dimensional measure zero and only ij the probability rneasure

p(E) = h({tE [O, 11 : tx E E)) is transverse to S.

Proof. For any y EX,

,@+Y) = Xl({t E [OJ]: tx E S+y)) = Xl({t E [0,1] : fa:-y E SI).

Thus the result foLlows.

The above lernma leads to the foliowing definition:

Definition 2.2.3 An element x E X is said to be transverse to a set S if

is Lebesgue measure zero for every y E X.

Again we can say that x is a test or a probe for S, using Ianguage that other aut hors have found convenient. It is convenient also to express this fact in a vaxiety of manners. Thus we Say that the subspace spanned by x is transverse to S or that the interual

[O, x] = {tx : t E [O, 11) CHAPTER 2. SHY SETS is transverse to S. By extension of this a collection {xl,xz, .. . , xk) of lineady independent elements of X is also said to be transuerse to S if

{(tl,tz,.--,tk) ER': tlz1-k ---+tktk+y~S) is of k-dimensional Lebesgue measure zero. Also we have a similar result as in the one-dimensional case. The proof is simiIar.

Lemma 2.2.4 A set S has the property that every translate of the k-dimensional subspace generated b y the independen! elements {XI,. . . , xh} (XI,. .- ,xk E X) meets S in a set of k-dimensional Lebesgue measure zero if and only if the measure

p(E) = Xr({(tl,t,,. .. tk)E [O. 1Ik : tlxl + -.. + E E}) is transverse to S, where Ak denotes the k-dimensional Lebesgue measure.

Occasiondly linear arguments fail to produce a transverse measure and one needs to seek other compact sets on which the measure is supported. Let F : [O, II -+ X be a continuous function. Then C = F([O,11) is a compact set and we Say C is transverse to S if (t é [O, 11 : F(t)E 5' + y) is Lebesgue measure zero for every y E X. In this case a probability measure transverse to S can be defined by

since p(S + y) = O for every y E X. We wiU give some applications in Section 3.4, Section 4.7 and Section 5.3.

2.3 Nul1 sets in a separable Banach space

The definitions in this section are generalizations of the notion of a set of Lebesgue measure zero in n-dimensional spaces to subsets of a separable Banach space. CHAPTER 2. SHY SETS

We assume t hat S is a universaily measurable subset of a separable Banach space X. The following definition is a version from [12] in a separabIe Banach space. Its extensions to non-separable Banach spaces and Polish groups can be found in Sec- tion 2.12 and Section 2.9 respectively.

Definition 2.3.1 A universaily measurable subset S of a separable Banach space X is said to be a Christensen nul1 set (shy) if there is a Borel probability measure on X that is transverse to S.

This concept is our central concern throughout. The article [27] has popularized the term sh y for Borel sets that are Christensen null and all their subsets and we shall make use of this term too. The complernent of a shy set is said to be prevalent . In his original paper Christensen [12] cailed shy sets Haar zero sets and other authors have used the term Haar nul1 sets. The remaining definitions in this section are narrower than Definition 2.3.1. Each of the following classes of sets is a proper subset of the shy sets in general, but may coincide in speciai cases.

Definition 2.3.2 A universally measurable set S in a separable Banach space X is said to be an s-nul1 set if there is a partition of S into universdy measurable subsets

i=l such that for each i there exists an element e; in X that is transverse to Si.

The following definition is from Aronszajn [2].

Definition 2.3.3 A Borel set S in a separable Banach space X is said to be an Aronszajn nul1 set (Borel sense) if for every sequence {el, ez, e3, .. .) whose linear span is dense in X there is a partition of S into Borel subsets 00 CHAPTER 2. SHY SETS

such that for each i the element e; is transverse to Si.

It is convenient for contrast to give a similar definition motivated by that of Aron- szajn [2]. The original paper studies the Borel version of Definition 2.3.3-we give a universally measurable version as wel for comparison.

Definition 2.3.4 A set S in a separable Banach space X is said to be an Aronszajn nul1 set (uniuersally measurable sense) if for every sequence {el, ez, es, - .. ) whose linear span is dense in X there is a partition of S into universaDy measurable subsets

such that for each i the element ei is transverse to Si.

Perhaps the nmowest version of a nul1 set in this spirit is that from Preiss- TiSer [46].

Definition 2.3.5 A universally measurable set S in a separable Banach space X is said to be a Preiss-TiSer null set if every element of the space is transverse to S.

Definition 2.3.1, 2.3.2 and 2.3.5 al1 have similar forms in the Borel sense, Le., replacing the universally measurable condition by the requirement that the sets are Borel. The above definitions in the universdly measurable sense are parallel to the definitions in the Borel sense but they are different. A Christensen null set need not be Borel. The following simple example can show this.

Example 2.3.6 In R there is a Christensen ndset which is not Borel. We know that in R there is an analytic set B which is not Borel (see [Il, pp. 4921). So there is a Borel set F 2 B such that B \ F is Lebesgue measure zero. Thus the set B \ F is universally measurable and so is Christensen nuil but it is not Borel from the following statement. CHAPTER 2. SHY SETS

In the space Rn a universally measurable set is Christensen null iff it has n-dimensional Lebesgue measure zero.

The above result wili be given and proved in the next section. From the above it is easy to see that, in the real line R, Preiss-TiSer nufl sets, Aronszajn null sets, s-nul1 sets, and Christensen null sets are equivalent to Lebesgue measure zero sets that are universdy measurable. However, in Rn (R 2 2) the ciass of Preiss-TiSer null sets is much smaller than the class of Lebesgue measure zero sets that are universally measurable. This is because any proper subspace of Rn (n 2 2) is Lebesgue measure zero but is not Preiss-Tiger null. Wenow compare in generd t hese different sets in the universally measurable sense. The foilowing inclusions are obvious:

Preiss-TiSer nul1 a Aronszajn nul1 + s-nul1 => Christensen null.

The following are two examples to show that, in general, the first two inclusions are proper.

Exarnple 2.3.7 In the plane R2 the set {Xel : X E W) is obviously Aronszajn null where el = (1,O). However it is not Preiss-Tiljer null since the element el itself is not transverse to it.

Exarnple 2.3.8 There is an s-nul1 set that is not Aronszajn null. In fact, in Exam- ple 2.4.4 the set K is compact in the infinite dimensional sepaxable Banach space X. So it is s-nuil (see Theorem 3.4.2) . Note the set K is not Gaussian null in Phelps sense (see Gaussian null sets in Section 2.4 and Example 2.4.4). Since an Aronszajn null set must be a Gaussian ndl set in Phelps sense [45], thus the set K is not Aron- szajn null. Since an s-null set is Christensen null so the set K is also an exarnpIe of a Christensen nuii set that is not Aronszajn null. CHAPTER 2. SHY SETS

Remark. In the plane IR2 a Christensen ndset is an s-null set (see Theorem 3.2.4). However we do not know whether there is a Christensen null set that is not s-null in

Rn (R > 2).

PROBLEM 1 h a separable Banach space is there a Christensen nul1 set that is not s-null?

2.4 Gaussian null sets

The main purpose of this section is to cornpâre the null sets in Section 2.3 with two other kinds of smd sets, Gaussian null sets in Phelps sense and Gaussian null sets in the ordinary sense. For convenience we require Gaussian measures defined on Borel sets. The following three dehitions are reproduced from [45].

Definition 2.4.1 A non-degenerate Gaussian measure p on the real line B is one having the form p(B) = (2x4-' /B exp[(-26)-'(t - t~)~]df, where B is a Borel subset of W and the constant b is positive. The point a E W is called the mean of p.

Definition 2.4.2 A Borel probabiiity measure A on a Banach space X is said to be a Gaussian measure of mean xo if for each f E Xu,f # O, the measure p = X O f-' is a non-degenerate Gaussian measure on the real line P as above, where a = f (xo).

Definition 2.4.3 (Phelps) A Borel subset B of a separable Banach space X is called a Gaussfan nul1 set in Phelps sense if p(B) = O for every non-degenerate Gaussian measure p on X. CHAPTER 2. SHY SETS

Recall that a Borel subset B of a separable Banach space X is an ordinary Gaussian null set if there is a non-degenerate Gaussian measure p such that p(B) = O. Here we also Say that the set B is Gaussian null in the ordinary seme. It is easy to see that a Gaussian null set in Phelps sense is Gaussian nuli in the ordinary sense. However the converse is not tme. We know that there does not exist a positive a-finite measure on l2 (the space of square summable sequences) whose nul1 sets are translation invariant (see Theorem 2.14.3 or [51, pp. 1081). So for any non-degenerate Gaussian measure p there is a Borel set S C ez such that p(S) = O but p is not transverse to S. Thus the set S is not Gaussian null in Phelps sense. We compare the two kinds of Gaussian null sets with the null sets Section 2.3. A translate of a non-degenerate Gaussian measure is again such a mesure [34], so a Gaussian nu11 set in Phelps sense is Christensen nulI. In [45] it is shown that a Aronszajn nul1 set is Gaussian nd. Thus we have the foliowing implications:

Aronszajn null =+ Gaussian null in Phelps sense + Christensen null.

We know that, in Rn,Gaussian measures are mutuaily absohtely continuous with respect to the n-dimensional Lebesgue measure and that a Borel set in IRn is Chris- tensen null iff it is Lebesgue rneasure zero (see Theorem 2.7.5). Thus in Rn Gaussian nul1 sets in two senses and Christensen nuil sets in Borel sense are al1 equivalent to the Borel sets of Lebesgue measure zero. The following example from [43] shows that the second inclusion is proper in an infinite dimensional sepaxable Banach space.

Example 2.4.4 In an infinite dimensional separable Banach space X there is a compact set which is not Gaussian nul1 in Phelps sense. Let (w,) C X have dense linear span and satisfy llwnll + O, then its symmetric closed convex huil K is compact but is not Gaussian nuü. En fact, we define L : t2+ X by setting, for x = {x,) E &, CHAPTER 2. SHY SETS 15

It is clear that L is Iinear and has dense range. Let U denote the unit ball of e2. Then LU C K. Since K is the closed convex huil of the compact set {+w,)- u {O), it is compact. By the continuity of L we know that if p is any non-degenerate Gaussian meaçure on t2 then A = p O L-' is a non-degenerate Gaussian measure on X. Note

U Ç L-'K and so X(K) = (pO L-' )(K)= P(L-' K)3 p(U). It is known [47] that any non-degenerate Gaussian measure on & assigns positive measure to any non-empty open set. Thus A(K) > O and K is not Gaussiaa nul1 in Phelps sense. However the set A' is s-nul1 and of course Christensen null (see Theorem 3.4.2).

Wenow Ieave the following as an open probIem.

PROBLEM 2 In an infinite dimensional separable Banach space is a Gaussian nul1 set in Phelps sense necessady Aronstajn ndl?

2.5 Sets of Wiener measure in Co[O,l]

En this section we surnmarize some material from Kuo [34] on the Wiener measure in the space Co[O,11. We compare the nul1 sets, especialiy Christensen null sets with the null sets of zero Wiener measure on the space Co[O,11 of red-valued continuous functions x(t) with x(0) = O. Co[O,I] is a Banach space with the supremum norm. Let R denote the Borel o-field of Co[O,LI. A subset I of Co[O,11 of the following fom

where O < tl < t2 < < t, 5 1 and E is a Borel sübset of Rn, will be called a cylinder set. CHAPTER 2. SHY SETS

Definition 2.5.1 Let I be a cylinder set. Define

The unique extension of w to the Borel a-field R is cded the Wiener rneasure on top, 11-

Remark. The measure w does indeed possess a unique extension to the O-field R (see (34, Theorem 3.2, pp. 431). When n = 1 this Wiener measure is a Gaussiân measure of mean O on Co[O,11 (see Kuo [34, pp. 1281).

Exarnple 2.5.2 If0 < t 5 1, then from the definition of w

Example 2.5.3 Let O < s < t 5 1 be fixed and consider the randorn variable x(t) - x(s). Let E C W2 be the set E = {(x, y) : a 5 1: - y 5 6). Then

W({X : a I~(t) - ~(s) 5 6)) = w({x : (x(t),x(s)) E E))

By making a change of variables and simpiifying we have

~({x:a 5 z(t)-x(s) 5 b}) = J- J- Now we discuss the Wiener measures of the following sets for a > O,

3a = a(x) such that for vt,s E [O, 11, - z(s)l 5 alt - sIQ CHAPTER 2. SHY SETS

For convenience of discussion we use the following notations: 1 S = dyadic rational numbers in [O, 11,

3sI,s2 E S such that

Iz(s1) - +2)l > alsl - s21P

The following theorem is reproduced from Kuo [34].

Theorem 2.5.4

(a). w(C,) = 1 if O l/2.

Proof. (This is a sketch of a proof from Kuo [34].) (a). For O < cr < 1/2, it is known that Lm,,, w(H,[n]) = O (see a proof in [34, pp. 421). It is easy to see that

where Ha denotes the cornplement of Ha and so on. Thus

(b). For constants a > 1/2 and a > O, let

Clearly for al1 n, H.[u] 2 Ja,=,,,.Kuo [34] asserted that the random variables CHAPTER 2 SHY SETS 18

are independent and each is normally distributed with mean O and variance 1/2". Thus

Hence for each a > O and a > 1/2, lim,,, w(J, ,,,,) = 0, and so w (7H,[a] = 0. Therefore w(Ca) = lim,,, w (-1H, fn] = 0. w Corollary 2.5.5 In the space Co[O,11 the set of continuous, nowhere Hdder continuous functions with exponent a,O < u c 1/2, has Wiener measure zero.

Proof. For every continuous function f which is Helder continuous with exponent cr (O < a < 1/2), there exists a M > O such that for any t, s E [O, 11, 1f(t) - j(s)l 5 Ml t - si" and so f E Ca.Using similar methods as for the set of continuous, nowhere differentiable functions in Section 4.2 it can be shown that the set of continuous, nowhere H6lder continuous functions with exponent ct (O < a < 1/2) is universaüy measurable. By Theorem 2.5.4, the set of continuous functions that are not somewhere Holder continuous with exponent a (D < ci < 1/2) has Wiener measure zero. Thus the result follows. H CHAPTER 2. SHY SETS

This result contrasts sharply with that the set of continuous, nowhere H6lder continuous functions with exponent O < a < 1/2 is prevalent (see Section 4.2).

2.6 Non-measure- t heoret ic variant s

Our concern will be with the various notions of nul sets presented in Section 2.3, in particular, with the notion of a shy set (Christensen nui1 set). Al1 of these are based on measure-theoretic concepts. Many require that the intersection of a set with a iine in some direction have Lebesgue rneasure zero. One could ask for this intersection to be smailer. Following Aronszajn [2, pp. 156-1571 let S be a class of Borel sets of the real line that forms a a-ideal and is closed under translation and affine transformation. The natural choices for B are the 0-ide& of first category sets, or of countable sets, or of a-porous sets. May more choices are possible- We can Say that a Borel set S in a separable Banach space X is B-nuii if for every sequence {el, ez, es,. . . ) whose linear span is dense in X there is a partition of S into Borel subsets

such that for each i the set of real nurnbers

belongs to B for every x E X. Let us focus just on the case where B is the collection of al1 countable subsets of reals. Then we can define a class of nuiI sets that are sharper than Aronszajn null sets and due to ZarantoneUo [62].

Definition 2.6.1 A Borel set S in a separable Banach space X is said to be a Zaran- tonello null set if for every sequence {el, ez, e3,. . .) whose linear span is dense in X CHAPTER 2 SHY SETS there is a partition of S into Borel subsets

such that for each i every line in the direction of the element ei meets Si in a countable set.

Even in finite dimensional spaces the Zarantonello null sets are smder than our ot her classes of nul1 sets. For example take a collection of Cantor sets {Cl, C2?.. ., Cn ) and consider the product set

in Rn. Such a set cannot be a Zarantonello nuli set for any choice of Cantor sets. In fact, on each Ck we can construct a positive Borel measure pk without point mas, Le., pk({x)) = O for any x E Ck. Thus on the set S the product measure p = pl x p2 x . x p,, is also positive without point mass. But for any Zarantonelo null set A C Rn we have A = U7=l Ai and A; n (x + Xei) is countable for every x E Rn where e; = (0, . .. ,1, . . .,O) with 1 in the ith position. Thus pi(Ai n Xe;) = 0. By Fubini's theorem we have p(Ai) = O and thus p(A) = O. So the set S is not a Zarantonello nul1 set. This narrower notion of nuli sets plays a role in the differentiability structure of Lipschitz mappings on Banach spaces. For example a real-valued Lipschitz function on a separable Banach space is Gâteaux differentiable outside of a Borel set that is Aronszajn null. If the function is also convex thea it is Gâteaux differentiable outside of a Borel set that is Zarantonello nuil (see Aronszajn f2, pp. 1731). CH.4PTER 2. SHY SETS

2.7 Basic theory

In this section we present the basic parts of the theory of Christensen nuIl sets (shy sets) in a separable Banach space X with an indication of proofs. Some of these extend to more generai settings and some of the other notions of ndsets share one or more of these properties. Some of the proofs here use the original methods of Christensen [12].

Theorem 2.7.1 Christensen null sets have empty interior.

Proof. Suppose a Christensen ndset S C X had non-empty interior. Then there are a non-empty bail B S and a Borel probability measure p such that p(S+z) = O for any x E X. So B + x C S + x and p(B + x) 5 p(S + x) = O for any x E X.

Therefore p(B + x) = O. Note that X is separable. There is a sequence (2;) E X which is dense in X. So 00

This is a contradiction. Hence the result foUows. rn

Theorem 2.7.2 A countable union of Christensen null sets is itself a Christensen nul1 set.

Proof. (This proof is reproduced from Christensen [l2].) Let Sn be a sequence of Christensen null sets and let pn be their corresponding transverse Borel probability measures. Through translating and norrnalizing and induction a sequence of Borel probability measures pk can be found in a neighborhood of zero such that xs, = O

1- and 113 - x * pkll 5 1/2" where x is the convolution of different p,, z = 1,2, .. . , n - 1. Thenp =p,*/t,*---II iswelldefined. ~incep=x~*~~*~~wherex~=~~*~~-*~~-, and yn = p,,, *pi+.+?* . , so xs,, *p = O and therefore ~s *p = O where S = UE, Sn* rn CHAPTER 2. SHY SETS 22

Theorem 2.7.3 Every translate of a Christensen nul1 set is also a Christensen nul1 set.

Proof. This conclusion follows directly £rom the definition of Christensen null sets.

Theorem 2.7.4 In an infinite dimensional separable Banach space every compact set or a a-compact set is Christensen null.

The proof of this theorem WU be given as Theorem 3.4.2 where it is used to illustrate some of the methods of the subject.

Theorem 2.7.5 In the space Rn a universally rneasurable set is Christensen null if and only if it is Lebesgue measure zero.

Proof. Let S C !Rnbe a Christensen nul1 set and p be its transverse Borel probability measure. Then by Fubini's t heorem we know

Then the result follows easily. H

These five properties show that the Christensen null sets can be expected to play a role in the study of inhite dimensiond Banach spaces analogous to the role that sets of Lebesgue measure zero play in finite dimensional spaces. From their definitions and the cornparisons in Section 2.3 and Section 2.4 we see that Preiss-Tider nul1 sets, s-nul! sets, Aronszajn null sets and Gaussian nul1 sets in Phelps sense al1 have empty interior, and are translation invariant in a separable Ba- nach space. From the following theorem we see that they all also satisfy the countable union property (that is, the countable union of these sets of same kind is also a set of this kind) in a separable Basach space. CHAPTER 2. SHY SETS

Theorem 2.7.6 Let X be a separable Banach space, Sn G X be a sequence of sets of one of classes of Preiss-TiSer null, s-null, Aronszajn null and Gaussian null in Phelps sense. Then U Sn is also in the same class as Sn-

Proof. (i). Let Sn be Preiss-TiSer nuli sets. Then for every x f X, z is transverse to Sn,n = 1,2, - .- . Thus {t E W : tx + y E Sn)is Lebesgue measure zero for every y E X. Note that

m t€W: ~X+YE tt.+yESn). n=l Thus {t E R : tx + y E Uzl Sn)is Lebesgue measure zero for every y f X. So U:ll Sn is dso Preiss-TiSer null. (ii). Let Sn be s-nul1 sets. Then Sn = UEl Sni such that for each i there exists an element en; E X that is transverse to Sn;. Thus the countable union of Sn has a partition as 03 Ca 00

Thus un=, Sn is &O s-null. (iii). Let Sn be Aronszajn null sets. Then for every sequence {el, e2, - - - ) whose linear span is dense in X, there are partitions of Sn into universdy measurable sets

such that for each i, the element e; is transverse to Sni- By the same argument as in (i) we know that e; is transverse to Sn;. Thus the countable union

is Aronszajn nuli. (iv). The result for Gaussian nuil sets in Phelps sense follows directly from the definit ion. rn CHAPTEX 2. SHY SETS

2.8 The completeness assumption

Throughout we have taken our space to be a Banach space. One might have thought the theory does not use the completeness in any fundamental way and that the same definitions would be useful in normed linear spaces. A simple example shows that this is not the case. Let tf denote the subspace of e, composed of sequences that have oniy finitely many non-zero elements. Then ej is an infinite dimensional incomplete, normed linear space. It is clex that ef is separable since the set of sequences of rationd numbers that have finitely many non-zero elements must be dense in tj. Write Sn for the mernbers of ef that have zero in all positions after the fist n. Then &, = Ur="=,n.Each Sn is a closed proper subspace of lj and so elementary arguments (d Section 3.3) show that each Sn is shy in the space ej. In this case a countable union of shy sets comprises the whole space Bf and our theory loses one of its main features. This same flaw would be apparent in any space that could be expressed as a count able union of proper, closed subspaces. Thus, throughout, the theory will be developed in Banach spaces or, more generally, in completely metrizable topological groups. The same featue applies, of course, to category arguments.

2.9 Haar null sets in Polishgroups

The theory of Christensen null sets was origidiy expressed in Abelian Polish goups and motivated by an attempt to generalize to non-locally compact groups the notion of sets of Haar measure zero: Since a non-locally compact group does not have a Haar measure some different approach is needed. CHAPTER 2. SHY SETS 25

In order t hat many theorems fiom Harmonic analysis carry over to the case of non- Iocally compact A belian Polish groups and the new concept "Haar zeron coincides wi th the Haar measure on locally compact Abelian Poiish groups the following definition was introduced in [12].

Definition 2.9.1 Let G be an Abelian PoIish group. A universally measurable set S C G is a Haar zero set if there is a Borel probability measure p on G such that p(S + x) = O for nny x E G.

This definition satisfies our aim, and has proved useful in the differentiability theory for Lipschitz mappings between Banach spaces. We know that an infinite dimensional separable Banach space is also an Abelian Potish group. So the theory in the setting of Abelian Polish groups fits the theory for infinite dimensional separable Banach spaces. Theorem 2.7.1, Theorem 2.7.2 and Theorem 2.7.3 remain valid in this new setting. Theorem 2.7.4 also remains valid but the proof needs modification. We state here and give a brief proof by using the ideas from [12]. '

Theorem 2.9.2 In a non-locally compact Abelian Polish group G a compact set or a a- compact set is Haar zero.

Proof. (This proof is sketched from [12].) We will show that for any universally measurable set A that is not Haar zero we have

F(A,A) = {g E G : (g + A) n A is not Haar zero ) is a neighborhood of zero element. Lf this is already proved, suppose there was a corn- pact set S which is not Haax zero. Then the set S - S would be both a neighborhood of zero element and a compact set. This contradicts that G is not locally compact. Thus the set S is Haar zero. 'A simiiar result for "total-porosity" in an infinite dimensional Banach space was given in [l]. CHAPTER 2. SHY SETS

Now we show that F(A,A) is a neighborhood of zero for a non-Haar zero set A. Suppose this were not the case. Then we may choose a sequence g, in G not belonging to F(A, A) such that d(z,z + g,) 5 112" where z = x:,' gi and d is an invariant metric on G compatible with the topology. Set

Then A' is not Haar zero. Now we dehe a mapping 6 £rom the Cantor group h' = {O, 1INto G by

Since A' is not Haar zero then there exists g E G such that + A') has non-zero Haar rneasure in K. Then + A') - + A') = LI is a neighborhood in K. Then for large v, e, E Li where eV = (0,- - - ,O,1,0, - ,O) with 1 in the v-th position. So O(ev) = g, € A' - A' and hence (fi + A') n A' # 4. This is a contradiction. Therefore F(A, A) is a neighborhood of zero.

The theory can also be developed in the setting of a non-Abelian Polish group. Christensen in [14,pp. 1231 indicated the extension of Christensen nul1 concept to such setting by using the two-sided invariance, and also pointed out that such extension does not fit any non-separable metric group. Here we will introduce four definitions in a completely metrizable separable group G, also called Polish groups. Since we are not assuming that the group operation is commutative we shail write the operation as a multiplication. In cases where we explicitly assume an Abelian group we revert to additive notation.

Definition 2.9.3 (Christensen) A universally measurable set X C G is cailed shy if there exists a Borel probability rneasure p such that p(gXh) = O for al1 g, h E G. We also say that p is transverse to X. CHAPTER 2. SHY SETS 27

Definition 2.9.4 A universdy measurable set X C G is cded lefi shy if there exists a Borel probability meaçure p such that p(gX) = O for dl g f G. We also Say t hat p is Ieft transverse to X.

Definition 2.9.5 A universdy measurable set X S G is cded right shy if there exists a Borel probability measure p such that p(Xh) = O for al1 h E G. We also say that L( is right transverse to X.

Definition 2.9.6 A universaily measurable set X G is called lefi-and-right shy if there exists a Bord probability measure p such that ~(xh)= O and y (hX) = O for a11 h E G. We aIso Say that p is left-and-right transverse to X.

From these definitions we cm easily see the following implications:

shy s left-and-~ghtshy =+ left shy (right shy).

Note that left-and-right shy is fonnally stronger than left shy and right shy. We show they are equivalent. This foilowing theorem seems to be not in the current literature.

Theorem 2.9.7 If X G is Zejt shy and right shy, then X is left-and-right shy.

Proof. Since X C G is left shy and right shy, then there exist Borel probability measures and pz such that

p,(Xh) = p2(hX) = O (Vh E G).

Now we define a new Borel probability measure v by

for dl universally measurable sets B E G. By Fubini's theorem, CHAPTER 2. SHY SETS and

and

So v is left-and-right transverse to X. The result follows. We will give several examples in Chapter 5 to show that, on the Polish group 31[0,1] (see Chapter 5), there exists (1) a probability measure that is both left transverse and right transverse to a Borel set X C X[O, Il, but is not transverse to X. (2) a probability measure that is left transverse to a Borel set X C 1i[0,1], but is not right transverse to X. (3) a probability measure that is right transverse to a Borel set X C X[O, 11, but is not left transverse to X. Also in the Polish group G we will give examples of left-and-right shy sets which can be decomposed into continuum many non-shy sets. Jan Mycielski [41] showed that the family S of shy sets is closed under finite union in an arbitrary completely metrizable group and also closed under countable unions in any Polish group. In the following theorems we will develop the corresponding properties for left shy sets and right shy sets and sorne further properties. The methods axe well known dthough the details may not be in the literature.

Theorem 2.9.8 If & and Y2are left shy sets. Then YiU fi is also a lefi shy set. CHAPTER 2. SKY SETS 29

Proof. Let pl and p2 be the probability measures on G left transverse to & and Y2respectively. We define

m(X)= pi x P~(((x,y) E G2 : x-'y, y-lx E X)) for universdy measurable sets X 2 G. Then m is a probabiiity measure. For any 9 E G, and

Thus m(gK)= m(g&) = O and hence rn(g(K u fi)) = 0.

The separability is needed in the next two theorems, but was not used in the preceding one.

Theorem 2.9.9 If G is a PoIish group and &, fi, . . . are left shy, then U:, If is also left shy.

Proof. Let mj be a probability measure left transverse to 5.Since G is separable there exists a compact set Ci with diameter 5 1/2j and mj(Cj)> O. Without loss of generality we can assume that mj(Cj)= 1 and that the unity of G belongs to Cj.

Since diameter of Ci5 1/2j, the infinite product glgz - - for gj E Cj converges in the sense of group multiplication. Let mn be the product measure of the measures mj in the product space n Ci-We dehe m(X)=

It is easy to see that m is a probability measure. We will show that m is left transverse to al1 K. Let ni C, = Cl x - x Ci-l x Ci+l x Ci+?x - and min be the product CHAPTER 2. SHY SETS 30 rneasure of ml,. .. mi+*,. . . in Cj. Since mi is left transverse to Yi, then for any h E G we have

T hus

So the result follows.

By similar arguments we cmhave the foilowing.

Theorern 2.9.10 If G is a Polish group and &, &, . . . are right shy, then Uzl I: is also right shy.

CoroIIary 2.9.11 If G is a Polish groap and &,Kt... are left-and-right shy, then UFl K is also left-and-right shy.

Theorem 2.9.12 If G is a Polish group then any non-empty open set is neither left shy nor rigfzt shy.

Proof. It is known that every metrizable topologicai group must have a left invariant metric and a right invariant metric (see [30, pp. 581). Thus we need only show the results for a Polish group with a compatible right invariant metric p. Let S be a non-empty open set in G and B(f, r) C S is a non-empty bali. Suppose

S is right shy. We need find a contradiction. Since G is separable, let {fi) C G be a countable set dense in G. Then

where CHAPTER 2. SLPY SETS

In fact, it is easy to see that the above inclusion holds if

In fact, for g E B(f,r) f -l,f<,then g = hf -'fi and p(h, f) < r. By the right invariance

where e is the unit element of G. So g E B(fi, r) and hence B(f,r)J-'fi C B(fi, r). For g f B(f;,r), ~@f?f,f) = ~(9f;~,e)= P(sA) < r*

So gfy' f E B(f,r) and g E B(f,r)f-L fi- Thus B(fi,r)C B(f,r)f-' fi. Since S is right shy, so t here exists a Borel probability measure p such that p(Sg) = O for al1 g E G. Thus

This contradicts p(G) = Z and so S is not right shy-

By sirnilar arguments we can show that S is not left shy. The theorem follows. I

In the foiiowing we give some basic properties of left shy sets. The right shy sets have the same properties. Their proofs are similar.

Theorem 2.9.13 IfG is a Poiish group, then Ieft shy sets have empty interior.

Proof. From the above theorem we know that every open set is not left shy. So any set with non-empty interior is not left shy. That is, a left shy set has empty interior.

Theorem 2.9.14 IfG is a Polish group, thetz any lefi translate or right translate of a left shy set is also left shy. CHAPTER 2 SHY SETS 32

Proof. Let S be a left shy set and p be its transverse Borel probability measure. Then for any x E G, p(xS) = O. For every y E G, by the associativity, p(z(yS))= p((xy)S)= O for al1 x E G, and so yS is left shy. For every h E G, we define a Borel probability pl by pl(A) = p(Ah) for any universaily measurable set A C G. Then for any x E G, ~l(.(Sh)) = p,((xS)h) = p(xS) = 0 by the transversality of p to S. Thus pl is transverse to Sh &d Sh is left shy.

Theorem 2.9.15 Let G be a Polish group. Then (i) A set S & G is shy if and only if

is shy. (ii) A set S Ç G is left shy if and only if S-' is right shy. (iii) A set S E G is right shy if and only ij S-' is lefi shy.

Proof. It is easy to see that a set S is Borel or universally measurable iff S-' is Borel or universally measurable respectively. Since (Swl)-' = S and (ii),(iii) coincide, we need only show the necessities of (i)and (ii). (i). Suppose that S is shy. Then for al1 f,g E G there exists a Borel probability measure p such that p( f Sg) = O. We define a Borel probability measure by

Then for al1 f,g E G,

E( js-'g) = ji(( j-')-'s-'(g-')-') = jl((g-'sj-y) = p(s-lSf -') = 0.

Therefore S-' is shy. CHAPTER 2. SHY SETS

(ii). Suppose that S is left shy. Then for al1 f E G, there exists a Borel probability measure p such that p( f S) = O. Use jï as in (i) we have

Examples (i), (ii) and (iii) mentioned in this section wili show that a measure that proves a set in a non- Abelian Polish group is shy on one side, does not prove t hat it is shy on the other side. Indeed even if that measure proves that it is shy on both sides it does not prove that it is shy since that requires more. Specificdy even if p(gX) = p(Xh) = O for aii g, h in the group it does not follow that p(gXh) = O for al1 g nnd h. But there may yet exist some other measure for which this is the case and that is the source of the problem. In a locally compact Polish group this problem does not aise because of the following theorem, proved in Mycielski [41,Theorern 1, pp. 311.

Theorem 2.9.16 In a locally compact Polish group, ifthere exists a Borel probability measure that ts left (right) transverse to a set S then S is a set of Haar measure zero (for any Haar rneasure on the group).

In Chapter 5, we wiil give examples of left-and-right shy sets without being shy in the non-locally compact Polish group.

2.10 Preservations of nul1 sets under isomorphisms

In this section we will discuss the preservation of our nul1 sets under isomorphisms. Here a mapping T from a Banach space X to a Banach space Y is an isomorphism if T is one-one, linear and continuous. CHAPTER 2. SHY SETS 34

Theorem 2.10.1 Let X be an infilrite dimensional separable Banach space and T be an isomorphism on X. Let S 6e one of sets of Gaussian null, Gaussian null in Phelps sense, Christensen null, s-null, Aronszajn null and Preiss-TiSer null. Then T(S) is also a nul1 set of the same kind,

Proof. Since T is oneone and continuous, by Theorem 11.16 in [ll], if a set S C X is Borel or universally measurable then T(S)is Borel or universally measurable respectively. We now need oniy show the assertion for each kind of Borel nul1 sets. (i). Let S be a Gaussian ndi set in the ordinary sense. Then there is a Gaussian measure p such that p(S) = O. We define a measure on Borel sets by p(E) = p(TV1E).

Since for any f E XU7 O f -'= (p O T-l) O f-l = p O (fT)-l and fT E X', thus ji is also a non-degenerate Gaussian measure. Since p(T(S)) = p(S) = 0, the set T(S) is also Gaussian null in the ordinary sense. (ii). Let S be a Gaussian null set in Phelps sense. From the proof of (i) it is easy to see that p O T is a Gaussian measure iff p is a Gaussian measure. Thus T(S)is Gaussian nul1 in Phelps sense. (iii). Let S be a Christensen ndset. Then there is a Borel ~robabilitymeasure p such that for every x E X, p(S + x) = O. As in (i) we define a measure ji on Borel sets. Then for every .y E X, there is some x E X such that .y = Tx since T is one-one. Thus

ji(T(S)+ y) = P(T(S)+ Tx)= F(T(S+ 2)) = p(S + 5) = 0, and hence T(S)is Christensen nd. (iv). Let S be an s-nul1 set. Then t here is a sequence {ei) such that S = U Si and Si n (x + Re;) is Lebesgue measure zero for every x E X. Here Rei is the one dimensional space spanned by ei. It is easy to see that T(S) = UT(Si). For any y E X, there is some x E X such that y = Tt. Note

T[Si n (X + Wei)] = T(Si)n (Tx+ WTei) = T(Si) n (y + WTei). CHAPTER 2. SKY SETS 35

FVe can define a probability measure on universally measurable sets by p(E) = X1(T-'E). Thus F(T(Si)n (y+ Wei))= 0 for any y E X and hence T(S)is s-null. (v). By using the sarne method as in (iv) we can get the results for Preiss-TiSer nul1 sets and Aronszajn nul1 sets.

The statements are true if T is an isomorphism fkom an idnite dimensional separable Banach space to an infinite dimensional separable Banach space. We can also extend one of the assertions to Abelian Polish groups in the following.

Theorem 2.10.2 Let G1 and Gz be Abelian Polish groups. Let T be a homeomorphism from G1 to G2 such that

for ail XI, x2 E Gi, where + and + denote the operations on G1 and G2 respectively. Then a set S C G1 is shy in G1 if and only ifT(S)is shy in Gz.

Proof. Since T is a homeomorphism, both T and the inverse T4' of T map Borel sets and universally measurable sets into Borel sets and universally measurable sets respectively. T-' also satisfies

for all y,, y2 E G2.This cm be seen from the assumption by applying T to its both sides. So we only need to show the necessity. If S G1is shy in G1,then there is a Borel probability measure p such that for al1 x E Gl, p(S + x) = O. Now we define a Borel probability rneasure on G2 by CHAPTER 2. SHY SETS for al1 universally rneasurable sets P C G2.Then

So jï is a Borel probability rneasure. We now check that ji is transverse to T(S).For any y E G2there is some x f G1such that y = Tfx). Then

Thus T(S)is shy in Gz and the result follows.

2.11 Measurability issues

Our definition of a shy set (Le., Christensen null set) requires that the set be universally rneasurable. This requirement is essential in view of the following theorem of Bogachev [5] cited by Preiss @6].

Theorem 2.11.1 (Bogachev) Let X be an infinite dimensional separable Banach space and let {ei) be a sequence of elements whose linear span is dense in X. Then there is a decomposition of X into a sequence of sets Xisuch that the intersection of Xi with any Iine in the direction e; has linear rneasure zero.

From this t heorem, if we do not require the universal rneasurability in the definition of shy sets, every separable Banach space would be the countable union of shy sets. So we are unable to omit the reference to the universal measurability of the sets in Section 2.3 as regards sby sets or s-nuii sets. If we remove the requirement of universal measurability in the definition of a shy set, the same problem &ses in Poiish groups. The following argument of Dougherty 1181 shows this. Let G be an uncouutable Polish group. Let 5 be a wetl-ordering of G in minimal order type. Then 5 and its cornpiement are subsets of p. Let p = pi x pz and p' = pz x pl where pl is an CHAPTER 2. SKY SETS 37

atomless measure on G and $2 is a measure concentrated on a single point of G. if the continuum hypothesis holds then p and p' are Borel measures transverse to 5 and its complement re~pectively.~This violates the additivity of shy sets- Thus the universal measurabiiity cannot be omitted. There are four possible ways of requiring measurability in defining a shy set S. (a) S is Borel. (b) S is universd1y measurable. (a') S is Borel or a subset of a Borel shy set. (b') S is universally measurable or a subset of a universally measurable shy set. The case (a) is too narrow, It does not include some sets we often meet. The case (b) is our choice. It has been proved very useful in many respects. Tbe case (a') is the choice of Hunt et al. [27]. This choice is not compatible with our choice in case (b). If we follow Hunt et al. [27] and cda set shy only if it is contained in a Borel shy set then we arrive at a different theory. The follo:&g argument, also £rom Dougherty [la], shows this. Let X be an uncountable Polish group with an invariant metric which is not Iocally compact. If the continuum hypothesis holds, then there is a subset S C X such that Sn A has cardinality less than 2'0 whenever A is a a-compact set but S n A has cardinality 2'0 whenever A is a Borel set which is not included in a a-compact set (see (241). Since any Borel probability measure on X is based on a-compact subsets of X and the continuum hypothesis implies that sets of cardinality less than 2No have measure zero under any atomless measure, then the set S is universally measurable and shy. However, any Borel set which includes S must be the complement of a a-compact set and therefore prevalent. Thus this set S cannot be contained in any Borel shy set. The case (b') is more general than the cases (a), (b) and (a'). A subset of a universally measurable shy set need not be universally measurable. For example, on

*This can be shown under weaker hypothesis (e.g., Martin's axiom). CHAPTER 2 SHY SETS 38 the real line, there exist Bernstein sets, sets such that the sets and their complements intersect every perfect set. Choose a Bernstein subset of a Cantor set of measure zero. Then this set is Lebesgue measure zero, but is not measurable with respect to al1 finite Borel measures on the red line. Such a set would be shy according to the case (b'), but not shy according to cases (a) and (b). The set S in Iast paragraph is shy according to the case (b'), but is not shy according to the case (a'). There are other interesting questions that can be posed and which are related to this discussion. In [18] Dougherty asked whether an anaiytic shy set must dways be included in a Borel shy set and whether any analytic non-shy set must include a Borel non-shy set. Solecki [52] answered the first question affirmativeIy and the second question negat ively in the following.

Theorem 2.11.2 (Solecki) Let G be an Abelian Polish group, then (i) If A C G is analytic and Haar null then there ezists a Borel set B G which is Haar null and A C B. (ii) Assume that G is not locally compact and admits an invariant metric then there exists an analytic set A such that AAB às Haar null for no CO-analyticset B.

2.12 Extension to non-separable Abelian groups

The non-separable case requires some attention to the measure theoretic details. Here is a simple example showing what can go wrong.

Example 2.12.1 Consider the non-separable Banach space lm, the space of al1 bounded sequences with the supremum nom. The cube

admits a transverse Borel probability measure. Take the measure p defined on P, as follows

w here pi = -' ëzI2dz J2n~/ for Borel sets B C IR, i = 1,2,.. . . Then p(C,) = O for ail n and indeed, p(C, + x) = O for dl x E t,. Thus each set C, has a Borel probability measure transverse to it. But

so that the whole space is a union of countably many sets that we might have ben inclined to cal1 "shyn.

One of the features of finite Borel measures on Polish spaces that is lacking for non-separable spaces is the close co~ectionwith the compact sets. An analysis of the proofs in Example 2.12.1 shows that this property of Borel measures on Polish spaces is what is missing in general. We will see in the next section that the measure p we constructed in the last example is not "tightn even though it is a Borel probability measure. One way around this is to restrict attention to measures that have this property. The simplest approach, followed in Hunt et al. [27], is to consider just those Borel probability measures with compact support. In a Polish space the shy sets would not change. For if there is a Borel probability measure transverse to a set S then there must also exist a Borel probability measure with compact support that is also transverse to t hat set. We will see this from the following t heorem.

Theorem 2.12.2 Any ~robabilityrneasure p, on a Polish space G is tight. CHAPTER 2. SHY SETS

Proof. (This proof is a sketch of some of the main ideas frorn [44, pp- 28-30].) Since G is separable, for each natural number n we can find countably many balls Bnj of radius l/n such that G = U, Bnj- So G = U, B,, where Bnj is the closure of B,,. For arbitrary E > O there exists an integer kn such that

Let Xn = U Bnj and K = n Xn.

Then

n n We now need only show that IC is compact. Let {x,) E IC be an infinite sequence then there is nl < kl such that K n BI,, = KI contains inhite points of (2,). Note h; c uf=,hj . There is an integer 722 5 k2 such that fi n &, contains infinite points of {x,). Induction yields a sequence of sets {Km)such that KI 2 K2 1 .. > Kn 2 . . . and each IC, contains infinite points of {x,). Note the diameter of Kn is less than 2/n and G is complete. So nnKn is a singleton. Thus there is a subsequence of {x,,) which converges to that single point. Therefore K is compact and the result follows. R

Another development is given in Borwein and Moors [SI for shy sets (Haar nul1 sets) in non-separable Abelian topological groups. Their method overcomes the short- comings that an open set may be shy by, again, restricting the class of rneasures to which the definition of shy sets is ailowed to apply (see the GIossary for the definition of Radon measure).

Definition 2.12.3 (Borwein-Moors) Let G be a completely metrizable Abelian topological group. A universally Radon measurable set A C G is called a Haar nuIl CHAPTER 2 SHY SETS 41 set if there exists a Radon probability measure p on G such that p(g + A) = O for each g E G.

From Theorem 2.12.2 we know that in an idnite dimensional separable Banach space the definition of Borel Christensen null sets coincides with the definition of Borel shy sets in [27],and also coincides with the above definition of Borwein and Moors. In a non-separable Banach space, by Theorem 2.12.2, the definition of Borel shy sets defined by Hunt et al. [27] stiil coincides with the definition of Borel Haar null sets by Bonvein and Moors 181, but the definitions of Borel shy sets defined in [27] and [SI are different from Christensen Definition 2.9.3. Example 2.12.1 illustrates this. In the next section we wiil discuss a further extension to completely metrizable topoIogical semigroups and give the main properties for the corresponding Haar null sets.

2.13 Extension to semigroups

Much of the material developed so far can be extended to certain topological semigroups. An account appears in Topsae and Hoffman-Jgrgenson [56, pp. 373-3781. We suppose that we are given a topological (usudy completely metrizable) semigroup G for which the operations x -, ax and x -, xa are continuous. Topsge and Hoffman-Jwgenson cdsuch semigroups "separately continuous semigroups". We assume aIways that G has a unit element, denoted as 1. Thus lx = xl = x for al1 x E G. A Radon probability measure p defined on G is said to be transverse to a universally Radon measurable set S in G if the set

has p-rneasure zero for every x, y E G and the element 1 is in the support of p. CHAPTER 2. SHY SETS

Here are the details needed to see that this defmition extends the notion of sets of Haar measure zero in locally compact groups. Let G be a locally compact group with right Haar measure p, p be a Radon probability measure transverse to a Radon universally measurable set S Ç G. Then by Fubini's t heorem we have

Since p(Sy-') is continuous with respect to y E G and 1 E suppp so p(S) = O. Thus S is Haar measure zero. The original definition of Haar null sets in Topsrae and Hoffman-Jargenson [56, pp. 3'741 just requires the existence of a r-smooth probability rneasure. The class of r- smooth probability measures is larger than the class oi Radon probability measures. However the definition of a Haar nul1 set by requiring the existence of a r-smooth probability measure transverse to a universaUy Radon measurable set coincides with the definition of a Haar null set by requiring the existence of a Radon probability measure transverse to the set. In fact let p be a T-srnooth probability measure transverse to a universally Radon rneasurable set S. Then the support of p:

suppp = {x E G : p(U) > O, VU a neighborhood of x) is closed and contained in every cbsed F with p(G \ F) = O. So G \ F G G \ suppp. Thus by the definition of T-srnoothness of p we have p(G \ suppp) = O. Note that a probability measure admits at most countably many disjoint sets of positive measure. Thus suppp is separable. We define a measure jï by

for a universally Radon measurable set A G. It is easy to see that ji is transverse to the set S. Since suppp is closed and separable, by Theorem 2.12.2, the measure F is tight on suppp. Note p(G \ suppp) = O. So fi is a Radon probability measure on G which is transverse to S. CHAPTER 2. SNY SETS 43

From the above discussion it is easy to see that in a non-separable Abelian topological group the definition of Haar nuU sets by Borwein and Moon [$] is equivaient to the definition of Haar nul1 sets by Topsae and Rohan-J~rgenson[56]. The following axe the main properties of Haar ndsets on completely metrizable semigroups. See Topsae and Hohan-Jargenson [56, pp. 373-3781 and Borwein and Moors [8] for proofs.

Theorem 2.13.1 Let G be a completely metrizable topological semigroup, then (i) If A is a Haar null set thea xAy is also Haar null for any x, y E G. (ii) A Haar null set has no interior. (iii) The union of countably many Haar nul1 sets is also Haar null. (iv) If G is an Abelian gtoup and A C G is not Haar null then the unit element O is an interior point of A - A. (v) If G is un Abelian group, then every compact set of G is Haar null.

PROBLEM 3 In a non-locally compact, non-Abelian Polish group, are compact sets left shy or right shy?

If ive impose the multiplication as an operation on the space C[a,b] of continuous functions with supremum nom it will become an Abelian Polish sernigroup with unit element for which the operation f -r fg is continuous. We will discuss the "multiplicatively shyn sets in the space C[a,b] and their relations to the "additively shyn sets in Chapter 4. CHAPTER 2. SHY SETS

2.14 Why not measure zero sets of a single measure

Our nuii sets in aii cases were required to have the property of transIation invariance. One might ask whether we could not have achieved this more directIy by taking the measure zero sets for an appropriate measure. There are many characterizations possible for the sets of Lebesgue measure zero in a finite dimensional space. The simplest cne to conceive is merely that these are the nul1 sets for a single measure (Lebesgue measure) that happens to be translation invariant. This gives immediately a class of sets that is translation invariant. In a compact group there is a unique translation invariant probability measure (Haar rneasure) which plays the same roIe, En a non-compact but locally compact, Abelian topological group again Haar measure can be used. The measure is unique up to multiples and translation invariant and so the sets that are of zero Haar measure play the role that we require. The foliowing theorem and its proof are sketched £rom Pl

Theorem 2.14.1 Let G be a locally compact, Abelian topological group. Then a set S E G is Haar nul1 ifl S is of zero measure Ior any Haar measure on G.

Proof. Let h denote a Haar measure on G. Since h is a-finite, we can use Fubini's theorem to show that for any Borel probability measure p on G,

That is,

where xs is the characteristic function of the set S. If S is Haar nuii and p is a test-measure for S, then the right hand and the left hand of this equation are zero. CHAPTER 2. SHY SETS

Since h is translation invariant, S is zero for the Haar rneasure h. Conversely, if S is zero for the Haar measure, we can obtain a test-measure p by choosing p with density with respect to the Haar mesure.

In a locally compact non-Abelian group there is a right Haar measure and a left Haar measure. One might worry that the measure zero sets are only invariant on one side but that is not the case. The measure zero sets for a tight Haar measure again serve as Our class of sets invariant under the group operations (see [41]). Why have we been unable to pursue the sarne course in an infinite dimensional Banach space? The first problem is that there is no nontrivial translation invariant finite or a-finite measure on such a space.

Theorem 2.14.2 There is no nontrivial translation invariant Jinite or a-finite measure on an infinite dimensional Banach spcace.

Proof. The proof of this tbeorem in detail is very compLcated and long (see [6l, pp. 138-1431). For our purpose we only show that there is no translation invariant

Radon probability measure C( on an infinite dimensional Banach space. Suppose that there were a translation invariant Radon probability mesure p on an infinite dimensional Banach space X. Then there are a compact set K and an open bal1 B(z,E) contained in X such t hat O c p(K) < oa and p(B(x, E)) < oo. Since the space X is infinite dimensional, we cmconstruct an infinite sequence of disjoint open bdls {B(y,44)) which are contained in B(x, €1. Since p is translation invariant, so the p measures of all such balls B(y, 44) are zero. Note

By the compactness of h' and the translation invariance of p there are finite many ti such that m CHAPTER 2. SHYSETS

This is a contradiction. Thus the result follows.

The above theorem can be generalized to any inhite dimensional topological space (see a proof in [61]). In order to describe our class of null sets we need not necessarily require that the measure be translation invariant, ody that the set of measure zero sets for that measure be translation invariant. Such a measure is said to be quasi-invanont. For exarnple the Gaussian measures in Rn are quasi-invariant and, indeed, the measure zero sets of such a measure are precisely the Lebesgue measure zero sets. But again this is not possible in an infinite dimensional space. In fact the following theorem shows that it would never be possible.

Theorem 2.14.3 Let X be a locally conuex, linear, inJnite dimensional topological vector space. Then there is no nontrivial o-Jinite quasi-invariant measure defined on the Borel subsets of X.

Proof. (This proof is reproduced from [61].) By using the remark following The- orem 2.14.2 we need only show that if a nontrivial quasi-invariant measure C( exists then a nontrivial translation invariant measure exists. We use B to denote the cIass of Borel sets and define a map T on X x X by

(x, y) -, (xy, y), Then T is an automorphism on fXx X, B x B) . By the Radon- Nikodym theorem there is a function f (x,y) > O such that for every F E B x B,

Consider the following three maps on X x X x X as foliows: CHAPTER 2. SHY SETS 47

We can easily see that Tlo T2= T3 O TzO Tl. Then from the uniqueness of the density function we get that for every (x, y, z) f X x X x X,

Thus there is some so E X such that f(zo, yz) = f (zoy, z)f (xo,y)- Set f (xo,&./) = g(y ). Replacing xoy by y we have that for every (y, z) E X x X, f (y, 2) = g(~z)/g(y)- Putting du = g-ld~.Then v is translation invariant. In fact, for any F E B x B we have

Thus for F = A x X E B x B we have

From the uniqueness of the density function we have v(Ay) = v(A) for any y E B. Since B f B is arbitrary we have that for any y E X, v(Ay) = v(A). H

In Section 3.4 we saw that on an infinite dimensional separable Banach space a Gaussian null set in Phelps sense is null for every non-degenerate Gaussian measure. However, in general a Borel shy set cannot be nuli for every a-finite Borel measure. In fact al1 Borel measures live on shy sets!

Theorem 2.14.4 (Aronszajn [2]) Given any a-finite Borel meusure p on an infinite dimensional separable Banach space X there is a Borel s-nuil set S su that p is concentrated on S, i-e., p(B) = p(S n B) for al1 Bon1 sets B.

Immediately £rom the above theorem we have the following CHAPTER 2. SHY SETS 48

Corollary 2.14.5 Giuen any a-finite Borel metrsvre p on an infinite dimensional separable Banach space X, for any Borel set A Ç X there is a Bo~els-nul1 set B and

Q Borel p nul1 set U such that A = BU U. in particular X can 6e decornposed into a p null set and a Borel s-nul1 set.

For a proof of the above theorem see (2, pp. 155-1561. The theorem provides another proof that the Borel shy sets cannot be described as the null sets for any one a-finite Borel measure.

2.15 Classification of non-shy sets

Hunt et al. [27] gave a number of variants of non-shy sets. Dougherty [18]gave more refined characterizations of non-shy sets. Let S be a universaily measurable subset of a Polish group. In the assertions below e will vary over positive real numbers, p over probability measures on G, and t over translation functions g -, glggz. The following eight properties from Dougherty [18] define different classes, in general, of non-shy

Epreualent] [lower density 11 fpositiue lower densit y] [observable] [ubiquifous] [upper density 11 [positiue upper densit y] [non-shy]. In [18]Dougherty mentioned the results and examples in the following (i), (ii), (iii) and (iv) but did not give proofs or explanations. Here we take this opportunity to verify these results and examples. Furthemore, we find non-implications (3) * CHAPTER 2 SHY SETS 49

(2') and (1') (4)' and give examples for them in (v) and (vi). From (i) to (vi) we will get a clear picture of relations among the eight classes of non-shy sets.

(i). Some implications. From the definitions we see that the implications (j) -+

(k) and (j') -+ (k') for j < k are tnvia.1, We now show that (j) -+ (j') for each j. Let S satisfy (1) then there exists a Borel probability measure po such that po(t(S)) = 1 for each t. By Fubini's theorem for any Borel probability measure p on G,

Thus there must exist g2,g f G such that p(Sg2g-') = 1, and (1') holds. Similar arguments can show that (j) + (j'), j= 2,3,4. (ii). (4) + (3'). For the intervd SI = [O, 1J and the Gaussian measure pl, defined by for al1 Borel sets B C R, the set SI satisfies (4) but does not satisfy (3'). In fact, for every E > O, we construct a function

and extend it evenly to (-oo, O), Then the measure pz defined by

is a Borel probability measure, but for any x E W, p2(2 + SI) < c.

(iii). (2) =+ (1'). The set S2 = (-CO, O) u (1, oo) satisfies (2) since for any E > O the measure p2 in (ii) satisfies p2(z + g2)< e for each z E B. Here 4 is the cornplement CHAPTER 2. SHY SETS 50 of S2. Note that W = (r + S2)ü (x + 5)and (x + S2) fi (x + g)= ). SO for the Gaussian measure pl as in (ii) and any x E W, pl(x + Sz) < 1. Thus S2 does not satisfy (1'). (iv). (2') + (3). The set S3 of positive real numbers satisfies (2') since for any c > O and any Borel probability measure p on R Ive can choose x small enough so that p(x + S3) > 1 - c. On the other hand, for any E > O and Borel probabiiity measure p we cm choose x Iarge enough so that p(x + S3) < r. Thus S3 does not satisfy (3)- (v). (3) + (2'). In Ut, let

For the Gaussian measure pl as in (ii) and every x E IR, it is easy to see that

and ' 1 °i(l+ Si) 5 l - 1,2,-rd, < L - < 1.

So S4 satisfies (3) but does not satisfy (2'). (vi). (1') (4). We will show that the sets &(A), &(Af) and &(A-) in Theorem 2.15.1 are al1 examples for this non-implication. From the above (i) to (vi) we can make the following conclusions. (2) + (1') implies (2) + (1) and (2') + (1'); (2') + (3) implies (2') * (2); (3) * (2') implies (3) * (2) and (3') * (2'); (4) * (3') implies (4) + (3); (1') + (4) implies (1') + (1) and (4') (4). Here we indicate all relations among (1)-(4) and (1')-(4') in a table as follows. We use -+ to denote + and ++ to denote +. We now study the following sets to justify (vi). Let A be a non-empty set of natural numbers,

3N > O such that if ra 2 N, s(n)> O for n E A and s(n) < O for n 4 A CHAPTER 2. SHY SETS

Figure 2.1: The relations of non-shy sets in general

&(A+) = {s E IRN : 3N > O such that if n 3 N, s(n) > O for n E A) and

Sl(A-) = {s E EtN: 3N > O such that if n 2 N, s(n) < O for n E A).

Note that

where

&,(A) = {s E IR": ifn > rn, s(n)>O for n E Aand s(n)

Sl,(Af)= ER^: ifn > rn, s(n) >Oforn~A} and

Slm(A-) = (s E EtN : if n 3 m, s(n) < O for n 4 A).

By using the methods as in Theorem 3.5.4 we can show that &,(A), &,(A+) and S1,(A-) are Borel sets. So the sets Si(A), SI(A+), SI(A-) and their complements are Borel sets. In the following theorem we show that ail &(A), &(A+), &(A-) and their complements are ubiquitous but not observable.

Theorem 2.15.1 Let A be a non-ernpty set of natural numbers. Then the sets &(A), &(A+) and &(A-) and their complements are al1 ubiquitous Borel sets but not observable. (That is, they satisfy (1 '/ but not (4).) CHAPTER 2. SHY SETS 52

Proof. Case 1. The set A is an infinite set. For a given Borel probability measure

p on llgN we can choose a number b, for each n such that

Then Ir({~E P~:IS(R)~ 5 bn for dl n 2 N})> 1 - 2-N.

Let 00 S = U {s E EtN : (s(n)l5 bn for al1 n > N}. N=I Then p(S) = 1. Define elements t, t+, t- E !RNas foilows,

t+(n) = lbnl + 1 for n E A and t'(n) = -Ibn[ - 1 for n E A . Then S + t C S1(A), S+t+ C &(A+) and S+t- E Sl(A-). Thus SI(A), &(A+) and &(A-) are ubiquitous. That is, they satisfy (1'). Now we consider their complements.

VN > O, E A, no 2 N such that s(no)5 0, - or 3mo 4 A, mo 2 N such that s(mo)2 O &(A+)- = {s E IRN : VN > O, 3n0 E A, no > N such that s(no)5 O}, &(A-) = {s E IRN : VN > 0, 3n0 E A, no 2 N such that s(no)3 O).

Now we define tl,tl E RNto satisfy the following.

and tr(n)= 1bnl + 1 for n E A. Then clearly we have CHAPTER 2. SHY SETS 53 - - - Thus Sl(A), &(A+) and Sl(A-) are ubiquitous too. That is, they aiso satisfy (1'). From the definitions of (4) and (1') it is easy to see that a set satisfies (4) iff the complernent of this set does not satisfy (1'). Therefore the sets &(A), Sl(A+) and Sr(A-) and their complements are ubiquitous but not observable. Case 2. The set A is a non-empty finite set. For the set &(A), by checking every step of the proof in Case 1, it is easy to see the conclusion still remains valid. For the sets SI(A+) and SI(A-) we choose an infinite set B C N \ A and define

Cl = {s E iRN: s(n) > O for n E AU B) and CZ={SEW~:s(n)

Dougherty [18] showed that the set

S(A) = (s E R" : s(n) > O for n E A and s(n) < O for n 4 A}, is upper density 1. Kere we sketch his proof and show more fiom Theorem 2.15.1.

Corollary 2.15.2 S(A) is upper density 1 but not observable.

Proof. For a given Borel probability rneasure p on R" and r > O we can choose b,, for each n such that r({s E R": Is(n)l > bn}) < r. CHrlPTER 2. SHY SETS 54

Thus p({s E RN: Is(n)l 5 bn}) > I - E and hence S(A) is upper density L. Note that S(A) C SI(A). From Theorem 2.15.1 we wiil see that &(A) is not observable and therefore S(A) is not observable. H

For the remainder of this section we wiiI try to describe in Rn(n 2 1) the classes of sets in (1)-(4) and (1')-(4'). In W" the set S in (1) has fui1 Lebesgue measure. This is proved in Theorem 2.7.5. Now we look at the sets satisfying (1'). (1') is equivalent to that Vp3t p(t (3))= O where 3 is the complement of S. Thus take p to be a Gaussian rneasure and so there is a t such that /r(t(g))= O. Since the Lebesgue measure and any Gaussian measure are mutually absolutely continuous, so ~.(t(S) ) = O and ~~(3)= 0 where An is the n-dimensional Lebesgue measure. Thus the set S also has full Lebesgue measure and hence al sets in (1) and (1') are equivalent to having full Lebesgue measure. If a set S satisfies (47, then, for every Gaussian measure p, there exists a t f Rn such that p(t(S)) > O. By the mutually absolute continuity of An and p we have An(t(S)) = An(S) > O. Again by the mutually absolute continuity of A, and p, for every Gaussian measure and t E Rn, we have p(t(S)) > O. So the set S satisfies (4).

Since (4) =S (4') so ail sets in (4) and (4') are equivalent to having positive Lebesgue measure. In Rn, for every set S satisfying (3') it is easy to see XnfS)> O. If not, A,(S+x) = O for any x E Rn. Thus for any Gaussian measure p, p(x + S) = O which contradicts (3'). Now ive can claim Xn(S) = m. If not then O < X,(S) < m. For any c > O we can construct a function F(x) by defining CHAPTER 2. SHY SETS

Then the probabiiity measure p induced by F(x),

satisfies that for any x f Rn, p(x + S) < é. This contradicts (3'). Thus for any set S satisfying (3') we have Xn(S) = oo. Therefore for any set S satisfying one of (21, (3)' (2') and (3') we have X,(S) = cc. However we currently have no exact characterizations for sets S in (2)' (3)' (2') and (3'). Based on the above discussions and (i)-(v) we can obtain the reIations among (1)-(4) and (1')-(4') as follows.

Figure 2.2: The relations of non-&y sets in Rn

2.16 Prevalent versus typical

Any finite dimensional space can be decomposed into a set of Lebesgue measure zero and a hst category set. The proof is entirely elementary. One shows that there is a dense open set (say one that contains al1 points with rational coordinates) of arbitrarily small Lebesgue measure. An appropriate intersection of dense open sets gives a measure zero, dense set of type Gs. In our more generd setting we do not have a notion of measure, just a notion of measure zero. Thus such an argument does not work, although one expects a similar decornposition should be available. The following theorem of Preiss and Tiler CHilPTER 2. SHY SETS 56

is remarhble because the measure zero part of the decomposition is in the strongest terms and the fist category part is given to be O-porous (see [54] for a-porous). a-porous sets form a smder class than the sets of the first category.

Theorern 2.16.1 (Preiss-Tiier) An inbite dimensional separable Banach space can be decomposed into a Preiss-TiSer null set and a set that is a covntable union of closed porous sets.

This result is of some interest in the st udy of derivatives on Banach spaces. From Aronszajn [SI we know that the set of points of Gâteaux non-differentiability of a real-valued Lipschitz function on an infinite dimensional separable Banach space X is Aronszajn null. However the Fréchet differentiability of a real-valued Lipschitz function on an infinite dimensional separable Banach space is completely different. One of the Preiss-TiSer results [46, pp. 222, Proposition 11 says that, on an infinite dimensional separable Banach space, there is a real-valued Lipschitz function which is Fréchet non-differentiable on a given countable union of closed porous sets. Thus from the above theorem there is a real-valued Lipschitz function which is Fréchet differentiable only on a subset of a Preiss-Tider null set in an infrnite dimensional separabIe Banach space. The decornposition in the above theorem is not possible in a finite dimensional Banach space because both a closed porous set and a Preiss-TiSer null set are Lebesgue measure zero. A natural question is to ask whether such a decomposition, even a weaker one, is possible in a non-separable Banach space. We leave it as a problem at the end of this section. In the following we consider the same problem in a Polish group. Note that, however, a discrete group could not have such decomposition. Also the discrete group cannot be decomposed into a shy set and a first category set. Any compact or locally compact Polish group with a difise Haar measure can be decomposed into a shy set CHAPTER 2. SHY SETS and a first category set by using the same proof that works for Rn. In the follorving we give a sufficient condition for a general Polish group to be decomposed into a shy set and a first category set.

Theorem 2.16.2 Let G be an Abelian Polisir group which permits a Borel probability measure p such that there is a constant b < 1 so thot

for every x E X and r 5 1/2. Then G can be decomposed into a shy set and a first category set.

Proof. Since G is separable, let {xi) be a sequence of points that is dense in G. Let

Then X is a dense Gsset and so G \ X is of the Tst cat egory. By the assumption for any x E G,

Thus the set X is shy in G and the decomposition G = X U (G \ X) is our desire.

Remark. Such a theorem is true for fmite dimensional spaces since Lebesgue measures can replace p in the proof of the above theorem. However we do not know whether an infinite dimensional Banach space permits a Borel probability rneasure satisfying the condition in the above theorem.

PROBLEM 4 Does every Polish group permit a decomposition into a shy set and a set of the first category? CHAPTER 2. SHY SETS

2.17 Fubini's theorem

Fubini's theorem is one of the most important theorems in measure theory. It is natural to ask whether there is a version of Fubini's theorem that holds for Haar zero sets. Christensen [12] gave the following example to show that a full version of Fubini's t heorem does not exist .

Exarnple 2.17.1 Let H be a separable infinite dimensional Hilbert space and let T be the unit circie in the complex plane. There exists in the product group H x T a Borel measurable set A such that (i) For every h E H, the section A(h) = {t E T : (h,t) E A) has Haar measure one in T. (ii) For every t E T, the section A(t) is a Haax zero set in H. (iii) The complement of A is a Haar zero set in the product group H x T.

Proof. (This proof is reproduced from [12].) Since an infinite dimensional separable Hilbert space is isometric to the space L2 of square integable functions. We assume H = L2,let A = {(f, t) : the Fourier series of f converges at t}.

Then from the famous result of Cadeson that the Fourier series for any L2 function is almost everywhere convergent, (i) follows. Note that for every t E T, A(t) is a closed linear proper subspace of H. So it is Ham zero and (ii) follows. Clearly the product measure of the one point measure in H with mass 1 at zero and the Haar measure in T is transverse to the complement of A.

From the above example we see that if a fuIi version of Fubini's theorem holds, then (i) and (ii) imply that A is Haar zero. This contradicts (iii). However, as indicated in [12], a weaker version of Fubini's theorem does hold. See Borwein and Moors [8, Theorem 2.31 for details. CHAPTER 2. SHY SETS

2.18 Countable chah condition

In this section we discuss a property called the countable chain condition of the o-ideal of shy sets in some spaces.

Definition 2.18.1 Let G be a Polish group. An ideal 3 of subsets of G is said to satisfy the countable chain condition if each disjoint family of universally measurable sets in G that do not belong to F is at most countable.

We also Say that the group G does not satisfy the countable chain condition if the ideal 3does not satisfy the countable chah condition. A completely metrizable space is separable if and only if every family of pairwise disjoint non-empty open sets of this space is countable (see [20]). Thus the above definition is only for Polish groups. In 1973 Christensen [12] asked whether any family of disjoint universdly measurable non-shy sets in a Polish group must be countable. This is obviously true in finite dimensional spaces. Dougherty [18]answered this problern affirmatively in sorne Polish groups. Theorem 3.5.4 exhibits a coUection {S(A) : A C N) that forms a family of pairwise disjoint non-shy sets in an Abeiian Polish group that is uncountable. The following theorem and its proof are reproduced from Dougherty [18J.

Theorem 2.18.2 The Abelian Poli& grou7 RN does not satisfy the countable chain condition.

Proof. Specificdy the collection

from Theorem 3.5.4 consists of 2*0 disjoint non-shy elements. So IkN does not satisfy the countable chah condition. CHAPTER 2. SHY SETS 60

Mycielski observed that if A and B are sets of natural numbers whose syrnmetric difference is infinite, then not only are the sets S(A)and S(B)disjoint, but also any translate of S(A)intersects S(B)in a shy set (se[18]). Thus we can get 2'0 non-shy sets of EtN which mutudy have this strong disjointness property, by taking S(A) for 2'0 sets A Ç W which have infinite symmetric difference with each other. In Abelian Polish groups Dougherty [18] obtained the following general result by unusual methods.

Theorem 2.18.3 (Dougherty) Let G be a Polish group which has an invariant metric and is not locally compact. Suppose that there exist a neighborhood U of the iden- tity and a dense subgroup G, of G such that, for any finiteiy generated subgroup F of G., F n U is compact. Then the ideal of shy sets of G does not satisfy the countable chain condition.

By using a result of Kechris 1291 that in a Polish group there is a Borel selector s : G/H + G for the cosets of H where H is a closed subgroup of G, Dougherty [la] showed the following result.

Theorem 2.18.4 Suppose (G, +) is an Abelian Polish group and H is a closed subgroup of G. If the ideal of shy subsets of H does not satisfy the countabie chain condition, then the ideal ofshy subsets of G does not satisfy the countable chain condition.

After obtaining Theorem 2.18-3, Dougherty [18]conjectured that a Polish group G satisfies the countable chain condition oniy if G is locally compact. Later Solecki [52] answered this problem perfectly in the following.

Theorem 2.18.5 Let G be a Polish group admitting an invariant metn'c. Then each family of universally meusurable or, equivalently, closed, pairnise disjoint sets which are not Haar nul1 is countable ifG is locally compact. CHAPTER 2. SilY SETS 61

In Chapter 5, we show that the c-ideal of shy sets in the non-Abeiian, non-locally compact space R[O,11 of autornorphisms does not satisfy the countable chah condition. So we conjecture that the above Solecki's result is tme for non-Abelian Polish groups. We will leave it as an open problem in Chapter 5. Theorem 2.18.3 implies that an infinite dimensionaI separable Banach space does not satisfy the countable chain condition. However, the proof of Theorem 2.18.3 is in general. We pose the following problem.

PROBLEM 5 Using techniques in Bunach space theory, show that any injnite dimensional separable space does not satisfy the countable chain condition.

Thickness of non-shy sets

In this section we wiii discuss a property known as thiclness. The definition is from [O6].

Definition 2.19.1 A subset A of a topological semigroup G is thick if the unit element is an interior point of AA-'.

In 1939 Ostrowski [42] showed the well-known fact that eveq- subset of the real line with positive Lebesgue measure is thick. That is, the non-shy sets in the real line are thick. Of course the converse does not bold. For exarnple, the Cantor set in the real line is shy but it is thick (see 156, pp. 367-368, Theorem 2.4.21). It was Christensen who first showed that non-shy sets are thick in any non-locally compact

Abelian Polish group (see our Theorern 2.9.2 for a proof). Hoffman-Jgrgensen 1561 extended it to completely metrizable, AbeIian topological semigroups in the following form. We have stated this as Theorern 2.13.1 (iv). For reference we repeat it here, using our new tenninology. CHAPTER 2. SHY SETS

Theorem 2.19.2 Let G be a completely metrirable, Abelian topological semigroup. Let A be a uniuersally Radon measurable subset of G. If A is not a Haar null set, then A is thick.

We discuss now, in a Banach space, the thidmess of non-Preiss-TiSer null sets, non-Aronszajn ndsets, non-s-nd sets, non-Gaussian null sets in Phelps sense and non-Gaussian ndsets in the ordinary sense. In W,all these sets are equivalent to the sets of positive Lebesgue measure and so they are thick. In Rn (n > l),each straight line is non-Preiss-TiSer null but not thick. However, in Rn (n > l),non-Aronszajn null sets, non-s-nul1 sets, non-Gaussian null sets in Phelps sense are equivalent to the sets of positive Lebesgue measure. So they all are thick. In an infinite dimensional separable Banach space we know that, from Exam- ple 2.3.8, there is a compact set K which is not Gaussian null in Phelps sense. Note that K - K is dso a compact set and it cannot be a neighborhood of the zero element in an infinite dimensional space. Shus, from the cornparison in Section 2.3 and Section 2.4 non-Aronszajn null sets, non-Preiss-TiSer nul1 sets and non-Gaussian nuIl sets in Phelps sense may not be thick in an infinite dimensiond sepârable Banach space. Non-Gaussian nul1 sets in the ordinary sense are thick (see [56, pp. 372-3733 for a proof). Recall (Problem 1) that we do not yet know whether d Christensen null sets are s-null. If we cannot answer this problem, perhaps we can answer the following problem.

PROBLEM 6 In an infinite dimensional separable Banach space, are al1 non-s-nui1 sets thick? Chapter 3

Probes

3.1 Introduction

To show that a set S in a Banach space X is shy (i.e., a Christensen ndset) we need first to establish that S is universally measurable (perhaps by showing that it is a Borel set or an analytic set in X) and then to exhibit an appropriate probability measure on X that is transverse to S. It is the finding of this testing measure or probe t hat requires some techniques. Proving that a set is prevalent amounts merely to showing that the complernent of the set is shy and so this does not introduce any new problems. However, showing that a set is non-shy dlrequire showing that every possible measure fails to be a testing measure or probe for the set. We will see that, occasiondy, this is not so hard and requires only a few measure-theoretic observations. Where possible, rather than constructing a testing measure, we would prefer to use Lemma 2.2.2 and Definition 2.2.1 and exhibit an element z f X transverse to S. This merely picks a direction in the space so that all lines in that direction intersect S in a set of linear measure zero. It is of some intrinsic interest to know if this is CHAPTER 3. PROBES possible. Asserting only that the set S is shy says much les. Failing that we might show that there is a decomposition S = U~"=,S;and find elements x; transverse to Si-In the language of Definition 2.3.2 we would say that the set S is s-null. Again there is some intrinsic interest in knowing whether this is possible. Where these ideas fail we may hope to find a finite dimensional subspace transverse to S. There would be some intrinsic interest in knowing the least dimension that could be selected when this technique works.

Where Iinear arguments fail (as they do in a variety of situations) perhaps we can construct a curve (a continuous image of an interval) that is transverse to S. Xgain knowing that the set S admits a curve all of whose translates intersect S in a set of measure zero (Le., measure zero along the curve) is of some intrinsic interest. In this chapter we survey some of the methods that have been used to establish that a given set is shy or non-shy and provide some concrete examples to illustrate the methods.

3.2 Finite dimensional probes

The language of probes in [27] was introduced to have a convenient way of expressing the techniques. Often to show that a set A is shy one hds a subspace (usually one or two dimensional) that proves that the set A is shy. That subspace was called a probe for the complement of A (see Hunt et al. [27]). Here we extend this to cal1 the subspace or the measure supported on it also a probe for the set A. A subspace P of X that is a n-dimensional probe for a shy set S C X is also a m-dimensional probe for the shy set S if m > n. In fact we can find m - n linear independent elements {xl,. .. , z,,,) such that the dimension of the span of

({XI,. . ., x,-,) U P) is m provided X has at least m linear independent elements. CHAPTER 3. PROBES

Thus it is necessary to give a new definition to describe shy sets-

Definition 3.2.1 A shy set S Ç X is said to be m-dimensionally shy if S ha a rn dimensional probe but no n dimensional probes for n < nt, and S is said to be inbite-dimensionalZy shy if S has no finite dimensional probes. The empty set is said to be O-dimensionally shy,

Theorem 3.2.2 In W2 there exist 2-dinzensionally shy sets.

Proof. In 1928 Besicovitch [4] constructed a set of E C R2of Lebesgue measure zero which includes line segments of length 1 in every orientation (see also Proposition 12.2 in [22, pp. 1631). Thus nny Borel set containing the set E and having Lebesgue measure zero is a 2-dimensionaliy shy set. R

Theorem 3.2.3 In Rn (n > 2) any shy set is at most 2-dimensional.

Proof. Let F be a n-dimensional Lebesgue measure zero set of Rn. By Theorem 7.13 [21, pp. 1063 there is a 2-dimensional subspace P of Rn such that every translate of P intersects F in a set of k-dimensional measure zero. Thus F is an at most 2- dimensionally shy set and hence F is an at most 2-dirnensionally shy set.

Theorem 3.2.4 Any shy set in R2 space can be decomposed into tuo at most 1- dimensionally shy sets.

Proof. By Theorem 2.7.5, a set A Ç IR2 is shy iff A is 2-dimensional Lebesgue measure zero. Let A be Lebesgue measure zero in R2. Then X2(A) = O and by Fubini's theorem we have a Lebesgue ondimensional measure zero set N C W such

that for di x2 E W \ N, Xl(A,,) = O where A,, is the section of A at xz (the second variable). Let B={(xi,x2) EA: x24N) CHAPTER 3. PROBES and

C = {(x,,x2) E A : ~2 E NI-

Then A = B U C. We shdl show that B and C are at most 1-dimensional shy sets. For the set B we choose el = (1,O). Thus for any c = (cl,c2) E R2,

is 1-dimensional Lebesgue measure zero when c2 E W \ N, and is ernpty if cz E N. Anyway it is 1-dimensionai Lebesgue measure zero. Thus the set B is an at most I- dimensionaily shy set, For the set C choose e = (0, l),then for any c = (CI, cz) f R2 the set {U E R : c + cre E C) is one-dimensional Lebesgue measure zero. Thus the set C is an at most 1-dimensionatly shy set. Hence the result follows.

PROBLEM 7 In Rn (n > 2), can any 2-dimensionally shy set be decomposed into two at most 1-dimensionally shy sets?

PROBLEM 8 Let X be an infinite dimensional separable Banach space. Does fhere exist a Borel shy set S C X that is not an n-dimensionally shy set for any n?

3.3 Element ary linear arguments

In many simple situations a crude linear argument sufices to show a set is shy. We illustrate wit h some elementary examples, mostly from the literature. Certainly a proper subspace of a Banach space X is shy if it is universally measurable. In fact if S Ç X is a proper subspace then every element x of X \ S is transverse to S. Choose x 4 S. For any y E S, let G = (X E R : y + Xx E S). Then G is a singleton. If not, thereexist XI and X2, Al # X2, such that y+Xlx E S and y+X2x E S. CHAPTER 3. PROBES

Since S is a linear space, then

This contradicts the choice of x. The result follows.

Example 3.3.1 The set S of differentiable functions is a proper linear subspace of C[O, 11. S. Mazurkiewicz in [40] showed that the set S is not Borel, but is CO-analytic (see also [ll, pp. 5031). So the set S is universaily measurable and therefore is shy.

Note, however, that in generai a linear subspace need not be a Borel set nor need it be universdy measurable. ' The following example is from Christensen [12].

Example 3.3.2 Let E be a separable Fréchet space and let a;, i E I, be an algebraic basis of E adb; the coefficient functionals. Then btl(0) are proper linear subspaces of E. If bf '(O) are universaUy measurable for some i's, these sets are shy sets. In fact, however, the subspaces b~'(0) are universaily measurable for at most finitely many i E 1. If not, there is a sequence {ij} of I such that each bcl(0)is universaily measurable. Set L, = Uj2,btl(0) Then Ln is a universally measurable proper linear subspace. So Ln is shy. Since ai, i E I is an algebraic bais of E, so E is the union of Ln's and hence E is shy, which is impossible.

For many examples of sets S that have a probe x, the set

that we are required to show to be Lebesgue measure zero for every y E X is, in fact, a singleton or empty set. Thus every iine in the direction x intersects S in at most one point. In many of the simplest applications this is the case and the measure-theoretic arguments reduce to simple computations.

Assuming Martin's axiom, Talagrand in [53] showed every separable infinite-dimensional Banach space has a hyperplane that is univedy of measure zero without being closed. CHAPTER 3- PROBES

In any example that illustrates a shy set it is of intrinsic interest to know that the set is not merely shy, but has very small intersections with ail lines in some particular direction.

Example 3.3.3 As an example (hom [2?, pp. 2261) consider the set S of ail convergent series in the space !!,for 1 < p < m. This is a closed, proper subspace and so, trivially, it is shy. For a specific transverse element we can take the element x = (1, I/2,1/3,. .. ). It belongs to tpbut diverges. The set S intersects each iine in the direction x in at most one element.

The same argument applies to a hyperplane provided it is universally measurable. It is not always the case that hyperplanes are universally measurable except in concrete examples as shown in Example 3.3.2.

Example 3.3.4 As another example (from [27, pp. 3261) the set S of functions f in Li[O, 11 for which J,' f (t) dt = O. This is evidently a dosed hyperplane in Li[O, 1] and so, trividy, shy. For a specific transverse elernent take fo = 1. It is easy to see that the set S intersects each line in the direction fo at most one element.

These same arguments do not need much linearity in the problem. For example if S is a universally measurable proper subset of a Banach space X that satisfies

6)Yl - Y2 E S if YI, Y2 E S, (ii) ty 4 S for Lebesgue a.e. t E W if y 4 S, then S must be shy in X. Indeed, let y 4 X \ S and let

for any x E X. If S is empty there is nothing to prove. If S is not ernpty, choose t~ f So. Then by (ii), (x + ty) - (x + tly) = (t - tl)y E S only for a Lebesgue measure zero set of numbers of t - tl. Thus Xi(&) = O and the set S is shy. CHAPTER 3. PROBES

Example 3.3.5 As a further example to illustrate simple methods of this type consider the Banach space C[O, l] of continuous functions equipped with the supremum norm. Let S denote the set of al1 continuous functions that are monotone on some closed subinterval of [O, 11. Write, for any closed subinteml I C [O, 11, S(I)for the continuous functions that are monotone on 1. Then

where the union is taken over the countable collection of al1 subintervals I of [O, 11 with rational endpoints. It is easy to see that each S(I)is closed and is shy. To sethat it is closed we just need note that the uniform limit of a Cauchy sequence of rnonotonic and continuous functions is also monotonic and continuous. So S(I)is cIosed+ To see that S(I)is shy take any element g E C[O,11 that is not a.e. differentiable on I. Then g is transverse to S(I). In fact the set

cmcontain at most one point for any f E C[O,11. If not, there are distinct ti, t2 E iR such that f + tlg, f + t2g E S(1). Then f + tlg, f + t2g f S(I)are a.e differentiable on I and so is (j+ tlg)- (J f t2g)= (tr- t2)gon I. This contradicts the choice of g. Consequently we have shown that S is shy, indeed that S is an s-nul1 set in the sense of Definition 2.3.2. In particular, we can express this observation in the following language:

Theorem 3.3.6 The set of functions in C[0,1] that are monotone in no ~~~e~al forms a prevalent set.

For more theorems of this type in the function space C[u,b] see Sections 4.2 and 4.3. CHAPTER 3. PROBES

In a Banach space, a universdly measurable convex set that does not contain a line segment in some direction is easily seen to be shy. If a convex set S does not contain a line segment in some direction x, then S intersects the line in at most one point. So the element x is transverse to S and S is shy. Thus convex sets that do not span the whole space are, in general, shy. It is of some interest to find out whether closed, convex spanning sets in certain spaces are non-shy, This will be discussed further in Section 3.5.

3.4 Compact sets

A nurnber of arguments can be used to show that a compact (or a a-compact) subset of an infinite dimensional Banach space must be shy. See, for example, the original article [El of Christensen where the argument uses the fact that A - A is a neighborhood of the zero element for any universally measurable non-shy set A. If a compact set A were non-shy then the set A - A is compact and aIso a neighborhood of the zero element. This is impossible in an idnite dimensional Banach space.

Example 3.4.1 For an elementary iilustration of a concrete example, here is how to show that the compact Hilbert cube in lais shy: We use = {z E e2 : lxnl 5 n-') to denote the compact Hilbert cube in e2. Choose y = {7~-~/'). Then y E G \ 1". It is easy to see that y is transverse to IO0. So I" is shy in &.

Hunt et al. gave an interesting proof in [27, pp. 2251 that uses a category argument to exhibit that every compact set in an idnite dimensional Banach space admits a transverse element, indeed that the set of transverse elements is residuai. This is of some general interest since it allows us to obtain transverse elements without the CHAPTER 3. PROBES necessity to construct one in advânce. This is aIso of sorne intnnsic interest since we see that "mostn directions in the space are transverse to the set. in the folIowing we give a similar but simpler proof suggested by J. Bomein to show every compact set admits a transverse element, an argument which also show that the transverse elements are residual.

Theorern 3.4.2 A compact subset of an infinite dimensional Banach space is shy.

Proof. Let V be an infinite dimensional Banach space and S C V be a compact set. Then, if the linear span of S is denoted by SpanS, we have

where

Since S is compact, a11 sets Sn are compact. Thus SpanS is a-compact and first category. Hence V \ SpanS f 4. Therefore S is transverse to every element x € V \ SpanS since the line Xz intersects S in at most one element. Thus S is shy. . Corollary 3.4.3 A u-compact set in an infinite dimensional Banach space is s-null.

The above method neither holds for Gaussian ndsets nor for Aronszajn nul1 sets since the fact that there is a Borel probability measure is transverse to the one- dimensional probe of the set V\ S does not guarantee p(S) = O for al1 non-degenerate Gaussian measures p. In fact there exist compact sets which are not Gaussian null sets in Phelps sense (see Example 2.3.8). From the fact that a translate of a non- degenerate Gaussian measure is also a non-degenerate Gaussian measure we know that a Gaussian null set in Phelps sense is also a shy set. Thus the class of shy sets is wider thaa the class of Gaussian nul1 sets in Phelps sense. Recall that an Aronszajn CHAPTER 3. PROBES nul1 set is also a Gaussian null set in Phelps sense* Thus the cIass of shy sets is also wider than the class of Aronszajn nuil sets. In non-locally compact groups with invariant metrics compact sets are shy (see [lS]). However we do not know whether the invariant metrics are needed for this statement.

PROBLEM 9 In a general non-locally compact Polislr groxp mithout invariant met- n'cs, are compact sets shy?

In the following we will give an interesting method to show that certain sets in an infinite dimensional Banach space are shy. The following theorem and lemma that we will display were introduced by Aronszajn 121 for separable Banach spaces. In fact they remain valid for non-separable Banach spaces. Vie ceproduce the theorem from 116, pp. 2191 (see [16]for a proof), and give a simpler proof, suggested by J. Borwein, of the lemma. Theorem 3.4.4 (Josefson-Nissenzweig) Let X be an infinite dimensional Banach space. Then there is a sequence {un) C X', llunll = 1 such that (un) converges to O in the weaka-topology.

Lemma 3.4.5 Let X be a Banach space. Suppose un are elernents of X' vith nom 1 and un + O in weap-topology, while S is a bounded set such that u,(s) -, O uniformly for x E S. Then there is an element a f X fransverse to S, indeed

contains at most one point for every x f X.

Proof. Since un + O unifodyon S, so 6j(un)G supZEsun(x) + O. Thus 62(un) + O where A r WSis the convex hdof S. We cIaim that SpanS C Ur=,(nA) # X where SpanS is the linear span of S. We need only show that Uz=l(nA) # X. If CHAPTER 3. PROBES 73

U:=f,i (nA) = X, then by the Baire category t heorem, the interior of the set A is not empty and hence 6Â(u,) * O. Thus there exists an element a E X \ Spa& that is transverse to the set S.

From Theorem 3.44 and Lemma 3-4.5 we can show that certain sets in an infinite dimensional Banach space are shy. We write this as a theorern as follows.

Theorem 3.4.6 Let X be an infinite dimensional Banach space, and {un} Ç X' be a sequence as in Theorem 3-44 If a set S is uniuersally measurable and bounded such that un(x)+ O unifonnly for x E S. Then the set ES is shy where âZZS is the convex hall of S.

Using the above theorem we can obtain Theorem 3.4.2 immediately. Let S be a compact set in an infinite dimensional Banach space X. Then S is bounded and closed. By Theorem 3.4.4 there is a sequence {un}C X', 11~~11= 1 such that Un + O weakly. Standard arguments show that the bounded sequence {un) converges to O weakly implies that u,(x) -, O uniformly for x E S. From the above theorern the set S is shy.

3.5 Measure-t heoretic arguments

To show that a set is non-shy "requiresnshowing that there are no transverse elements, indeed that there are no transverse probability measures at dl. The following simple fact from the theory of Borel measures on Polish spaces is very useful in showing a set to be non-shy. See Theorem 2.12.2 for a proof-

Euery probability measure defined on the Borel subsets of a Polish space assigns positive measure to some compact set.

This then gives an irnmediate proof of the folowing: CHAPTER 3. PROBES

Lemma 3.5.1 A set S in a separable Banach space X that contains a translate of every compact subset is non-shy.

Proof. Suppose that the set S were shy then there is a Borel probability measure p that is transverse to S. That is, p(S + x) = O for each z E X. According to the above basic fact and assumption there are a compact set K and x E X such that p(K) >OandK+zÇS. Then KÇS-zandO

For example (from [6]), the positive cone C of the space co of nul1 sequences is non-shy. (It is clear that C is nowhere dense in Q.) In fact, every compact set K in

co is contained in a set {y f a-, : lynl 5 x,, n = 1,2,. ..) for a certain x f CO. So A' + x 2i CC.By Lemma 3.5.1, the positive cone C is non-shy. Borwein and Fitzpatrick [7] use Lemma 3.5.1 dong with a characterization of reflexive separable Banach spaces to justify the foilowing.

Theorem 3.5.2 In any non-regexive separable Banach space there is a nowhere dense closed convez subset that is not shy.

Proof. A separable Banach space E is not reflexive if and only if there exists a closed convex subset C of E with empty interior that contains some translate of each compact set in E (see [37]). Thus, the result follows from the last lemma.

In contrast to the above theorem Borwein and Moors [8] and Matsot6kova in [36] showed that in any super-reiiexive space a nowhere dense closed convex set is shy. Here is another example, from Dougherty [18], of an argument t hat exploits t his same measure-theoretic fact. Let IRN denote the space of aii sequences of rdnumbers furnished with the metric CHAPTER3. PROBES 75

This is an Abelian Polish group, complete in this metric but non-locally compact. For any A C N, let

S(Aj G {s E IKN : s(n) < O for n 6 A and s(n) > O for n E A).

We reproduce the following lemrna £rom [18] and give a proof.

Lemrna 3.5.3 For any A Ç N, the set S(A) contains a translate of every compact subset of iRN.

Proof. For any compact set C the set Ci = {x(i)E IR : x E C) is a bounded closed set in W,i = 1,2, . .. . Let (ai,6;) be intervals such that Ci 2 (a;,b;). We define t E IKN

Then it is easy to see that C + t S(A). I From t his lemma we conclude immediately that the set S( A) in the Abelian Polish group RNcannot be shy.

Theorem 3.5.4 For any A N the set S(A) is a Borel set that is not shy in RN.

Proof. From Lemma 3.5.1 and Lemma 3.5.3 we need only show that the set S(A) is a Borel set. In IRN aay sequence sn + s implies s(i) -, s(i) for each i E N. Thus for each pair of integers (i,n), the sets 1 Ain = {S t R": s(i) 2 -n aod Bi, = {s E B~:s(i) < -i}n are closed sets. Note that

Thus S(A) is a Borel set. CH.4PTER 3. PROBES 76

Theorem 3.5.4 exhibits an interesting feature of shy sets that is not shared by the Lebesgue measure zero sets in a finite dimensional space. The coIlection {S(A): A C W) forms a farnily of pairwise disjoint non-shy sets that is uncountable. See Section 2.18 for a further discussion of this. The search for families of uncountably many pairwise disjoint non-shy sets in non-Abelian Polish groups is our primary goal in Chapter 5. Chapter 4

Prevalent properties in some function spaces

4.1 Introduction

During the middle of nineteen century, mmy farnous mathematicians tried to prove the differentiability of continuous functions. Frustrated, some of t hem gave examples of continuous functions that are not differentiable on a dense set or further on the set of irrationals. It was K. Weierstrass who first constructed a convincing exarnple of a continuous function that has no point of differentiability in 1875. After Weierstrass many examples of continuous nowhere differentiable functions were discovered by other mathematicians. In 1931 S. Banach and S. Mazurkiewicz separately gave similar existence proofs by using the Baire Category theorem in separate papers [3] and [40] respectively. Since then Baire category arguments have been widely used to prove the existence of functions which are difficult to visualize. It is now a common practice to show that certain classes of functions are "typicaln in spaces of functions by showing that they form residual subsets in those spaces. CHAPTER 4. PREVALENT PROPERTIES IN SOM3 FULVCTIONSPACES 78

The same program can be carried out using the measure-theoretic notion of prevalence, rather than the topological notion of category. The earliest such result is prob- ably that of N. Wiener [60]showing that the nowhere differentiable functions are full measure in the space C[O,11 of continuous functions x(t) with x(0) = O in the sense of the Wiener measure. In 1994 Hunt et ai. [27] applied the notion of shy sets (re- discovering the Haar zero sets of Christensen) to the same kind of problems. In this chapter we will show that certain classes of functions in various function spaces are prevalent.

4.2 Continuous, nowhere differentiable functions

In this section we wil sketch some of the ideas from the paper of Hunt [%]. Later sections in this chapter are devoted to sirnilar problems in some specific function spaces. Let S denote the set of al continuous functions on [O, 11 that are somewhere differentiable, Le., for which there is at least one point of differentiability. It is proved in [%] that S is shy in the space C[O,11. In fact the author does not address the measurability issue but shows that S is contained in a shy Borel set. (The definition used in that paper for a shy set is exactly this, that an axbitrary set is shy if it is a subset of a Borel shy set. Our definition requires us to prove that the set is universaily measurable if not a Borel set.) Let us show how to check that S is universaily ~easurable.A continuous function f has a finite derivative at some point x E [O, 11 if and only if for every positive integer n there is a positive integer m such that if O < [hl[, 1 hzl < l/m and x+h~,x + h2 E [O, 11 CHAPTER 4. PREVAL ENT PROPERTES IN SOME FUNCTIOlV SPACES 79

For each pair (n,m),let

E(n,m) = {(f,x) E CIO, 1] x [O, 11 : (*) holds }.

Then C[O, l]\Sis the projection into C[O, 11 of n:, um=, E(n,m). Note that E(n,m) is closed. So C[O, I]\S is anaiytic (111 and therefore the set S is universaiiy measurable (see [LI]). Furthemore, S is not Borel (see 1381 and [39] for details). We now tum to the method of proof that the set S of somewhere differentiable functions is shy. As pointed out in [28] there is no element transverse to S. Thus the simplest of the methods is not avaiiable here. To see this, let g be a continuous function. For f(x) = -xg[x) and every A, f + Ag is differentiable at x = A. Hence there is no element transverse to S. The reason here is that a linear combination of nowhere differentiable functions cm be differentiable. In [2S] it is shown that there is a two-dimensional subspace of C[O, 11 that is transverse to S. We sketch parts of the proof here. In [28] Hunt used two continuous nowhere differentiable functions

Through complicated and elegant computations he showed that t here exists a constant c > O such that for any a,B E IR and any closed interval I C [O, 11 with length 6 5 112,

By using this result he showed that the set

G = {(a,,B) E iR2 : f + ag + ph is Lipschitz at some x E [O, 11) is two-dimensional Lebesgue measure zero for any f E CEO, 11. It is easy to show that the set G is of type F, and that a function differentiable at a point r is Lipschitz at x. So the set S is contained in G. Lu [28] Hunt showed X2(G) = O as follows. Let 1V 2 2 be an integer and split [O, 11 into N closed intervals I of length c = 1/N. Split G into UIU:=f=, Sn where

Sn = {(a,p) E G : f + ag + +h is n-Lipschitz at some point x E 1).

Therefore, by using the resuit we first mentioned above we have

Trivially the result follows. Hunt's methods show that the somewhere differentiable functions form a shy set that is 2-dimensional in the laquage that we introduced in Section 3.2. It would be interesting to know if more can be said about this set. Weleave this again as an open problem.

PROBLEM 10 Is the set of somewhere differentiable functions in the space C[O,11 s-dl?

4.3 Prevalent properties in C[a,b]

To illustrate some of our methods we shail prove some of the simpler prevalence results in the space C[a,b] of continuous functions equipped with the supremum norm.

Theorem 4.3.1 For c E [a,b] and C E IR, let

Sc = {f E C[a,b]: f(c) = C).

Then Sc is closed, nowhere dense and shy. CEMPTER 4. PREVALENT PROPERTIES ZN SOME FULVCTIOLVSPACES 81

Proof. It is easy to see that Sc is closed and nowhere dense in C[a,b]. Take a non-zero constant hinction g = d # C. Then

can contain no more than one element for any f E C[a,b]. If not, there are XI, X2 E

[O, 11, # X2. f f Xig E Sc and f + Azg E Sc. Then f(c) + Xld = f(c) + X2d. So (A2 - Xl)d = O which contradicts d # O. Therefore the one-dimensional Lebesgue rneasure is transverse to Sc and Sc is shy. rn

A continuous function is called nowhere rnonotonic on an interd [a! b] if it is not monotonic on any subintervd of [a, 61.

A continuous function is called nowhere monotonic type on an interval [a, b] if for any 7 E IR the function f (z)- 7x is not rnonotonic on any subinterval of [a,b]. It is well known (see, e-g., [Il, pp. 461-4641) that nowhere monotonicity and being nowhere rnonotonic type are typical properties in the space C[a,b]. In this section we will show directly that these two properties are dso prevalent properties in C[a,b]. The following theorem follows easily frorn the result of Hunt sketched in Section 4.2. The proof, here, is more elementary.

Theorem 4.3.2 The prevalent function f E C[a,b] is of nozchere mmonotonic type.

Proof. Given any interval 1, let

G(I)= { f E C[a,b] : f is of monotonic type on I), and let

there exists a y E [-n, n] such that f - yx is monotonic on I CHAPTER 4- PREVALENT PROPERTIES IN SOME FUNCTION SPACES 52

FVe show that G,(I) is a closed set and therefore G(I)is a Borel set. For any Cauchy sequence {fi) C G,(I) t here exists a function f E CEO, i] such t hat fi -, f uniforrnly. Then there exist yi E [-n, n] such that fi(x) - y;x are rnonotonic on I. Then ive can choose a subsequence {y;,) of {y;) such that yi, + y E [-n, n]. It is easy to see that f - yx is rnonotonic on I. Thus G,(I) is closed. Now we show that G,(I) is a shy set. Choose a function g E C[a,b] that is nowhere differentiable on I. For any f E C(a,b], let

Then G,, is a singleton or empty set. If not there exist XI, A2 E G,,,, Xi # X2. Then there exist yl and y2 such that f + Alg - ylx and f + X2g - y2x are rnonotonic on 1. Therefore

is differentiable almost everywhere on I. This contradicts our assumption that the function g is nowhere differentiable. Thus G,, is a singleton or empty set and hence G,(I) is a shy set. It follows that the set G,(I) is a shy set. Let {Ik)be an enurneration of al1 intervals whose endpoints are rational numbers. Then each G(Ik)is shy and the union is also shy. Hence the result foliows. I

Corollary 4.3.3 The preualent function f E C[a,61 is nowhere rnonotonic.

Note that we have proved this earlier. As a corollary Our methods actually show more.

Corollary 4.3.4 The set of functions which are sonaewhere rnonotonic is s-nul1 in Cb,61. 4.4 Prevalent properties in bd, ~DU',b23'

Following Bruckner [9],Bruckner and Petruska [IO] we use bA, bDB1, bBL to denote the spaces of bounded approximately continuous functions, bounded Darboux Baire 1 functions and bounded Baire 1 functions defined on [O, I] respectively, ali of which are equipped with supremum noms. Ail these spaces are Banach spaces and form a strictly increasing system of closed subspaces (see [9], [IO]). In [IO] it was shown that for a given arbitrary Borel measure p on [O, 11 the typical function in F = bd, bDD1, or bB1 is discontinuous p almost everywhere. In this section we tvill show that such typical properties in these three spaces are also prevalent properties for any o-finite Borel measure.

Theorem 4.4.1 Let p be a o-Jinite Borel rneasure on [O, 11. The preualent function in 3 = bA, bDB1, or bB1 is discontinuous p almost everywhere on [O, 11.

Proof. Let

S = (f E 3 : f is continuous on a set FA. p(Fx) > 0).

We show first that the set S is a Borel set. Note

where and Cf is the set of continuity points of f. Let (fm} be a Cauchy sequence in AL, n then f, -, f E F. Let

N=l m=N Then C C Cf.Note So f E AL. Thus AL is closed and the set S is a Borel set. n n We now show that the set S is a shy set. It is well known that there is a function g E .F which is discontinuous p almost everywhere on [O, 11. See [IO, pp. 331, Theorem 3-41. We wiil use this function g as a probe. For any given function f E 3, let

We claim that Sg is Lebesgue measure zero. For distinct Xi,X2 E Sg,if p(FXIn FA^) >

wouid be continuous on FA,n &. This contradicts the choice of the function g. Thus for distinct Al and A2 the correspondhg sets fi,and 9, satisfy p(FA,nF,,)= O. Since p is o-finite on [O, 11 then [O, I] = Uül Xiwhere p(Xi) < cm and Xi n Xj = 4: i f j. Let

Then Sm, is finite. If not, there exist countably many Xi E Smnsuch that 00 00 00 = r(& n Xm)= r(X, n (UFA,)) 5 p(L)< 00. i= 1 i= l This is a contradiction. Hence Sm, is finite and Sg = Uz,,=, Sm, is at most countable. Thus S, is Lebesgue measure zero and the result follows.

In the proof of Theorem 4.4.1 we did not use any special property of functions in bA, bVB1, bB1 except that in al1 these classes there are functions that are discontinuous L( almost everywhere on [O, 11. Thus we can extend Theorem 4.4.1 in a general form as follows (see [32] for a typicai version).

Theorem 4.4.2 Let p be a u-fïnite Bore2 measure on [O, 11. Let 3 be a linear space of bounded functions f : [O, 11 + W with supremvm rnetn'c. Suppose that there is a function f E F that is discontinuous p almost everywhere on [O, 11. Then the prevalent function in 3 is discontinuow p almost eaeryvhere on [O, 11. CHAPTER 4. PREVALENT PROPERTIES IN SOME FUiVCTION SPACES 85

4.5 Prevalent properties in D[a,b]

In this section we use D[a,b] to denote the set of differentiable functions f whose derivatives are bounded and f (a)= O and furnished with the metric (for f g E D[a,b])

P(M= SUP lff(4-9w* XE[U,~] Then D[a,b] is a Banach space. We study prevalent properties in this Banach space.

Theorem 4.5.1 Both the prevalent function and the typical funcfioa f E D[a,6] are monotonie on some svbinterual of [a,61.

Proof. Let

DZ = {f E D[a,b] : ff(x)= O on a dense set of [a,b]).

Then DZ is a closed licear subspace of D[a,b]. In fact, let f, g E DZ,then the sets

{XE [a,b]:jf(x) =O} and {t E [a,b]: gf(x)=O} are of type G6 since f' and gr are Baire 1 functions. But the intersection of two dense sets of type Gs is also dense, so

is dense for any al,crz E W. Thus alf + azg E DZ. Similady we can show DZ is closed. Suppose { f,) is a Cauchy sequence in DZ such that fi -t f' uniformly where f E D[a,b]. Again since f, are all Baire 1 functions, the sets {x E [a,b] : fi(x) = O} are dense and of type Gs . So the set

is dense and of type G6.Thus f E DZ and hence DZ is a closed linear space. It is easy to see that DZ is a proper subspace of D[a,b]. For example the function x - a E Dia, b] but f 4 DZ. Thus from the results in Section 3.3 DZ is shy. For any differentiable, nowhere monotonic function f, the derivative f' is a Baire 1 function and so the set {x E [a,b] : f1(x) = O) is a G6 set. Further since f is nowhere monotonic then {x E [a,b] : f (x)= O) is dense in [a,b]. Therefore the set

G = { f E D[a,b] : f is nowhere monotonic on [a,b]) is a çubset of DZ. In fact the set G is of t*ype Gs in D[a,b]. Let 1 be an open subinterval of [a,61, and

F(I)= { f E D[a,61 : f is monotonic on I).

Fur any f E F(I), f'(x) > O on the entire 1 or j'(r) 5 O on the entire 1. So due to the metric defined on D[a,b] it is easy to see that F(I) is a closed set. Hence the set of functions in D[a,b] that are somewhere monotonic is the union of al1 those F(I) over intervals I with rational endpoints. So it is of type F, and therefore G is of type Gs in D[a,b]. The result follows.

We know that, for any o-finite Borel measure p, the derivative of the typical function f E D[a,b] is discontinuous p almost everywhere on [a,b] (see [IO]). By applying Theorem 4.4.2 we obtain the following theorem.

Theorem 4.5.2 Let p be a o-finite Borel measvre on [a,bj. Then the derivative of the prevalent function f E D[a,b] is discontinuous p almost everywhere on [a,b].

If we do not use the fact that for any a-finite Bore1 measure p there exists a function g E D[a,b] whose derivative is discontinuous p almost everywhere, we cm easily obtain a simpler result as follows.

Theorem 4.5.3 The derivative of the prevalent function f E D[a,b] is discontinuous on a dense set of [a,b] .

Proof. Let I be a subinterval of [a, 61 and let

A(I) = {f E D[a,b] : f' is continuous on 1). CHAPTER 4. PREVALElVT PROPERTIES IN SOLMEFUNCTION SPACES 57

Then £rom the definition of the metric on D[a,b] it is easy to see that A(I) is closed. AIso for any f,g E A(I) and a, E W, of + pg E A(I). So A(I) is a closed linear subspace of D[a,61. We wili see that A(I) is also proper in D[a,b]. Take c E 1. The funct ion

has as its derivative the function

2(x - c) sin - COS ;II;l ifx#c fW = { O ifx=c which is discontinuous at x = c. Thus A(I)is a closed linear proper subspace of D[a,61. So A(I) is shy. The set S of functions in Dia, b] whose derivatives are continuous on some subinterval is the union of the A(I) taken over al1 subintervds 1 with rational endpoints. Hence the set S is shy (as a countable union) and the theorem follows. rn

4.6 Prevalent properties in BSC[a,b]

The space BSC[a,b] of bounded symmetrically continuous funct ions equipped wi t h the supremum norm is a complete space (see E5.51). From [50] we know that the set of functions f E BSC[a,b], which have a c-dense sets of points of discontinuity, is residual. In this section we show that such a set is also prevalent. Here we Say that a set is c-dense in a metric space (X, p) if it has continuum many points in every non-empty open set. In [33] Pave1 Kostyrko showed the following theorem.

Theorem 4.6.1 Let (X,p) be a metric space. Let F be a linear space of bounded functions f : X + W furnished with the supremum norm 11 f 11 = suprEX{1f (2)1). Sup- pose that in F there exists a function h such that its set D(h) of points of discontinuity is uncountable. Then

G = { f E F : D(f) is uncountable} is an open residual set in (F,d), d(f,g)= Il f -gll.

By rnodifying the methods in [50] we can get a stronger result in separable metric spaces.

Theorern 4.6.2 Let (X,p) be a separable metric space. Let F be a complete metric linear space of bounded functions f : X -, B furnished with supremum nom 11 fil = supIEX{lf(x)l). Suppose that there is a function h such that its set D(h) of points of discontinuity is c-dense in (X, p). Then

G = { f E F : D(f) is c-dense} is a dense residual Gs set in (F,d), where d( f,g) = Il f - gll.

Proof. Given a non-empty set 0, we can show that

A(O)= {f E F: D(f)nOisofpowerc) is a dense open set by using the methods in [50]. In fact, let { f,) C F \ A(O) be a convergent sequence. Then there is a function f E F such that f, - f uniformly. Let en denote the set D(f,)nO. Then e, is at most countable and so the union Ur=P=,en is at most countable. We know that f is continuous at each point x E O \ UE, en, so f E F \ A(0). Hence F \ A(0) is closed and A(0) is open. Now we show that A(O) is dense in F. For every bdB(f, c) E F, if f E A(O) there is nothing to prove. We assume f E F \ A(O), then f has at rnost countably many points of discontinuity in O. From the assumption there is a function h E F such that h has a c-dense set of points of discontinuity in O. Let M be a constant such that Ih(x)l < M for all x E X and set CHAPTER 4. PREVALENT PROPERTES IN SOME FLrNCTlON SPACES 89

Then g E F is discontinuous in continuum many points of O and

Thus g f A(0)n B( f, c) and hence A(0)is dense. Let {ri)be a dense countable subset of X, then

i=1 m=l is a dense Cgset where B(x;,l/m) is the open ball centered at xi and &th radius l/m. Thus G is a dense residual Ggset in F. W

CorolIary 4.6.3 The typical functions in R[a,b], the space of Riemann integrable functions furnished with the supremum nom, have c-dense sets of points of discontinuity.

Proof. The set of points of discontinuity of any bounded symmetrically continuous function is Lebesgue measure zero [55, pp. 27, Theorem 2-31. Thus such a function is Riemann integrable by the well known fact that a bounded Lebesgue measurable function is Riemann integrable iff its set of points of discontinuity is Lebesgue measure zero [ll]. Tran in [57] has constructed a syrnmetricaiiy continuous function whose set of points of discontinuity is c-dense. Hence the result follows. 1

The following theorem gives a similar form of Theorem 4.6.2 in the sense of prevalence.

Theorem 4.6.4 Let (X,p) be a separable metn'c space, p be a a-finite Borel measure that is non-zero on every open set in (X,p). Let F be a complete metric linear space of bounded functions f : X 4 iR furnished with the supremum nom 11 fil = supZEX{[f (z) 1). Suppose that in F there ezists a function h such that its set of points of discontinuity is c-dense in (X,p). Then the prevalent function f E F has a c-dense set of points of discontinuity. CHAPTER 4. PREVALENT PROPERTIES IN SOLLIEFCTNCTION SPACES 90

Proof. Let G = {f E F : D(f) is c-dense in X).

By Theorem 4.6.2 the set G and its complement are Borel sets. We need only show that for every f E F the following set

f + Ah is discontinuous at most countably many I points in some non-empty open set OAC (X,p) is a Lebesgue measure zero set, For every X E S there exists a non-empty open set Ox E (X,p)such that f + Ah is discontinuous at most countably many points in Ox. If there are two distinct XI and X2 such that p(Oxrn Ox,) > O, then both f + Xlh and f + X2h have at most countably many points of discontinuity in the non-empty open set 01, n Clx,. Therefore (Xi - X2)h has at most countably many points of discontinuity on such non-empty open set. This contradicts the property of function h. Hence for distinct Al and Az their corresponding non-empty open sets O.\, and O,\, satisfy ,u(Ox, n 0A2)= O. Since p is a-finite on (X,p) then X = UT X, where /t(Xi)< oo and Xi n Xj = 4, i # j. Let

Then S',, is finite. If not, there exist countably many Ai in Sn, su& that

This is a contradiction. Thus Sm is finite and S = U&=, Sn, is at most countable. So the set S is a Lebesgue measure zero set. The span of h is an one-dimensional probe for the set G and the result follows.

Corollary 4.6.5 The preualent function f f BSC,[a, b], the space of bounded n-th symmetrically continuous functions furnished urith the supremurn nom,has a c-dense set of points of discontinuity. CHAPTER 4. PREVALENT PROPERTXES IN SOME FUNCTION SPACES 91

Proof. The space BSC,[a, b] is a complete metric space (see [33]). By Tran's results [57]there exist functions hl and h2 in BSCI[a,b] and BSCZ[a,b] respectively such that hl and h2 have c-dense sets of points of discontinuity on [a,bI. Also note that BSCl[a,b] = BSC[a,b] C BSC2k-1[a,b] and BSC2[a,b] C BSC*k[a,b] (see [33] for details). Thus the result foiiows £rom Theorem 4.6.4.

Applying Theorem 4.6.4 and Tran's results [57] we obtain irnrnediately a prevalent property in the space R[a,b], which is aiso a typical property in R[a,b] as shown in Corollary 4.6.3.

Corollary 4.6.6 The prevalent &actionf E R[a,b] has a c-dense set oj points of discontinuity.

In the following theorem ive display another prevalent property in the space BSC[a,61.

For X C R a function f : X -+ IR is cailed countably continuous if there is a countable cover {X,,: n E W) of X (by arbitrary sets) such that each restriction f 1 X, is continuous. By constructing an example of a non-countably continuous function Krzysztof Ciesielski [15] answered a question of Lee Larson: whether every sj-mmetrically continuous function is countably continuous. Wewill show that non-countable continuity is both a typical property and a prevaient property in the space BSC[a,b].

Theorem 4.6.7 Let F = {f E BSC[a,b] : f is countably continuous). Then F is a closed, nowhere dense, linear subspace of BSC[a,b].

Proof. First we show that F is a Linear space. For fi,f2 E F, there are countable covers {X,!,: n E N) and {X: : n E N) of [a,b] such that the restrictions fi(Xk are CHAPTER 4. PREVALENT PROPERTES IN SOhE FCINCTION SPACES 92 continuous. It is easy to see that

So {X;' n X; : i, j E W) is a countable cover of [a,b]. Also the restrictions f lx;'n are continuous. Thus for any a,,t3 E B,cr fi + B fi E F. We now show that F is closed. Let { fn) C F be a convergent sequence. Then there is a function f E BSC[a,b] such that f, -, f uniformly. For each f, there exists a countable cover {Xi")of [a, b] such that f, 1 Xi" is continuous. Since [a, b] Uz, XF,

Note that for any m the restriction f,l oz,Xi. is continuous. Thus the restriction f 1 nz=,Xi. is continuous since f is the uniform limit of f,. Therefore f E F and F is closed. Wenow show that F is nowhere dense in BSC[a,b]. For any bal1 B(f, e), if f 6 F there is nothing to prove. If f E F, then f is countably continuous. Choose a function g E BSC[a,b] that is not countably continuous (se Ciesielski's construction in [15]). Then f + (g/M)c/2 E B(f,e) where M = sup,,[,,q Ig(x)l. Obviously, f + (g/M)r/2 is also not countably continuous. ü not, f + (g/M)e/ 2 - f = (g/M)e/2is countably continuous since F is linear. Since F is closed, it is nowhere dense. The result foLlows. I

Corollary 4.6.8 Both the prevalent function f E BSC[a,b] and the fypical function are not countably continuous.

Proof. In [15]Krzysztof Ciesielski constnicted a function f E BSC[a,b] that is not countably continuous. Thus the set F in Theorem 4.6.7 is a closed, nowhere dense, proper and lineâr subspace of BSC[a,b]. Therefore the result follows. H CHAPTER 4. PREVA LENT PROPERTIES IN SOL\~E FuNCTION SPACES 93

4.7 Multiplicatively shy sets in CIO, 11

In the space C[O,11 of continuous functions on [O, Ij there is another aigebraic operation of importance - multiplication. C[O,11 is a Banach algebra with multiplication _Jgof elements f,g E C[O,11 defined in the pointwise sense. Under this operation

C[O,11 is a Polish sernigroup with unit. (The unit is the function f (x) G 1.) This allows for two distinct notions of shy sets. Let S E C[O,11 be universally measurable. Then S is said to be an additiueiy shy set if there is a Borel probability measure p so that

Xlso S is said to be a multiplicatiuely shy set if there is a Borel probabiIity measure containing the unit element in its support so that

In this section we propose to investigate this definition rather briefly. The ideas arise from purely forma1 considerations and may or may not be useful in applications.

Theorern 4.7.1 Let 1bf denote the functions in C[O,11 that are somewhere monotonic. Then M is 60th additiuely shy and multiplicatiuely shy.

Proof. We have already showed that M is additively shy (see Example 3.3.5). We now show that A1 is multiplicatively shy. For I Ç [O, 11 we use M(I) to denote the set of functions in C[O,11 that are monotonic on 1. Take a function g nowhere differentiable with O < g(z) 5 1. Write

Define a Borel probability measure p by

p(X) = Xf((t E [O, 11 : F(t) E X)). Then the support of p contains the unit element f(x) G 1. In fact, for every bal1 B(l,e),there exist O < S < 1 and O < 77 < 1 such that g(x) 3 6 for al1 z E [O, 11 and

1 - E < 6' 5 1 for al1 t > q. Thus p(B(1,~))> 1 - 77 > O and hence 1 E suppp. For any f E C[O, 11, consider the set

T = {t E [O, 11 : F(t)E fhl(1)).

Vie claim that T contains at most one element. In fact, if not, there are tl,t2 E [O, 11, tl < ta and rnl,m;!E M(I) such that

g(z)'l = Sml and g(~)'~= fmz.

T hus

This is impossible since the left side of the equality is nowhere differentiable on I but the right side is almost differentiable on I. Thus X1(T)= O and M(I)is shy. Note

where the union is taken over al1 subintervals of [O, 11 with rational endpoints. There- fore il1 is multiplicatively shy. m

Theorem 4.7.2 Let Z be the set of continuous functions on [O, 11 with at least one zero. Then Z is not additively shy, but is multiplicatively shy in C[O, 11.

Proof. Tt is easy to see that Z is closed. In kt,for any Cauchy sequence { fn) C Z there exists a function f E C[O, 11 such that f,, 4 f uniformly. For each n, there exists

2. 2. E [O, 11 sucb that f,(z,) = O. Thus there exists a subsequence zr -t XQ E [O, 11.

Let j -+ co in fnl (xn1) = O we have f (xo)= O. Therefore f E Z and Z is closed. Now we show the assertion in two steps. CHAPTER 4. PREVALENT PROPERTIES IN SOME FUNCTION SPA CES 95

(i). Wite F : [1,2] -+ C[O, 11 bÿ F(t)= t. Then for any f E C[O, 11,

So we can construct a Borel probability measure L( with support containing the unit by p(X) = X,({f E [1,2] : F(t)E X)).

Obviously L( is transverse to the set 2. Thus Z is multiplicatively shy. (ii). We show tbat Z is not additively shy. Suppose Z were additively shy. Note that every translation of Z is also additively shy, and

C[O, l] = U(Z+ r) r EQ where Q is the set of rational numbers in IR. That is, C[O, 11 is the countable union of countably many additively shy sets. This is a contradiction. Thus Z is not additively s hy.

Both the set Z in the above theorem and C[O, 11 \ Z are upper density 1 under the additive operation (see Section 2-15 for the definition of upper density 1). In fact, for every E > O and every Borel probabiiity measure p, there exists a compact set

A' C C[O, 11 such that p(K) > 1 - E since p is tight on C[O, 11. Choose functions g, h E C[O, 11 satisfying

g(0) = 1 + rnax rnax 1 f(z)j, g(1) = -1 - rnax rnax 1f(x)l /EK z€[0.1] fEK z@J] and h(x) = 1 + max max 1f(x)l f EK ~E[O,l] on [O, 11. Then Thus KÇZ-g ~dh'CC[O,ll\Z-h.

Therefore both Z and C[O,11 \ Z are upper density 1.

Corollary 4.7.3 The set Sc in Theorem 4.3.1 is akio multiplicatively shy.

Proof. For my function f E Sc, f -C has at least one zero point. By Theorem 4.7.2 Sc - C is multiplicatively shy, and so is Sc.

As we have seen, the non-invertible elements in the multiplicative semigroup C[O, 11 generate some curious results as regards multiplicatively shy sets. One, possibly better, way to investigate these notions would be to restrict our attention to the set Cf [O, 11 of strictly positive valued functions in C[O,11.

Lemma 4.7.4 With the operation of multiplication Ci[O, II becornes an Abeliun Po6 ish group.

Proof. We only need show that Cf [O, I] is Poliçh. On it we impose an invariant

metric d so that

Then the space C+[O, 11 is complete under the metric d. In fact, for âny Cauchy sequence {f,) C C+[O,11 and a fixed point x E [O, l], {f,(x)) and {l/f,(s)) are Cauchy sequences in the uniform metric. So the set (11f,(z)) is bounded and there exists a function f (x) > O such that

Therefore a function f is generated such that f, -+ f and L/ fn -t 1/ f pointwise.

Note { f') is a Cauchy sequence under the metric d. So f, + f and l/f, -, l/f uniformly. Thus f is continuous and f E C+[O, 11. The separability of C+[O, 11 is clear and hence C+[O, l] is Polish.

The shy sets in the space C+[O, 11 could be cdled again rn~ltiplicati~elyshy sets.

Theorem 4.7.5 A set S C_ Cf[O, 11 is nultiplicatively shy in C+[O,11 if and only if

InS= {lnf: f ES)

is additively shy in C[O, 11.

Proof. Note that the function f (x) = In x is a one-one, continuous mapping Gom C+[O: 11 onto C[O, 11 and satisfies

Thus by Theorem 2.10.2 the result follows.

Corollary 4.7.6 The prevalent function f E Cf [O, 11 is nowhere dzflerentiable.

Proof. Let S denote the set of functions in C[O, 11 that are differentiabie at some point of [O, 11 and let S+ denote the set of functions in Cf[O, 11 that are differentiable at some point of [O, 11. Then S = in S+. Since the set S is universally measurable and shy in C[O, 11 (see Section 4.2), by Theorem 4.7.5, S+ is multiplicatively shy.

Remark. We can construct a mapping F : [O, 11 x [O, 11 -+ C+[Ol 11 by

where g(x) and h(x) are as in Section 4.2. The Borel probability measure p defined by

AX) = h({(tl,t2)E [O, 11 x [O, I] : F(tl,tZ) E X)), with support containing the unit, is transverse to Sf. Similarly, we have the foiiowing results. Corollary 4.7.7 The prevalent function f E C+[O,11 is nowhere of rnonotonic type.

Corollary 4.7.8 The set exp Z is not multiplicatively shy in Cf[O, 11 where Z is as in Theorem 4.7.2. Chapter 5

Space of automorphisms

5.1 Introduction

The space %[O, 11 is dehed as the set of aU homeomorphisms h : [O, 11 + [O, 11 that fix h(0) = O and h(1)= 1. Note that these are exactly the strictly increasing contiauous functions leaving the endpoints fixed. This is a subspace of the cornplete metric space C[O, 11. We shaii show that X[O,1J admits a complete metric. There are discussions on the typical properties of functions in the comp1ete metric space W[O,11 in [Il], for example. We can impose some dgebraic structure on N[0, lj in severai ways. The most natural way is to consider the group operation, defined as composition of functions. In this chapter we shall study some kinds of prevalence and give some examples of non-shy sets by using the arguments we developed in Chapter 2 and Chap ter 3. For a different measure-theoretic study in the space W[O,11, see Graf, Mauldin and Williams [25] and [261. We foiiow ideas of MauIdin and UIam (see [58]), who have defined a probability measure P on %[O, 11 that is "natural" from a probabifistic point of view. Roughly, to generate a horneomorphism h f %[O, Il randomly with respect to the uniform distribution over [O, 11 one chooses h(l/Z) from (O, 1) at random wit h respect to the uniform distribution, then one chooses h(1/4) from (O, h(1/2)) and h(3/4) from (h(l/2), 1) at random again with respect to the uniform distribution. This continues defining h on ail dyadic rational numbers. With probability 1, h is strictly increasing and unifody continuous on dyadic rationa1 numbers and so defines a member h E N[O, 11. A measure P is defined on %[O, 11 to reflect these notions (see [25] for details). In this chapter that say the measure P means such a measure. It has the property that the expected values for h(t) is t, that is,

A number of useful properties are shown in [25] and [26]. For exarnple, P almost every h E ït[O, 11 touches the line y = x on the interval (O, 1). Note, however, that their study is of a different nature than our study here. The group structure plays no significant role in the notion of "prevalence" relative to the rneasure P. The measure P is detemined in a probabilistic manner and inside or outside composition by elements of 'H[O, 11 changes that determination. Thus that study can only be used to contraçt with Our study here. In this chapter we use P to denote such a measure. We find that some sets studied in [25] and [26] are very successful as examples to answer the following open problems. Jan hlycielski [41] asked whether, in a non-Abelian Polish group, the existence of a Borel probability measure left transverse to a set Y implies the set Y is shy- We shail answer this problem negatively in îf[O, 11 by showing that every set {h E %[O, l] : hY(O)= a) (a> O) is left-and-right shy without being shy. Slawomir Solecki showed that a Polish group admitting an invariant metric satisfies the countable chah condition if and only if this Polish group is locally compact- The problem is left whether ali non-locally compact, non-Abelian Polish groups or some of them do not satisfy the countable chah condition. %[O, 11 is a non-locaIly compact, CHAPTER 5. SPACE OF A I/TOMORPHISII,iS non-Abelian Polish group with no invariant metric that rnakes it complete. We shall show t hat this group does not satisiy the countable chain condition. The organization of this chapter is as foliows. In Section 5.2, we show that the group X[O, 11 is a non-locally compact, non-A belian Polis h group without invariant metrics. In Section 5.3, 5.4, 5.5, 5.8, we use compact curves to construct Borel probability measures to show that certain sets are shy, or left shy, or Rght shy. In particular, we show that, in R[O,11, there exists (1) a Borel probability measure p that is both Left transverse and right transverse to a Borel set X, but is not transverse to X. (2) a Borel probability measure p that is left transverse to a Borel set X, but is not right transverse to X. In Section 5.6, we give a characterization of compact sets in %[O, 11 and a compact set argument to show that a set is non-shy, or non-left &y, or non-right shy. In Section 5.7, 5.9, 5.10, 5.11, 5.12, we use the compact set argument of Section 5.6 to show that certain sets are non-shy, or non-left shy, or non-right shy. Specifically, we show that some sets of the first category are non-shy, some left-and-right shy sets are non-shy, and some typical properties are not prevalent. Fiaaily, in Section 5.13, using the results in Section 5.12, we show that the non-Abelian Polish group R[O,l] does not satisfy the countable chain condition. In Section 5.14, severd open problems are given.

5.2 Space of automorphisms

Weshall first prove that R[O,11 is a Polish group that is neither Abelian nor locally compact.

Lemma 5.2.1 The space %[O, 11 is a non-locally compact, non-Abelian Polish group. Proof. %[O, 11 is a Gs set in a complete space (see Ill, pp. 4681). Indeed, it is topoIogically cornplete with respect to the metric

which is topologically equivdent to the uniform metric p. (This is mentioned in Oxtoby [43].) The composition of two functions in N[O, Il wili usudy not commute. For example, f (x) = sin (:x) and g(x) = x2 E %[O, 11,

7r 2 f O g(x) = sin (Ex2) but g O f(z) = (sin - .

So X[O, 11 is not Abelian. Now we show that the Polish group 3-1[0,1] is not locally compact. For any function f E N[0,1] and E > O there exists a 6 > O such that a(f,g) < E for all g E R[O, 11, p(f, g) < 6. Thus we can find a closed rectangle contained in B( f,E) such that its sides are pardel to axes and two of its vertices are on the graph of f. Let a and b be x-coordinates of the two vertices with a < b. Choose a < c < b and define a sequence of functions (fn) for large n. Precisely, let fn(x) = f (x) when O 5 x 5 a; fn(x) = f(x) when b 5 x <_ 1; then f, is the segment function connecting (a, f(a)) and (c, f (b) - l/n) when a 5 x 5 c; and fn is the segment function connecting (c, f(b) - I/n) and (b, f(b)) when c 5 x 5 6. Then {fn) and al1 its subsequences fail to converge in %[O, 11, and so %[O, 11 is not locaiiy compact. It is clear tbat %[O, 11 is separable since it is a subspace of the space C[O, Il. Thus N[O, 11 is Polish. I

The sequence (f,) Ç B(f, E) in the above proof aiso fails to converge in the uniform metric. So the uniform metric p does not give a complete metric for the set of al1 homeornorphisms. Also we can daim that the metric p is not invariant. In fact, choose functions x, x', sin (Zr) E H[O, l] and let f(x) = x2. It is easy to see that the function x - sin (fx)attains maximum at 4 on [O, l] where O < BO < 1 and CHAPTER 5. SPACE OF A UTOMORPHISMS

cos (:Bo) = 2/7r. By computation Bo 0.560664, Note

X 7r K A f (r)- f (sin ?z) = (r - sin -1) (z + sin -r) and 4 + sin -Bo * 1.331842. - 2 2 3Y

z,sin %)2 < p (f(x), j (sin iz)) Therefore the uniform metric p is not (left) invariant. It is, trivially, right invariant since for any f,g, h E R[O,11,

that is p(f 0 h,go h) = p(f,g). One rnight ask whether a different equivalent metnc on 'H[O, 11 might be found that is invariant. Our next theorem is cited in Christensen [El where it is attributed to Dieudonné, who mentioned it in [17] as an example but did not give a detailed proof. It follows from this theorem that there is no invariant metric on this space, since it is known that an invariant metric on a Polish group must make it complete (see [31, (2.3)j).

Theorem 5.2.2 There is no lefi invariant metric on the Polish group %[O, 11 that makes the group complete.

Proof. We know that every metrizable group must have a left (right) invariant metric (see [30, pp. 581). For a Polish group, if it has an invariant metric then it is complete in this metric (see [31, (2.3)]). Suppose that X[O, 11 is complete in a left invariant met& d. Now consider a sequence {un}: Then un E R[0,L] and

Since the uniform metric p is top~logicdlyequivalent to the metric u as in Lemma 5.2. l7 by Theorem 9.24 in [Il, pp. 3881, for every u E R[O,11 and c > O there is a 6 > O such that for al1 v E N[O,11,

p(u,v) < 6 d(u,v) < E and d(u,u) < 6 * p(rs,v) < é.

Since p is right invariant,

Then there exists a N > O such that p(u,u;', x) < 5 if n, m > N. Thus d(u,u;', x) < r if n,rn > N,and so {un) is a Cauchy sequence in the d metric but it cannot converge in the d metric. In fact, if {un)converges to an elernent u E 'H[O, 11, then (un') would converge to u-' in the d metric since the d is left invariant. Thus again by Theorem

9.24 in [ll, pp. 3881, (u;') would converge to u-' in p metric (uniformly). But (un') converges pointwise to the function

which is discontinuous at x = O. This is a contradiction. Thus there is no Left invariant metric on X[O, 11 that makes it complete. The result follows.

5.3 Someexamplesofprevalent propertiesinX[O,l]

In this section ive will give some prevalence resuits by considering the folIowing probability mesure. Define a mapping F : [1/2,1] -, R[O,11 by F(t) = st. It is eaçy to CHAPTER 5. SPACE OF AUTOMORPHISMS see that F is continuous, and so F([L/2,1])is a compact set. As in Section 2.2 we define a probability measure p by

p(X) = 2h({t f [1/2?11 : F(t)E X)). (5.1)

We use this example of a measure to show that a rneasure may be left transverse to a set without being right transverse, and that a rneasure may be left-and-right transverse wit hout being transverse. We begin with a simple theorem in order to contrast it with the more surprising results that follow.

Theorem 5.3.1 For a, b E (O, l),let Sa,b denote the set of functions h E R[O,11 such that h(a) = b. Then the probability measure p defined by 5.1 is transverse to Sa,b.

Proof. It is easy to see that the set SaVbis cIosed and nowhere dense in 'H[O, 11. In particular Sa,bis a Borel set. Now for any g, h E 3-1[0,1],consider the set

If t E T, then on 1, g(ht(a))= b and hence ht(a)= g-l(b) where g-' is the inverse function of g. So the set T can contain at most one element and hence the probability measure p is transverse to Sa,b.

Corollary 5.3.2 The set Salbis closed, novrhere dense, and shy.

In particular, from this theorem we see that for a given function f the set of elements h E %[O, 11 such that h(x) = f (x) for al1 x in sorne subintewal of [O, l] must be both ktcategory aad shy. The reason is that this set can be expressed as the countable union of the sets in the theorem taken over al1 rational nurnbers a. In contrast to this simple result we shd show how a Borel probability measure may be left transverse or right transverse or left-and-right transverse and yet not CHAPTER 5. SPACE OF A LITOL~~ORPHISMS 106

be transverse to a set. LVe start with an elernentary example that illustrates the distinctness of left/right transverse notions in X[O, 11.

Theorern 5.3.3 Giuen an intemal I with the closure TC (O, l), let G(I) denote the set of functions in the space %[O, 11 that are linear on 1. Then the probability measure p dejned by 5.1 is left-and-right transverse to G(I), but is not transverse to G(I).

Proof. First we show that G(I) is a Borel set. It is easy to see that

G(I) = { f E %[O, 11 : f is linear and slop f > O on I)

where slopf is the slope of f on 1. Let

f E 'H[O, 11 : f is Iinear and slopf 3 n

PVe will show that F,,is closed. For aay Cauchy sequence ifk) C Fn there exists

f E x[O,11 such that fk + f uniformly. On I we denote fk(x) = ûkz + Bk where uk 2 l/n. Choose ttvo distinct points XI,x2 E 1 \ {O). Then

So ~kk-, a E R and therefore pk -, P E W. Then akx +Bk -, ax + B = f(x) on 1. Since al1 ak >_ l/n so a 2 l/n and f E Fn. Thus each F, is closed and the set G(I) is a Borel set. Further, we can show that Fn is nowhere dense and hence G(I) is of the first category. In fact, since the complete metric r~ on X[O, 1) is equivalent to the uniform metric p, for any non-empty open bal1 B(f, €1 C 'H[O, 11, there exists a 6 > O such that for any h E %[O, 11, p(f, h) < 6 we have o( f, h) < e. It is easy to construct a non-linear function g E X[O, 11 such that p( f,g) < 6. Then g E B(f, E) but g 4 Fm. So Fn is nowhere dense and G(I)is of the first category. CHAPTER 5. SPACE OF A LITOMORPHIS&IS 107

For any g E X[O, 11, ive use the Borel probability measure p defined by 5.1 and consider the set

Tl = {t E [1/2, I] : g O F(t)E G(I)}.

If t E Tl then on 1,

where a(t)> O. Let y = xt, then g(y) = a(t)yl't+P(t)for y on a subinterval of [O, Il. Suppose XI(Tl)> O. Then the functions y = xt (t E Tl)map I into uncountably many subintervals of [O, 11. Thus there must exist tl,t2 E Tl,tl < t2 such that for y on some interval J, 1 a(tl)yq + P(h) = 4tz)yk + B(t2)-

Then by differentiating both sides of the equality above with respect to y E J we have --- ha(t2) yt1 t2 = t24tl)

This is impossible for a11 y E J. So XI(Tl)= O and hence p is left transverse to G(I). Lknow show that p iç dso right transverse to G(I).For any h E %[O, 11, consider the set

T2= {t E [1/S,Il : F(t)O h E G(I)}.

where a(t)> O. Differentiating both sides with respect to t on I we have

From the above equality we can know that the set T2has at most two elements. If not, there exist tl,tz E Tz,tl < t2 such that on I This is impossible since the function h(x) is strictly increasing. Thus XL(T2)= O and hence p is right transverse to G(I). We now show that the probability measure p is not transverse to G(i). Choose 9,h E X[O, l] such that g(x) = 1 + ahz and h(x) = ez-l on I where cr is a positive constant depending on 1. Then on [1/2, IJ the function

g O F(t)o h(x) = g(ht(x))= crtx + 1 - ut is linear for any t E [l/2,1]. Thus

Therefore p is not transverse to G(I).

Corollary 5.3.4 The set of functions in the space R[O, 11 that are somewhere linear (i.e., linear in at least one subinterval of [O, 11) is a left shy and right shy set, and is oj the fist category in R[O, 11.

Proof. For any intervak I with endpoints in (O, I), G(I) is left shy, right shy and first category from the above theorem. The set of functions in %[O, 11 that are somewhere linear is the union of al1 G(I)over al1 subintervals with rational endpoints in (O, l), and so the result follows. rn

Corollary 5.3.5 The set of piecewise linearfunctions in 3.1[0,1] is lefi-and-right shy, and first category.

Proof. Let S denote the set of piecewise linear functions in X[O, 11. From Theo- rem 5.3.3 we know that the set S is contained in a set that is left-and-right shy. So we need only show that S is Borel- Let

It can be checked that CU wmw

?Ve will show that the sets Sm,n,l,pare closed and nowhere dense. For any Cauchy sequence {fk) C Sm,n,l,pthere exists f E %[O, 11 such that fk + f uniformly. Then there exist a-, 8 and partitions ([a:, bf])zlof [O, 11 with minl<;<,- - Ibf - a:] 2 l/n such that m and 111 5 a: 5 1, -p < pf < p, i = 1, - - ,m. By standard arguments ive can choose a subsequencz {Ccj) of N such that

and

a? 4 a; E [O,l], 6p -+ bi E [O,l].

It is easy to see that ([ai,bi])Zl is a partition of [O, 11 and m I(+)= C(aix+ P~)X[~.L.I(X)E Sm.n.i.p- i=l Thus S,,,,,J,is closed. Similar arguments as for F, in Theorem 5.3.3 show that Sm,,,r,p is nowhere dense. Thus S is Borel and the result follows. W

Remark. For any cIosed interval I C (O, 1) and any function f E 7@, 11, choose g(x) = 1 + a ln x and h(x) = f-'(e'-') on I where a is a positive constant depending on I. Then

g O ft O h(x) = g(f (h(x)))= atx + 1 - crt CHAPTER 5- SPACE OF AUTOMORPHISMS

is linear for any t E [1/2,1].So we cannot expect to choose some function f E %[O, 11 such that the compact cuve F(t)= (t E [1/2,1])would be transverse to the set of somewhere linear functions in R[O,11. There still remains the following problem. The measure which proves that the sets G(I)are left shy and right shy does not prove that they are shy. Some other probe would be needed. In fact, the existence of a left-and-right transverse measure does not imply the existence of a transverse measure as we shall show in Theorem 5.12.3.

PROBLEM 11 Let S be the set of functions in the space 'H[O, 11 that are somewhere linear on [O, 11. Is the set S shy?

5.4 Right transverse does not imply left transverse

In this section we will show t bat a Borel probability measure right transverse to a set need not be left transverse to the set Y. kVe will use again the compact cuve F(t) and the Borel probability measure p defined by 5.1 to verify this in the following two t heorems.

Theorem 5.4.1 Givcn an interval I wïth the dosure TC (0, l),let G(I) be the set of functions in the space %[O, 11 ihat are of the fonn cr hx+ j3 on I vthere a (> O), p are constants depending on the corresponding finctions. Then the probability measure p defined by 5.1 is right transverse to G(1) but is not lefi transverse to G(I).

Proof. Sirnilar arguments as in the proof of Theorem 5.3.3 show that G(I)is a Bore1 set and is of the first category. For any h E R[O, 11 consider the set We claim that T3 contains at most one element. If not, there exist tl,t2 E T3, tl < t2 such that F(tl)o h, F(t2)O h E G(I).Then on 1

By differentiating both sides of the above two equalities with respect to z, it follows that on 1,

This is impossible since h(x) is strictly increasing on I. So X1(T3)= O and the probability measure p defined by 5.1 is right transverse to G(I). We now show that p is not left transverse to G(I).Choose a function g E X[O, 11 such that on 1, g(x) = l + a ln x. Then for any t E [1/2,1],

Thus g O F(t)E G(1)and so

Hence p is not left transverse to G(I).

Similm arguments as for Corollary 5.3.4 show the following corollary.

Corollary 5.4.2 The set of functions in the space 3[0,1] that are somelohere of the form a ln x + B is right shy and fist category where a > O, P are constants depending on the conesponding functions.

If a Borel probability measure pl is right transverse but not left transverse to a set S contained in a Polish group G, then the measure pz, defined by p2(X) p~(X-l), is left transverse but not right transverse to the set S-l. Thus £rom Theorem 5.4.1 we cm obtain a Borel probability measure that is left transverse to a set but not right transverse to this set. Here we supply a further concrete instance of a measure that is left transverse but not right transverse to a set.

Theorem 5.4.3 Given an interual I with the closure 7 C (0, 1) and an ct > O, let G(I)denote the set of functions in the space %[O, 11 thot are of the fom (1 + cr ln x)' on I where p > O is a constant depending on the corresponding functions. Then the probability rneasure p defined by 5.1 is left transverse to G(I) but not right transverse to G(1).

Proof. To show that G(I)is a Borel set we modify part of the proof of Theorem 5.3.3. Note that G(I)= {f E%[o,~]:f =(l+alnx)~~on I).

Let

Fn = {f E G(I) : I/n 5 5 n).

T hen

We will show that Fn is closed and nowhere dense. That F, is nowhere dense in %[O, 11 can be shown in a similar way as in Theorem 5.3.3. We now show that each F, is closed. For any Cauchy sequence {fk) C Fn, there exists f E %[O, 11 such that fk -t f uniformly, and l/n 5 5 n. SO ive cm find a subsequence {kj) of N such

on I and hence f f F,. Therefore Fn is closed and it follows that the set G(I) is Borel. Xow we show that the probability measure p defined by 5.1 is left transverse to G(1). For any g E %[O, 11 consider the set

Let y = xt then

Suppose X1(T4)# O. Then the iunctions y = xt (t E T4) map I into mcountably many intervals contained in [O, 11. Thus there exist tl,t2 E T4, tl < t:, such that for y on some interval J £ [O, 11,

By differentiating both sides of the last equality with respect to y, it is easy to see that on J,

From the last two equalities we have

Again by differentiating both sides with respect to y E J, it is easy to see that on

J, t2 - tl = O. This contradicts that t2 > tl. Thus X1(T4)= O and hence the probability p is left transverse to G(1). We conclude by showing that the probability measure p is not right transverse to G(I).Choose h E X[O, 11 such that h(x) = 1+ a In x on I. Then for any t E [1/2,1],

So F(t)o h E G(I)and

Hence p is not right transverse to G(1). M

Again similar arguments as for Corollary 5.3.4 show the following corollary.

Corollary 5.4.4 For any a > 0, the set of functions in the space H[O, 11 that are oj the fom (1 + orlns)' somevhere is left shy and jirst category, where fi > O is a constant depending the corresponding functions.

5.5 A set to which the measure p is left-and-right transverse

In the following let us look at one more example of left-and-right shy sets in X[O, 11.

Theorem 5.5.1 Let O < E < hl < w, and

Then the Borel probability measure p defied by 5.1 is left-and-right transverse to SC,M*

Proof. For any Cauchy sequence { f,) Ç SctMthere is a function f E %[O, 11 such that fn + f uniformly. Note that Therefore f E SCsMand Sc,hfis cIosed. Since the complete metric a on X[O, 11 is equivalent to the uniforrn metric, thus for any non-empty open bail B(f, e) N[O, Il, there exists a 6 > O such that for any h E R[O, 11, p(f, h) < 6 we have a(f,h) < E. So ive can constmct a function g E 3-1[0,1] such that p( f,g) < 6 and g(x) = 42x for sufficiently small x E (O, 1). Then g E B(f, e) but g @ Sc,M.SO Sc,Mis nowhere dense. We now show that Scy is left shy and right shy. For any g E E[O, Il, consider the set

Tl = {t E [1/& 11 : g 0 F(t) E S€,M).

If t E Tl then

Since the function g is differentiable alrnost everywhere, we have

for almost every c E [O, 11. Suppose that Tl contains two or more elements. Then there exist tl,t2 E [1/S,11, tl < t2 such that

and

for almost every z E [O, 11. Thus for almost every z E [O, 11. Since l/t2 - l/tl c 0, the above inequality is impossible. Thus Tlcontains at most one element and hence X1(TI)= O. It follows that the Borel probabili ty measure p defined by 5.1 is left transverse to Se,hf. We now show that p is ais0 right transverse to Se,M.For any g E X[O, 11, consider the set

T2 = {t E [1/2,1] : F(t)og f Se,icr)-

If t E T2,then €1~- YI 5 lgt(x>- gt(y>l5 VIX -Y/-

Since g is differentiable almost everywhere,

for almost every x E [O, 11. If T2 did contain two or more elements, there exist tl: t2 E [O,ll, tl < t2 such that

and

€ < lt2gt2-1(x)g'(x)l 5 1bI for almos t every x E [O, 11. Thus

for almost every x E [O, 11. This is impossible since g(x ) + O as x 4 O. So AL(T2) = O and the probability measure p is right transverse to Sc,,ti. By Theorem 2.9.7 the result folIows. I

Corollary 5.5.2 The set S

is left-and-right shy, and first category in %[O, 11. Proof. Since the set S = Un=,m Slfn,n, the result follows. H

PROBLEM 12 Is the Borel probabdity measvre p dejîned by 5.1 transverse to the set in Theorem 5.5.1?

5.6 A compact set argument

In this section we will give a characterization of compact sets in 'H[O,11. By applying such a characterization of compact sets we shali find examples of sets in 31[0,1] that are neither Left shy nor right shy. Some of these results contrast sharply with those in [23] and [26]. The following Iemma contains a simple method that can be used to show the non-shyness of some sets.

Lemma 5.6.1 If a set S in R[O,11 contains a two-sided translate (left, or right) of eveq compact set, then S is not shy (left shy, or right shy).

Proof. For any Bore1 probability mesure p there is a compact set K C %[O, 11 such that p(h') > O (see Section 3.5). Since there functions g, h E %[O, 11 such that g 0 k 0 h E S for all k E K, then K 2 g-lSh-' and so p(g-'Sh-') 2 p(K) > O. Thus S is not shy. The non-left shy result and the non-right shy result can be shown in the sarne way. H

We first give a characterization of compact sets in R[O,11 by applying the Arzeli- Ascoli theorem to the space R[O,11.

Theorem 5.6.2 A set K C 'K[O,l]is compact ifl (i) K is closed, (ii) K is equicontinuous, and (iii) for every non-empty closed set Ko C i1T,

inf h(x) and sup h(x) hEKo &ho are both in %[O, 11.

Proof. Let K C %[O, 11 be a compact set. Since 'H[O, 11 is Hausdorff, (i) follows.

By Theorem 9.58 in [Il] K is totally bounded. Let c > 0, and let fi,- - - ,fn be an ~/3-netin K. For every f E K, there exists j 5 n such that

where p is the the uniform metric. Thus for any x, y E [O, 11,

Since each f, is uniformly continuous on [O, 11, there exists a 6 > O such t hat Ifi(r)- f;(y)I < c/3 for al1 1 5 i 5 n if lx - y1 < 6. Therefore 1 f(x) - !(y)[ < c, and hence (ii) is true. We now show that (iii) is true. For any non-empty closed set & C IC, let

g(x) = sup f (x) and h(x) = inf f (x). /Eh-O f €Ka

It is easy to see that both g and h leave x = 0, I fixed. Since ho E ïi' is closed, so it is compact. For any XI,xz E [O, 11, xl < xz, there exist fi, fz f KO such that fi(x1) = g(x1) and f2(x2)= h(x2). Since fi, fi are strictly increasing, Thus g(x) and h(x) are strictly increasing functions. Since 11' is equicontinuous from

(ii), for every é > O there exists a 6 > O such that for ait f E Ii', 1f(x) - f(y)[ < ~/2 if lx - y1 < 6. So for al1 +,y E [O,l], 12- y1 < d and f E Ih,

T hus 1 sup k(x)- sup k(yj < sup Ik(x) - k(y)l 5 -€- kEKo ~EKO kEKo 2 That is, g(x) - g(y) < E. Changing the positions of s and y yields g(y) - g(x) < c.

Therefore Ig(x) - g(y)1 < 6 and so g is continuous. For the continuity of hl note that for all t,y E [O,1J, lx -y( <&and f E &,

and hence by the arbitrariness of f E &,

That is, h(x)-h(y) 5 !c < c. Changing the positions of z and y yields h(y)- h(x) < e.

Thus 1 h(x) - h(y) f < E and hence htx) is continuous. Therefore (iii) is true. To show the sufficiency of the conditions (i), (ii) and (iii), let {f,) be any infinite sequence in K. Since K is equicontinuous and uniformly bounded, (f,) has a uniformly convergent subsequence {f,,) such that f,, -, f by the Arzeli-Ascoli The- orern. We claim f E %[O, 11 and the theorem is proved. Suppose f 4 RB,11. Then f must be constant on some subinterval [a,b] C [O, 1j. Fix c E (a,b), and consider the This set is then closed. If it is non-empty and infinite, then for a11 x E (a,c),

SUP fnk (4= m? fn, €Pl which violates the condition (iii). Thus PL is finite. Similarly the set

is also closed. If it is non-empty and idnite, then for al1 x E (c, b),

which also violates the condition (iii). Thus Pz is finite. But both Pl and P2 cannot be finite, and so we have a contradiction. Thus f E X[O, 1) and so Ir' is compact. .

5.7 Exarnples of non-left shy, non-right shy sets

In this section we apply Theorem 5.6.2 to show that some sets discussed in [L5] and [26] are neither left shy nor right shy.

Theorem 5.7.1 For any function q(x) E X[O, 11, let

Then G1 is a Borel set that is neither left shy nor right shy.

Proof. Wekst show that Gz is a Borelset. Note Since al1 h are continuous and the metric on %[O, 11 is equivalent to the uniform metric, the sets are open. Hence Gris a Borel set. We now show that Giis neither left shy nor right shy. For any compact set I<, by Theorem 5.6.2 there exists a function g E %[O, 11 such that k(x) < g(x) for al1 k E K and al1 z E [O, 11. Choose

Then for al1 k E K,

lim sup f (k(4 = O, x-O+ Q(X) and so f o k E Gr.By Lemma 5.6.1, G1is not left shy. To show that G1is not right shy, choose

Then for al1 k E K,

T hus k(f lim sup (41 = O, x-O+ ~(4 and so k o f E Gz.By Lemma 5.6.1, Giis not left shy.

Corohry 5.7.2 Let SI = {h E R[O,11 : h'(0) = 0).

Then Si is neither left shy nor right shy. CHAPTER 5. SPACE OF A UTOMORPHIS~US 1'2'2

Proof. Let q(x) = x in Theorem 5.7.1. Then G1= Si and the result follows. . Theorern 5.7.3 For any function q(x) E X[O, 11, let

Then G2 is a Borel set that is neither left shy no+ right shy.

Proof. We first show that Gz is a Borel set. Note

Since al1 h in H[O,11 are continuous and the metric on X[O, 11 is equivalent to the uniform metric, so the sets

are open and lience Sz is a Borel set. LVe now show that S2 is neither left shy nor right shy. For any compact set K, by Theorem 5-62, there exists a function h E X[O, 11 such that k(x) 2 h(z) for al1 k E K and dlx E [O, 11. Choose CHAPTER 5. SPACE OF A UTOM ORPHISMS 123

and so f 0 k E G2.By Lemma 5.6.1, Gz is not left shy. To show G2 is not right shy, choose f (x)= h-1(q1/2(x))E 'H[O, 11.

Then for dl k E K,

and so k O f E S2. By Lemma 5.6.1 again S2 is not right shy.

Corollary 5.7.4 Let

Then S2 is neither left shy nor right shy.

Proof. Let q(x) = x in Theorem 5.7.3. Then G2= S2 and the result follows. W

From Theorem 5.7.1, Theorem 5.7.3, Coroilary 5.72 and Coroilary 5.7.4 irnmedi- ately we have the following result.

Corollary 5.7.5 The sets Gl, Gp, Sl and S2 in Theorenz 5.7.1, Theorem 5.7.3, Corollary 5.7.2 and Corollary 5.7.4 are al1 neither shy nor prevalent.

Proof. From Theorem 5.7.1 and Theorem 5.7.3, both Gland G2are neither left shy nor right shy. So they dl are not shy. Since each one of both G1 and G2is contained in the complement of the other, so both Gl and G2 are not prevalent. Similarly, we can show that both SI and S2 axe neither shy nor prevalent.

In [25] and (261, Graf, Mauldin and Williams showed that P,(S1)= 1 where Pa is the right-average of the measure P as in the introduction part of this chapter, that is CHAPTER 5. SPACE OF A UTOMORPHISMS 124 for Borel sets B E R[O, 11. in the foilowing part of this chapter we always use Pa to denote the right-average. However, we showed in Corollary 5.7.5 that Siis neither shy nor prevalent. From this point we know that both P and Pa are neither left transverse nor right transverse to the complement of SI.

5.8 Examples of left-and-right shy sets

In order to compare with the results in last section we show here the following theorem.

Theorem 5.8.1 For any function q(x) E N[O, 11, let

Then S is a Borel set that is left-and-right shy in %[O, 11.

Proof. LVe first show that S is a Borel set. Note

Since al1 h f R[O, 11 are continuous and the metric on %[O, 1] is equivalent to the uniform metric, al1 sets

are open, and so the set S is a Borel set. We now consider the set

S., = {h t %[O, 11 : o c lirn inf 3< lirnsup - .-O+ q(x) =-O+ q(4 for a11 a,P > O, a < p. Once we show that the set S,J is left shy and right shy, then

m 5 = (Jsi,. .. n=l CHAPTER 5. SPACE OF A UTOMORPHISMS 125

is also left shy and right shy. FVe now show that the Bore1 probability rneasure p defined by 5.1 is right transverse to for any a,P > 0, CY < B. For any h E RIO, 11, consider the set

R = (t E [l/fz, lj : F(t)0 h E Sad).

If tl E R, t hen for sufiiciently small x E [O, 11,

This means that, for any t2 # tl,

as x -, Of. Consequently t2 R. Thus R contains at most one element and p is right transverse to sap.

To prove that S is left shy we choose a mapping Fl: [1/3,1] -+ R[O, 11 bÿ Fl(t)= qt(x) and define a Borel probability measure pi by

For any h E %[O, 11, consider the set

SE s f L, tben for sufficiently small x € [O, 11,

That is, aq(x) < h(qs(x))< Bq(x) for sufficiently small x E (O, 1). Let y = q'(x). T hen lh(y) lDY+ for sufficiently smdI y f (O, 1). Suppose t hat L contains two or more elements. Then there exist SI,sz E [1/2,1], sl < sz such that and 1 1 ay" < h(y) 5 ,ûyz for sufficiently smdl y E (O, 1). Thus

for sufficiently small y E (O, 1). This is impossible since l/s2 - l/sl < O. Thus the set L contains at most one element and hence the probability pl is Left transverse to Sa,p. By Theorem 2.9.7 SpVBis left-and-iight shy, and so the set S is also left-and-right shy.

Let q(x) = x. Immediately we have the following.

Corollary 5.8.2 For any a, O < a < cm, the set

is lefi-and-right shy in %[O, 11.

This conclusion contrats sharply with Corollary 5.7.2 and Corollary 5.7.4. In Section 5.12 we will see that the set {h E X[O, 1] : h'(0) = a} is not shy in 71[0,1]. As a contrast to the reçults in Theorem 5.7.1 and Theorem 5.7.3 we show that the sets G1in Theorem 5.7.1 and Gz in Theorem 5.7.3 are of the first category in X[O,11. It follows, too, that the sets Siin Corollary 5.7.2 and S2 in Corollary 5.7.4 are also of the first category in %[O, 11.

Theorem 5.8.3 The sets G1 in Theorem 5.7.1 and G2 in Theorem 5.7.3 are of the first category in R[O,11.

Proof. Note that The sets

are closed and, hence, so is

for any m 2 1. We claim that N is nowhere dense in X[O, 11. It foliows that G1is first category. Since the complete metric a on R[O,1) is equivalent to the unifom rnetric, thus for any non-empty open bdB( f, c) C %[O, LI, there exists a 6 > O such that for all h E R[O,11, p( f,h) < 6 we have a( f,h) < 6. If f 4 1V there is nothing to do. If f E N, then we cmconstruct a function y E 8[O, I] such that p(f, g) < 6 and g(x) = (S/p)q(x)for sufficiently small x E (O, 1)- Then g E B( f,E) but g $! N. Thus N is closed, nowhere dense. For the set G2,we know

oomm h(Tn) GZ=~u n (hE~[o.l]:- p=~m=1 n=m Wn1 a}.

SimiIar arguments as for G1 show that, for any rn, the set

is closed and nowhere dense in %[O, 11, Therefore Gz is of the first category. m

5.9 Non-shy sets that are of the first category

In the following theorems we wiil give further sharp contrasts between our measure- theoretic notion and category, and also between our resulis and results in [25] and [26]. The geometric pictures of the following sets are clear. CHAPTER 5. SPACE OF A UTOhfORPHISIVlS

Theorem 5.9.1 For any function q(x) E X[O, 11, let

and

= {h E %[O, 11 : h(x) 5 q(x))

Then both S> and Sc are closed, nowhere dense sets that are neither lefi shy nor right sh y.

Proof. It is easy to see that both S, and Sc are closed. Since the cornplete metric o on R[O,L] is equivalent to the uniform metric p, for any non-empty open bal1

B(j,E) Ç %[O, 11, there exists a 6 > O such that for a11 h E %[O, 11, p( f,h j < 6 we have O( f,h) < c. If f $ S,, there is nothing to prove. If f E S,, we can construct a function g E X[O, 11 such that p( f,g) < 6 and g(x) = q2(z)for sufliciently smail x E (O, 1). Then g E B(f,c) but g # S>. Thus S> is closed and nowhere dense in %[O, 11. SimiIar arguments show that S< is also closed and nowhere dense in RIO, 11. Ive now show that both are neither left shy nor right shy. For any compact set K, by Theorem 5.62, there exist g, h E R[O,11 such that h(x) 5 k(x) 5 g(x) for al1 k E I\' and al1 x E [O, 11. Choose functions

Then, for any k E K and ail x E [O, 11, CHAPTER 5. SPACE OF AUTOMORPHIShIS

and

(k0 f4)(4 = k(f4(4! L g(f4(4)= ~(4-

Thus fi O k, k O f2 E S> and f3 O k, k O f, E S, . Therefore, by Lemma 5.6.1, both S, and S< are neither left shy nor right shy.

Theorem 5.9-2 For any m E (O, 11, 1 E [1, +CO) and q(x)f %[O, 11, the sets

are closed, nowhere dense sets that are neither shy nor preualent.

Proof. It is easy to see that al1 sets Sm,>and Si,

B(f,E) 'H[O,11, there exists a 5 > O such that for al1 h E R[O,11, p(f,h) < 6 we have a( f,h) < e. If J 4 Sm,>,there is nothing to prove. If f E Sm,>,we can construct a function g E X[O, 11 such that p(J,g)< 6 and g(x) = mq2(x)for sufficiently smdl

x E (0,l). Then g E B(f,E) but g 4 Sm,,. Thus Sm,, is cIosed and nowhere dense in 'H[O, 11. Similar arguments show that Sr,and SL,= S> and Sr,<= S< are neither Ieft shy nor right shy. Since

S, n S, = {h E '~[o.i]: h(t) = q(t)) is a singleton and hence a shy set, so both Sr,>and Si,3 SI,, and Sr,< > Si,<,so Sm,>and Sr,< are not shy. On the other hand, where Gi and G2are sarne as in Theorem 5.7.1 and Theorem 5.7.3 respectively. By Corollary 5.7.5, G1and Gzare not shy in X[O, 11. Thus Sm,>and Si,< âre not prevalent in H[O, 11.

In [26] it was shown that for every m E (O, Ij and 2 E (1, +m),

E({hE %[O, 11 : h(x) >_ mt)) = O and

where P. is the right-average of the measure P. However, according the above theorem, these two sets are neither shy nor prevalent and yet they are closed, nowhere dense sets in R[O,11.

5.10 Non-prevalent properties that are typical

In this section we give several non-prevalent properties that are typical. Again we use the sets discussed in [25] and [26] and compare our results with the corresponding results in [25] and [26].

Theorem 5.10.1 For any function q(x) Ç W[O,11, let

and

Then 60th Ci and C2 are neither shy nor preaalent.

Proof. Note Ci > G1and C2 > Gz where G1and G2are as in Theorem 5.7.1 and Theorem 5.7.3. By Corollary 5.7.5 both Gl and Gz are not shy, and so both Ci and C2 are not shy. LVe now show that their cornplements 5 and are also not shy. Let Q be the set of rationd numbers in (O, 1). Then

and

In fact, let h E H[O, 11 such that lim - z c > O. Then thme exists a 6 > O such that for al1 x E (0,6), h(z)> cq(x). Let r E Q such that r < min(h(b),c). Then h(x)3 rq(x) for dl x E [O, 11. Thus

The converse is obvious, and so the first equality of the above two holds- FOCthe second equality, let h E H[O, 1] such that lirnsup,-, < M < +CO- Then there exists a 6 > O such that for al1 x f (0,6), h(x) < hfq(x). Let r E Q such that r < min(l/M, q(6)). Then h(x) < l/rq(x) for ail x f [O, 11. Thus

The converse is obvious and so the second equality holds. Thus Cland Cz are Borel sets. By Theorem 5.9.2, and are not shy, and hence the results follow. 1

Let q(x) = x in the above theorem. Then the sets

are neither shy nor prevalent. By way of contrast, note that in [25] and [26] it was shown that these two sets have P, rneasures 1 (see [26,Theorem 5.101). From the following theorem we see that these two sets are dense Gs sets in X[O, 11. Theorem 5.10.2 The sets Ci and C2 in Theorem 5.10.1 are dense Gs sets in %[O, 11.

Proof. From the proof of Theorem 5.10.1 we know

and

CI C2 = U {h E H[0,1] : h(x) C i/rq(z)} r EQ where Q is the set of rational numbers in (O, 1). By Theorem 5.9.2 we know that for any r E Q, the sets and

{h E 'FI[& 11 : Mx) 5 l/rq(x)) are closed, nowhere dense sets. Thus Cland Cz are dense G6 sets in %[O, 11.

Theorem 5.10.3 For any q(x) E X[O, 11, the typical function h E X[O, 11 sotisjies h(4 h(4 lim inf -= O and lim sup -= +m. =-O+ q(x) =-O+ ~(4 Remark. Although the sets Gi and G2 in Theorem 5.7.1 and Theorem 5.7.3 are also neither shy nor prevalent, that Cl and C2 in Theorem 5.10.1 are dense G6 sets contrasts sharply with the fact that G1and Ga are of the first category.

5.11 Corresponding results at x = 1

LVe now carry these results over to the right hand endpoint of the interval [O, 11. Since the group operation on %[O, 11 is the composition of functions, we do not know whether the mapping T : h + h by i(x) = 1 - h(1- x) changes the shyness of sets. We will leave it as an open problem. Here we will deal with some sets involving the behavior of functions at x = 1 individually. CHAPTER 5. SPACE OF A UTOMORPHISMS

Theorem 5.11.1 For any function q(x) E %[O, 11, let

and 1 - h(z) S, = {h E R[O,11 : iim = - z-1- 1 - q(x) +-} Then both S3 and S4 are Borel sets that are neither left shy nor right shy in ?f[O, 11.

Proof. Again we show that both S3 and S4 are Borel sets. Note that

1 - h(1 - Y') si = {h E %[O, ij : iim = O} n-00 1 - q(1 - 2-9

and

1 - h(l - 2-") s4 = {h E %[O, 11 : iim = +m) n-Ca 1 - q(l - 2-n)

Since a11 h E %[O, 11 are continuous and the metric on X[O, I] is eqaivalent to the uniform metric, then

and

are al1 open sets. Thus both 5'3 and S4 are Borel sets. We now show that both S3 and S4 are neither Ieft shy nor right shy. 3y The- orem 5.6.2, for any compact set K, there exist functions g,h E 'H[O, I] such that h(x)5 k(x) 5 g(x) for al1 k E K and al1 x E [O, 11. CHAPTER 5. SPACE OF A CrTObfORPBIShIS

For the set S3, choose

fi(.) = 1 - (1 - q(h-'(~)))~,h(x) = h-'(1 - (1 - q(~))~).

Then for al1 k E IC,

and

Hence 1 fl(W1 1 k(f2b)) lim sup - = limsup - = o. r-L- 1 - q(x) r-1- 1 -dx) Thus fi 0 k, k 0 f2 E S3. By Lemma 5.6.1 S3 is neither left shy nor right shy. For the set S4,choose

fi(x) = 1 - (1 - q(s-'(r)))1f2and f2(x) = g-l(l - (1 - q(x))'f2).

Then for al1 k E K,

and Thus Lim inf 1 - fdkb)) = lim id 1 - k(fz(4) = +m. r-I- 1 - dx) r-1- 1 - q(~)

Hence fi o k, k O f2 f S4. By Lemma 5.6.1 S4 is neither ieft shy nor right shy. W

Using similar arguments as in Corollary 5-7.5 we obtain the following.

Corollary 5.11.2 The sets S3 and S4 in Theorem 5.11.1 are neither shy nor prevalent.

Similarly as for Theorem 5.8.1 we show the following theorem. Note that the transverse curves are different.

Theorem 5.11.3 For any function q(x) E %[O, 11, let 1 - h(xf L - h(x) ~={ht~[o,l]:0

Then G is a Borel set that is lefi-and-right shy in %[O, 11.

Proof. Similarly as for S in Theorem 5.8.1 we can show that the set G is a Borel set. We now show that the set 1 - h(z) &,O = {h EX[~,I]: o c iirninf < Lm sup -1- 1 - x)- - 1 - q(x) is Ieft shy and right shy for a.ny a, P > O, o < P.

Wnte F2 : [1/2,1] -, R[O,11 by

and define a Borel probability measure pz by

For any h E 3-1[0,1], consider the following set: for al1 2: < 1 sufficiently close to 1. Set y = 1 - (1 - q(x))'. Then

for al1 y < 1 sdEciently close to 1. We claim that L contains at most one element. If not, thereexist tl,t2E L, ti < t2 such that

and 1 a(1- y)G < 1 - h(y) < P(1 -y)$ for al1 y < Z sufficiently close to 1. Thus

for al1 y < 1 sufficiently close to 1, which is impossible since I/tl - l/tz > O. Thus L is a singleton or empty set. So the set S,,J is left shy.

To prove is right shy, ive choose F3 : [1/2,1] -, R[O,11 by

and define a Borel probability measure p3 by

For any h E %[O, 11, consider the set for al1 x < 1 sufficiently close to 1. That is,

for al1 x < 1 sufficiently close to 1. We claim that R contains at most one element. If not, there exist tl, t2 E R, tl < t2 such that

and

for al1 x < 1 sufficiently cbse to 1. Thus

for al1 x < I sufficiently close to 1, which is impossible since l/tl - l/t2 > O. Thus R is a singleton or empty set, and hence Sa,@is right shy. By Theorem 2.9.7, the set is left-and-right shy in XIOt 11. Thus

is left-and-right shy.

Let q(x)= x. hmediately we have the following.

Corollary 5.11.4 For any a, O < a < +cm, the set

{h E %[O, 11 : hr(l)= a) is left-and-right sh y.

Theorem 5.11.5 For any function q(x) E 'H[O, 11, let i - h(5) CS = { h E H[O, 11 : lirn inf z-1- 1 - q(2) = O} CHAPTER 5. SPACE OF A UTOhfORPHIShlS and

Then both C3 and C4 are neither shy nor prevalent.

Proof. Since C3> S3 and C4 > S4 where S3 and S.4 are as in Theorem 5.11.1, by Theorem 5.11.1, both C3 and C4 are not shy. Similarly as for Theorem 5.10.1 ive can show

and

Thus C3 and Cq are Borel sets. Note

and

By Theorern 5.10.1, and Ca are not shy and hence C3 and CI are not prevalent.

In contrast, in [25] and [26], it was shown that both C3 and C4 have Pa measures 1 when q(x) = x. Similar arguments as for Theorem 5.8.3 show that both the sets S3 and S4 in Theorem 5.1 1.1 are of the first category. Also sirnilar arguments as for Theorem 5.9.2 show that for any rational number r E (O, l), the sets CHAPTER 5. SPACE OF A UTOsVfORPHIShiS and {h E %[O, 11 : 1 - h(x) < l/r(l - q(x))) are closed, nowhere dense sets in R[O, 11. Thus we cm have the following.

Theorem 5.11.6 For any q(x)E R[O, 11, the typical function h E W[O, l] disfies

1 - h(x) 1 - h(x) lim inf = O and lim sup = +m. r-1-1 1 - q(~) r 1 -Q(x)

PROBLEM 13 Let T : R[O, 11 -, 31[0,1] by Th(x)= 1 - h(l - x), is it tme that S 5 X[O, 11 is shy or leWight shy ifT(S)is shy or left/right shy'?

5.12 A non-shy set that is left-and-right shy

Jan Mycielski in [41] posed a problem, denoted by (Po),whether the existence of a Borel probability measure left transverse tu a set Y implies that Y is shy in a non- locally compact, completely metrizable group. In this section we give examples of non-shy sets that are left-and-right shy in %[O, l] and so answer the problem (Po) negatively. These examples also allow us to conclude immediately that the a-ideal of shy sets in R[O, 11 does not satisfy the countable chain condition. Corollary 5.8.2 and Corollary 5.11.4 show that the sets

and {h E %[O, 11 : h'(l) = a) are left-and-right shy sets. We shd show that these two sets are non-shy sets in 'H[O, 11. The first two lemmas are basic facts about monotonic functions. Lemma 5.12.1 For any O 5 a 5 +oc, if f E %[O, 11 und satisfies

for a decreaszng sequence {ç,} satisfying G/G+~+ 1 and c, -, Of as n + CO, then f'(0) = a.

Proof. For asy x

Note that and

Thus by the conditions we have

Thot is, f'(0) = a.

Lemma 5.12.2 For any O B 5 +m, i/ f E %[O, 11 and satisfies

-4 for an increasing sequence {dn} satisfying (1 - d,)/(l - dn+l)-, 1 and d, + 1- as n -, oc, then f'(1) = p.

exists a d, such that d, 5 x 5 d,+l. Then CH.4PTER 5. SPACE OF A UTOMORPHISlbS

and

Thus from the conditions we have

lim 1 - f(4= B.

That is, f'(1) = B.

Theorem 5.12.3 Let O 5 a, P 5 +oc, and

Then DaVpis not shy.

Proof. To show that Da8 is not shy, we shdl show that, for any compact set

Ii 2 %[O, 11, there exist functions g, h E X[O, 11 such that g o h o k E Das for any h E K. Fint we choose a decreasing sequence {G) C (0,1/4) such that al1

the intervals [h,c,+ l/nçJ are pairwise disjoint, c, + O+ and ~,,+~/c,+ 1 as n -, m. Second we choose an increasing sequence {d,) C (3/4,1) such that dl the intervals [d,, d, + (1 - dn)2dn]are contained in (3/4,1), pairwise disjoint, dn -, 1 and (l-dn+l)/(l-dn) -, 1 as n -, m. For example, c, = lin, d, = 1-l/n (n > 5) satisfy the requirements. This can be verified by simple computations. By Theorem 5.6.2

there exist functions fi, fi E 7-1[0,1] such that

for al1 h E K and al1 x E [O, 11. Now we choose a sequence of line segments {I,} contained in (0,114) x (0,112) and a sequence of line segments { 3.) (3/4,1) x (112, l), as in the figure below, such that (i) for any n, the lower end of Inis above the upper end of In+2, and the lower end of Jn+1 is above the upper end of J,, (ii) the corresponding point xf, of I, tends to O from right, and the corresponding point xi of Jn tends to 1 fiom left. (iii) for any n, the line segment connecting (ri,fi(zk)) and (xn,f2(zn)) is con- taincd in In,and the line segment connecting (xt,fl(x:)) and (xn,f*(x;)) is contained

Figure 5.1: The construction of the autornorphisms g and k

We now construct functions g, k E %[O, II in nine cases, and show that g o h o k E Dm,pfor any h E K for each case. Case 1: O < a, p < +m. We require that 4cla < 1 and 2@(1 - dl)< 1 so that, for any n, [QG,Q(~+ I/~)G]S (0,112) and [l - B(1 - dn): 1 - p(1- (1 + (1 - dn)2)dn)]G (1127 1)-

From the choice of {G)and {dn) it is easy to verify that aii these intervals are pairwise disjoint. So we construct a function g E %[O, 11 such that

and

We construct a function k E %[O, 11 such that k(~)= xk and k(dn) = 2:. Then for any h E K,

and g(h(4')) 2- = a. c, c, T hus Lm dh(k(4)) = a. n-m Ga So by Lernma 5.12.1, (g O h O k)'(O) = a. On the ot her hand, for any h E %[O, 11,

and CHAPTER 5. SPACE OF AUTOMORPHISMS

T hus

So by Lemma 5.122, (g O h O k)'(l) = 8. Therefore g O h O k E Da,p- Case 2: cr = O, p =O. We claim that, for any n,

are pairwise disjoint, and

are also painvise disjoint. Since [G,(1 + l/n)~]are painvise disjoint, the first part of the cIaim is obvious. For the second part of the claim, if there were some m such t hat

which contradicts that [d,, d, + (1 - dn)2d,]are pairwise disjoint. Since d, > 314, so 1 - (1 - d.)' > 314 and the claim follows. Now we construct a function g E %[O, 11 such that dIn) = [cg,(1 + l/n)'c',l CHAPTER 5. SPA CE OF A UT0L~IORPHISMS 145

Wealso construct a function k E 31[0,1] such that k(~)= xi and k(d,) = xz. Then for any h E K,

and

and 1 - lim dh(k(U))= o. n-oo 1-6

By Lemma 5.12.1 and Lemma 5.12.2, (g O h O k)'(O) = O and (g O h O k)'(l) = 0.

Therefore g O h O k E Da,p for any h f K. Case 3: a = +a,p =+W. We require 16cl < 1. Then we daim that, for any n,

are pairwise disjoint, and

are dso disjoint. Since [c,,, (1 + l/n)c,J are pairwise disjoint, the fbst part of the claim is true. For the second part of the clairn, if there were some rn such that then 1 - dm+1 < 1 - dm and

which, as sarne as in Case 2, yields a contradiction. Thus the daim is true. Now ive construct a function g E 'H[O, 11 such that

and

We also construct a function k E X[O, 11 such t hat k(c)= xn and k(d, ) = x:. Then for any h E Ii',

and

T hus Lim dh(k(~)))= +O0 n-oo C, and lim

By Lemma 5.12.1 and Lemma 5.12.2, (g O h O k)'(O) = f m and (g O h O k)'(l) = +m.

Therefore g O h O k E Da,p for any h E K.

Case 4: a = 0, O < < +W. From Case 1 and Case 2 we require 2@(1- dl) < 1 and construct a function g E %[O, 11 such that and

g(Jn) = [1- B(1- dn), 1 - P(l- (1+ (1 - dn)*)&)]- Also we constmct a function k E X[O, 11 such that k(~)= th and k(d,) = x:.

Then from Case 1 and Case 2 we know that for any h E K,(g O h O k)'(O) = O and

(g O h O k)'(l)= p. Therefore g 0 h o k E Da8 for any h E K. Similarly as Case 4 we can use the corresponding parts of Case 1, Case 2 and Case 3 to constmct the co~espondingfunctions g E %[O, l] and k E %[O, 11 so that for any h f K, g O h o k belongs to the corresponding Da,@for Case O < a < +a,@ = 0, Case0

Corollary 5.12.4 The sets S in Theorem 5.8.1 and G in Theorem 5.1 1.3 are not sh y.

Proof. Since the set D,,, (O < a < oo) in Theorem 5.12.3 is contained in both S and G. Thus the result follows.

From Corollary 5.8.2 and Corollary 5.11.1 and Theorem 5.12.3 we know that the set Do,p is left-and-right shy, but is not shy. This negatively answers the problem (Po)posed by Jan Mycielski [U].

5.13 a[0,1] does not satisfy the countable chain condit ion

From Theorem 5.12.3 we immediately have the following theorem for the non-locally, non- Abelian Polish group %[O, 11.

Theorem 5.13.1 The o-ideal of shy sets in R[O,l] does not satisfy the countable chain condition. Furthemore, there ezist sets that are lefi-and-right shy, and which can be decomposed into continuum many disjoint, non-shy sets in R[O,11.

Proof. For each 0 (O < a < oo), by Theorem 5.12.3, the set D, {h E %[O. 11 : h'(0) = a) is non-shy. For distinct QI, 02 (O < 01: a2 < cm),Da, and D,, are disjoint. So %[O, 11 contains continuum many disjoint, non-shy sets Da,O < a < cm. The result follows. I

5.14 Functions with infinitely many fixed points

In this section we study the set of functions in %[O,11 that have infmitely many fixed points.

Theorem 5.14.1 The set offunctions in R[O,11 which cross the line y = x in (O, 1) infinitely many times is neither shy nor prevalent.

Proof. Let

h crosses y = x infinitely many times in (O, 1)

CVe first show that F is a Borel set. Let

It is easy to verify that Fn,*,are closed sets and the functions in Fn,p,qcross y = x in (0,l) at least 2n + 1 times. Note

So F is a Borel set. Since F C X[O,l] \ {h E W[O, 11 : h(x) < z2}, and by Theorern 5.9.1 the set {h E %[O, 11 : h(x) 5 x2} is neither left shy nor right shy, so is the complement of F. We now show that F is neither left shy nor right shy. For any compact set h' E 'H[O, 11 we use the ideas in Theorem 5.12.3 to choose 1, and xi. Choose a monotonie çequence {y,) C (0,l) and functions g, k E 'H[0,1]such that k(y,) = s: and for any h E I<,

I g(h(z:n)) > xkn, g(h(&+l)) < X2,+11 h(k(y2n)) > yzn h(k(y~n+i))< ~2n+1-

Then g 0 h, h o k E F. Therefore F is neither Ieft shy nor rïght shy in W[O,11. Hence F is neither shy nor prevalent. rn

Theorem 5.14.2 For any rn E (0,1] and function q(x) E W[O,11, let

S = {h E W[O,11 : 32 E (O, 1) such that h(x) = mq(x)).

Then S is neither shy nor prevalent.

Proof. Let

Sn = {h E W[O,I] : 32 E [lin, 1 - l/n] such that h(x) = mq(x)}-

s= us*.

We now show that Sn is a Borel set. In fact, for any Cauchy sequence {fk) Ç Sn, there exist xk E [l/n,1 - l/n]and f E 7-1[0,1] such that fk(xk)= mq(xk) and fk + f unifordy. Thus there exists a subsequence {xk,} of {xk) such that zk, + XI, E

[lln, 1- l/n] and q(xkJ)+ q(xo). So f (xo) = mq(xo)and hence f E Sn*Sn is closed and hence the set S is a Borel set. Since

and by Theorem 5.9.2 CHAPTER 5. SPACE OF A UTOLMORPHIS~MS 150 is not shy, then S is not prevalent. Since the set F in Theorem 5.14.1 is a subset of S, by Theorem 5.14.1 the set S is not shy. R

In [25] and [26] it is shown that the set S has P measure 1 when q(x) = x in the above theorem. In the above theorem we have shown that the set of a.ll h E 'H[O, 11 such that h(x) = q(x) for some x f (O, 1) is not shy. As we have remarked earlier (see page 105), however, for any q(x) E %[O, LI, the set of al1 h E X[O, 11 such that h(x) = q(x) for al1 x in sorne subinterval of (O, 1) is shy.

5.15 Concluding remarks

The theory of shy sets in an Abelian Polish group seems to be completely understood (CE], [13], [14], [56], [27], [18], [SI, [a]). Only partial results are available, however, for non-Abelian (non-locally compact) compIetely metrizable groups ([la], [4l], [52], and some results in this tbesis).

It had been an open problem (see MycieIski [41]) whether in a non-Abelian Polish group there could exist sets left shy and/or right shy that are not shy. We have answered this in the Polish group %[O, 11 by proving the existence of non-shy Borel sets that are left-and-right shy. In addition ive showed that the Polish group RIO, I] does not sat isfy the countable chain condition. Solecki [52] showed that the a-ided of shy sets in an Polish group admitting an invariant metric satisfies the countabIe chain condition iff this group is locally compact. The following problem is stiil open.

PROBLEM 14 Is it true that the o-ideai ofshy sets in a non-Abelian Pokh group satisfies the countable chain condition ifl the group is locally compact?

The results in this chapter suggest the following problems. We conclude the thesis wi t h t hese problems. CHAPTER 5. SPACE OF A UTOMORPHISMS 15 1

PROBLEM 15 Does there ezist a leff shy (right shy) set in R[O,lj that is not right shy (lejl shy) ?

PROBLEM 16 Does there ezist a shy (lejt/right shy) set in %[O, 11 that is of the second category.

PROBLEM 17 Does the a-ideal of left shy (right shy) sets in R[O,11 satisfy the countable chain condition?

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additively shy set See page 93. almost every See page 10. algebraic basis -4 subset E is an algebraic basis of a Fréchet space F if every eIernent in F can be expressed as a unique linear combination of finitely many elements in E. analytic Let X be a complete, separable metric space. A set E C X is said to be analytic if E is the projection of a closed subset C of X x NN to X where @ is the product of countably many copies of the space N of natural numbers and equipped with the usual metric

for al1 m= (ml,mZ,m3,--a),n=(nl,n2,n3,..-) EW. approximately continuous A function f is said to be approzimately continuous on an intervai [a,b] if for every x E [a,61, there exists a set E with x as its density point such that

Aronszajn nuII See Definition 2.3.3 and Definition 2.3.4. atomless measure A measure p is an atomless measure if for every z E X, ~((2))= 0. CL OSSARY 159

automorphism A homeomorphism h of an interval [a, b] that satisfies h(a) = a, h(b) = b is called an automorphism of [a,61. Baire 1 function A function is said to be a Baire 1 hnction if it can be written as the pointwise limit of a sequence of continuous functions. Banach algebra A Banach algebra A is a Banach space that is also an algebra and satisfies that llxyll 5 11x11 - llyll for al1 z, y E A. Besicovitch set A Besicovitch set is a pIane set which contains a unit segment in every direction.

Borel measure A measure C( on a topological space X is called a Borel measure if p is defined on all Borel sets of X. In this thesis we consider p defined on ail universdly measurable sets. Borel probability measure A Borel measure on a topological space X is called a Borel probability measure if p(X) = 1. Borel selector Let G be a Polish group and H C G be a closed subgroup. Then a mapping s : GIH -+ G is a Borel selector for the cosets of H if it is Borel measurable and s(xH) E xH. Borel sets If X is a topological space, the smallest o-algebra containing the closed sets is called the class of Borel sets. left-and-right shy See Definition 2.9.6. Cantor group The Cantor group is the set of al1 sequences of 0's and 1's equipped with the metric

and the usual group structure xy = (x,y*, - - ,x,y,). c-dense A set S C W is c-dense in an intervai I if the intersection of S with every subinterval of I contains continuum many points. Generally, a set S of a metric space is c-dense in an open set O if the intersection of S with every non-empty open subset GLOSSARY of O contains continuum many points. Christensen nul1 See Definition 2.3.1, Definition 2.9.1 and Definition 2.12.3. CO-analytic A set is co-analytic if it is the complement of an analytic set. compact A subset K of a metric space is called compact if every open covering of Ii has a finite subcovering. compietely metrizable A topological group is called completely metrizable if its topology can be derived from a complete metric on it. continuum hypothesis The assumption that c = N1 is cded the continuum hypothesis, where c is the cardinal of the set R and N1 is the next cardinal after the cardinal of an infinite countable set. convolution A convolution of two measures p and v on a topological group G is defined by

for every universaily measurable set A. A convolution of a measure p with a characteristic function of a set A is defined by

countable chain condition See Definition 2.18.1. countably continuous See page 91. Darboux function A real-valued hinction defmed on an interval [a,b] is said to be a Darbouz function on [a,b] if for each x, y E [a,b! with x < y and c between f (x) and f(y) there is z E (x,y) with f(r) = c. discrete group A discrete group is the group with the discrete topology. Fréchet differentiable A mapping f from a Banach space to a Banach space Y is Fréchet differentiable at x E X if f (x th)- f (4 lim + t-O t GLOSSARY exists for every h E X and the Limit is uniform for llhll 5 1. Fréchet space A completely metrizable, Iocally convex space is called a Fréchet spa ce Gâteaux differentiable A mapping f from a Banach space X to a Banach space Y is directionalIy Gâteaux diflerentiable at x E X if

exists for every h E X. Further, the mapping f is Gâteaux difeerentiable at x E X if there exists an eletnent g E X' such that f(. th) - g(h) = iim + f (4 t-O t for every h E X. Gaussian measure See Definition 3-42. Gaussian nul1 in the ordinary sense A set S in a Banach space is said Gaussian null in the ordinary sense if there exists a Gaussian measure p such that p(S) = 0. Gaussian nul1 in Phelps sense See Definition 2.4.3. Haar measure An invariant rneasure on a locally compact group in calIed a Haar measure. Haar null, Haar zero See Definition 2.3.1, the comments foilowing Defini- tion 2.3.1, Definition 2.9.1, Definition 2.9.3 and Definition 2.12.3. Holder continuous A function f,defined on an interval [a, b], is said to be Helder continuous at x f [a,b] if there exist M, 6 > O such that for all y E [a,b], ly - XI < h1 I~(Y- f(x)I I~3 - YI. Holder continuous with exponent ct A function f,dehed on an interval [a, b], is said to be Hiilder continuous with exponent cr at x E [a,bJ if there exist fil,6 > O such that for al1 y E [a,b], ]y - x)< 6, [ f(y) - f(x)l 5 M~x-y)". homeornorphism A bijection h from a metric space X to a metric space Y is called a homeomorphisrn if both h and h-' are continuous. GL OSSARY

hyperplane A hyperplane in a Banach space B is a set of the fom {x E B : f(x) = a) where ,f is a non-zero Linear functional on B and a is a reaI number. isomorphism A mapping T hom a Banach space X onto a Banach space Y is an isomorphism if T is one-one, linear, continuous. left Haar measure A Haar measure is a Zej? Haar measure if it is left invariant. left shy See Dekition 2.9.4. left transverse See Definition 2.9.4. Lipschitz (continuous) A function f, defined on [O, 11, is Lipschitz at a point x E [O, l] if there is a constant M > O such that for al1 y E [O, 11, 1 f(x) - f(p)I 5 lMlx - y 1. Once this inequality holds we say that f is M-Lipschitz. locally finite A Borel measure is locally finite if every point z E X bas a neighborhood U with p(U) < oo. lower density 1 See page 48. m-dimensionally shy See Definition 3.2.1. metrizable A topological space (X, 7)is metrizable if there is a metric d on it so that 7 is the topoIogy of (X, d). multiplicatively shy See page 93. nowhere rnonotonic See page 81. nowhere rnonotonic type See page 81. n-th symmetrically continuous A function f is said to be n-th symmetn'cally continuous at a point x of an open interval (a:b) Ç W if

observable See page 48. Polish group A Polish group is a separable completely metrizable group. Polish space A Polish space is a separable completely metrizable space. GLOSSARY

positive cone In an ordered vector space E, the set (x E E : x 3 O) is cded the positive cone. positive lower density See page 48. positive upper density See page 48. Preiss-TiGer nul1 See Definition 2.3.5. prevalent See page 10. probe See page 64. Radon measure A Borel measure p on a Hausdorff space is called a Radon measure if p is locally finite and tight. Radon probability measure A Radon probability measure p on a Hausdorff space X is caiied a Radon probability neasure if it is a Radon memure with p(X) = 1- right Haar measure A Haar measure is a right Haar rneasure if it is right invariant. right shy See Definition 2.9.5. right transverse See Definition 2.9.5. semigroup A semigroup is simply a set G with an associative operation: G x G + G. In this thesis we always assume that our semigroup has a unit element. shy See Definition 3.3.1, Definition 2.9.3 and Definition 2.12.3. s-nul1 See Definition 2.3.2. spanning set In a Banach space X, a set C is a spanning set if the whole space X is the ffie hull of C. Here the fiehull of a set A is the set

super-reflexive A Banach space E is said to be super-refïezive if each Banach space which is finitely representable in it is reflexive. We say a Banach space El is finitely representable in E if for each hite dimensional subspace L E El and a > O there exists an embedding i : L -, E such that a-'11x11 5 llixll < (~IIxll (x E L). GLOSSARY 164 support of a measure The support of a measure p on a metric space X is dehed by supp(p) = {x E X : p(U) > O, VU a neighborhood of x). symmetricaiIy continuous A function f is said to be symmetrically continuous at a point x of an interval (a,b) Ç W if

T-smooth A Borel probability measure p is T-smooth if p(U) = sup, p(U,) for every family of open sets {U,) filtering up to the open set U. thick See Definition 2.19.1. tight A Borel measure p is cded tight if p(B) = sup p(K) for ad Borel sets B, where the sup is taken over ad compact sets K C B. topological group A topological group is a group (G, -) endowed with a topology such that (2,y) -+ xy-L is continuous from G x G to G. topological semigroup A topological semigroup is a completely metrizable semigroup for which the operations s -, as and x -+ xa are continuous. topological space A topological space is a pair (X, I),where X is a set and T is a collection of subsets of X such that 4, X E T and T is closed under arbitrary unions and finite intersections. transverse See Section 2.2. typical function If a set of functions with some property is a residual set in a function space, then every function in this set is called a typical function typical property See page 1. ubiquitous See page 48. universally measurable A set of a topological space X is universally measurable if it belongs the p-completion of each finite Borel measure p on X. GLOSSARY universally Radon measurable A set of a topological space X is uniuersally Radon measurable if it belongs to the p-completion of each finite Radon measure p on X. upper density 1 See page 48. well-ordering A strict linear ordering < on a set X is calied a well-ordering for X if every non-empty subset of X contains a first eIement. Wiener measure See Definition 2.5.1. Zarantonello nul1 See Definition 2.6.1. I ivi~urCVHUJN I IUN TEST TARGET (QA-3)

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