119 ^9/1 A/0. / 5Tl'/y

MEASURABLE SELECTION THEOREMS FOR PARTITIONS OF

POLISH SPACES INTO G s EQUIVALENCE CLASSES

DISSERTATION

Presented to the Graduate Council of the

North Texas State University in Partial

Fulfillment of the Requirements

For the Degree of

DOCTOR OF PHILOSOPHY

By

Harry S. Simrin, B.S., M.S

Denton Texas

May, 1980

'A'-t,

Simrin, H. S., Measurable Selection Theorems for Parti-

tions of Polish Spaces into Equivalence Classes. Doctor

of Philosophy (Mathematics), May, 1980, 86 pp., 1 figure,

bibliography, 24 titles.

A partition Q of a topological space X is said to be

measurable if the O-saturation of each open set is Borel. Let

R be the equivalence relation determined by Q. A Borel map

f : X + X is a Borel cross section for Q if (1) the graph of

f is a subset of R and (2) f(x) = f(t) whenever (x,t) e R.

A Castaing Representation for Q is a sequence { f : n > 1 }

of Borel cross sections for Q such that { f (x) : n > 1 } is n —

dense in R(x) for all x.

Let X be a Polish space and Q a measurable partition of

X into Gg equivalence classes. In 1978, S. M. Srivastava

proved the existence of a Borel cross section for Q. He

asked whether more can be concluded in case each equivalence

class is uncountable. This question is answered here in the

affirmative. The main result of the author is a proof that

shows the existence of a Castaing Representation for 0.

In the process of proving the above theorem, a side

issue concerning the nature of Borel sets in topological

spaces arose. The author proved a new characterization of

Borel sets that should be of very general interest. In this

theorem, mathematical expression is given to the process of

constructing a Borel set from Borel sets of lower classes,

descending down to the open (or closed) sets. The importance

of this characterization is that it allows certain properties

of Borel sets to be reflected upwards from their roots in the

open (or closed) sets. This provides a method for proving

properties of Borel sets in the absence of a conventional

transfinite induction hypothesis.

This reflection principle is illustrated by a proof that

the Q-saturation of a particular Borel set B is Borel. It is

known that the Q-saturations of some Borel sets are not Borel,

and it is not expected that this property is true for the

Borel sets of lower classes from which B is constructed.

Thus, there is no conventional transfinite induction argument.

It is known that the 0-saturation of an open set is Borel,

and this property is reflected by transfinite induction back

up to B by means of the above-mentioned Borel characterization.

Several parametrization theorems are also proven. An

invariant version of a theorem of R. D. Mauldin is proven.

Let X and Y be Polish spaces, and let B e S(X x Y). The

theorem equates the existence of (1) a Borel parametrization

of B, (2) the existence of a certain conditional probability

distribution, and (3) the existence of a Borel subset of B

that has nonempty compact perfect sections. A surprising

result is obtained: the existence of noninvariant param-

etrization conditions implies the existence of invariant

parametrization conditions in case (1) B is the equivalence

relation generated by a measurable partition and (2) the

equivalence classes are G^. An alternative proof of a param-

etrization theorem of Srivastava is also presented.

TABLE OF CONTENTS

Page

Chapter

I. INTRODUCTION 1

II. PRELIMINARIES 6

Fundamental Concepts History of Selection Theory of

Partitions

III. DISJOINT CROSS SECTIONS: A SPECIAL CASE . . . 26

IV. DISJOINT BOREL CROSS SECTIONS OF A MEASURABLE PARTITION 37

Definitions 6a-Trees Borel Schemes Borel Cross Sections for Measurable

Partitions

V. BOREL PARAMETRIZATIONS 63

VI. UNSOLVED PROBLEMS 81

BIBLIOGRAPHY 84

i n

CHAPTER I

INTRODUCTION

The primary purpose of my research was to extend Theorem

2.10. In this theorem, Srivastava proved the existence of a

measurable selection of a measurable partition of a Polish

space into G$ equivalence classes. It seemed (and still

seems) invitingly obvious that if each equivalence class has

uncountably many points, then there are uncountably many dis-

joint selections, possibly filling up Q. I was able to show

(Theorem 4.8) the existence of a Castaing Representation

(i.e., infinitely many disjoint selections) for the partition

in question, and I conjecture the existence of an uncountable

family of disjoint measurable selections.

In the process of deriving this proof, I solved a prob-

lem (Theorem 4.5) dealing with the nature of Borel sets: a

problem that is somewhat afield from measurable selection

theory. I foresee many applications of this theorem in areas

unrelated to selections. What this theorem does is to give

mathematical expression to the fact that a Borel set may be

traced down through the Borel classes until one reaches the

open (or closed) sets. The reason this new characterization

of Borel sets is important, is the following. Sometimes a

property will be true for a particular Borel set, but not

necessarily for any other Borel set. It is not possible,

then, to prove the truth of this property by induction on

the Borel sets of lower classes. My characterization of

Borel sets allows one to take the given Borel set, trace the

construction of the Borel set down through the Borel classes,

find the seeds of the property, and reflect them back.

I illustrate this process (Corollary 4.6) by analysis of

the saturation operator. I have a particular Borel set (Y fl W) ,

for which I conjecture that the saturation is Borel. I do not

make this conjecture for all Borel sets, nor for the Borel

sets of lower classes from which it is obtained. I cannot

proceed by induction. By use of Theorem 4.5, I am able to

reflect back up to the given set the "Borelness" of the sets

that are the saturation of open sets.

While studying the further problem of extending ny

Castaing Representation to uncountably many disjoint selec-

tions, I perceived that, with an additional hypothesis, I

could obtain a Borel parametrization (Theorem 5.6). I found

some difficulty in expressing my perception and, finally,

devised the concept of "regular finite binary tree." I was

able to use these trees as an indexing set in a scheme that

yields the parametrization (Theorem 5.5). Shortly thereafter,

I learned that Srivastava had obtained this theorem (with a

different proof) and had included it in his dissertation

[_23 j . He had omitted it in the published version of his

dissertation [22J and so I was unaware of it. Because my

indexing scheme appears suitable for other, yet unsolved,

parametrization problems, I have included my proof of this

theorem in this manuscript.

The dissertation is organized as follows. Chapter II

contains the preliminaries. This begins with some special

symbology, and is followed by definitions, fundamental

selection-theory and descriptive-set-theory tools, and a

short history of the selection theory of partitions.

Chapter III is somewhat unusual in that I prove, here,

a restricted case (Theorem 3.3) of the main theorem (Theorem

4.8). The restricted theorem is not used in proving the full

theorem. It is included here because its proof is consider-

ably easier to follow than the full proof, and yet it

contains all the essential elements of the full theorem,

including the rudiments of the Borel characterization. The

Borel characterization itself requires new notation and

3-dditional vocabulary. Moreover, the proof of the restricted

theorem allows me to motivate my more complicated proof of

the full theorem. By presenting the restricted case first,

tiie reader is allowed to concentrate one at a time on the

major obstacles to the proof.

Chapter IV contains both of my main theorems: the Borel

Characterization Theorem (Theorem 4.5) and the Castaing Rep-

resentation Theorem (Theorem 4.8). Game theoretic notation

and the concepts of 6a-trees and rank functions are intro-

duced. Some preliminary lemmas dealing with 6a-trees,

especially the kernel of a 6a-tree, precede the main theorems.

Besides Theorem 4.5, Lemma 4.4 is the key to the proof of the

Castaing Representation Theorem. This lemma shows that under

special circumstances the saturation operator will commute

with intersections. A small digression to compare the new

characterization of Borel sets with the analytic operation,

Operation A, is inserted between the main theorems.

Chapter IV is concluded with a discussion of some funda-

mental properties of selection theory. Several examples are

presented to show that the circuitous route taken was, in

fact, necessary. Specifically, I discuss the interrelation-

ships between Borel transversals, Borel cross sections, Borel

equivalence relations, and the saturation of Borel sets.

Probably the most important result of this section is Corol-

lary 4.15, where the observation is made that every nontrivial

measurable partition of a Polish space into G. sets contains 0

a Borel set whose saturation is non-Borel.

In Chapter V, I obtain an invariant version (Theorem

5.2) of a theorem of Mauldin (Theorem 5.1), and then show

(Theorem 5.3) the surprising result that, in the case of

measurable partitions, invariance is equivalent to non-

invariance. The chapter is concluded with the parametriza-

tion theorem, Theorem 5.6, discussed earlier. Following the

style of Srivastava, I prove this theorem as a corollary of

a more general invariant parametrization theorem (Theorem

5.5) for multifunctions.

The dissertation is concluded with Chapter VI, a short

discussion of unsolved problems.

CHAPTER II

PRELIMINARIES

This chapter contains the notation and principal concepts

that are needed in the remainder of the manuscript. Because

a considerable amount of fixed notation will be used, it is

best to begin with a glossary of symbols. Following this are

definitions of the terms used, interspersed with a sprinkling

of fundamental results. The chapter is concluded with some

known theorems that provide a springboard for my results.

Fundamental Concepts

The following notation will be used throughout this

dissertation.

R The set of real numbers.

E The set of even whole numbers.

0 The set of odd numbers.

W The set of whole numbers.

W The set of natural numbers.

Q. The set of rational numbers.

J N^, the set of infinite sequences of natural

numbers--homeomorphic to the space of irrational

numbers.

W The set of ordered k-tuples of natural numbers.

Seq { 0 }U U N , the set of finite sequences of keW

natural numbers.

2^ { 0,1 }^, the set of infinite sequences of zeroes

and ones--homeomorphic to the Cantor Set.

2k { 0,1 }k.

" J " ] £

2" { 0 }U U 2 , the set of finite sequences of keW

zeroes and ones.

Ord The collection of ordinals, according to Von

Neumann |_8, page 2] .

u) First infinite ordinal. o

First uncountable ordinal--also equal to the

set of predecessors of

If X is a topological space, the following notation is

used.

P(X) The power set of X; i.e., the collection of all

subsets of X.

8(X) The collection of all Borel subsets of X.

Z (X) The collection of Borel subsets of X of addi-Y

tive class y.

n (X) The collection of Borel subsets of X of multi-

plicative class y.

A(X) The collection of analytic subsets of X.

CA(X) The collection of coanalytic subsets of X.

Let s = (n1, n2, n3, ••• , nfc) and t = (n^, m2, m3 , ••• , m.)

be elements of Seq. We say s_ extends t if k > j and m^ = n^,

m2 = n2, m3 = n^, • • • , m = n . The juxtaposition of s and

t: is denoted (s,t). By this we mean the element

(ni« n2' n3' > nk> m

1» m

2' *'* ' 111 j) °f Seq. In particular,

if n e M, (s,n) means (n1, n2, n3, •••' , n , n) and (n,s)

means (n, n^, n2, ••• , n ). Every element of Seq is con-

sidered to extend 0; nevertheless, 0 is not explicitly called

out in juxtaposition. For example, (0,n) is written simply

as (n). The length of s is k. We denote this by |s|. If

a = (n1, n2, n3> •••) e J, then a|k = (nx, n2, n , ••• , n )

is the restriction of a to k. Similarly,

s|j = (n1, n2, n3, ••• , n.) if j < k. By a(j) ( s(j) ), we

mean the j th component of a (s); namely, n..

When a specific equation is referenced, LHS will denote

the expression to the left of the equality and RHS will denote

the expression to the right.

A a-field of sets is a collection of sets that is closed

under complements and countable unions. Let X be a topologi-

cal space. The class of Borel sets of X, B(X), is the

smallest a-field of sets containing the open sets. The Borel

sets can be constructed internally, beginning with the open

and closed sets. If £ and II denote the open sets and closed o o

sets, respectively, then the other Borel classes can be defined

inductively. For y > 0,

I (X) = { IM : A„ e U nQ(X) } and Y n $<y

n (X) = { DA : A e (J Z0(X) }. Y n n • ' 3

n 6<y

Then 8(X) = U { Z^(X) : Y < ^ > = U { II (X) : y < ^ }.

A Polish space is a topological space that is homeomor-

pnic to a complete separable metric space. The following

facts about Polish spaces are well known ) 12J .

(1) A metric space is Polish if and only if it is

homeomorphic to a Gg subset of the Hilbert cube.

(2) If X and Y are Polish spaces, f : X -*• Y is a one-

to-one Borel measurable map, and A is a Borel sub-

set of X, then f(A) is a Borel subset of Y.

(3) If B is a subset of a Polish space X, the following

are equivalent:

10

(a) B is a Borel subset of X.

(b) B is the one-to-one continuous image of a

closed subset of J.

(c) B is a Borel subset of any Polish space con-

taining (a homeomorphic image of) it.

Let X be a Polish space. An analytic subset of X is a

set that can be expressed as a continuous image of J. A

coanalytic set is a set whose complement is analytic. Every

3orel set is both analytic and coanalytic.

A partition of a set X is a collection of pairwise dis-

joint subsets of X (called equivalence classes) whose union

fills X. If Q is a partition of X and A c X, then

sat^(A) = U { E e Q : E n A ^ 0 } i s the Q-saturation of A.

When Q is unambiguous, we shorten this to sat(A). If X is a

topological space, we say the partition Q is measurable if

sat(U) e 8(X) for each open set U in X. We write R(Q) for

the equivalence relation on X generated by Q; that is,

R(Q) = U { E x E : E e Q } . We make frequent application of

the following observation. If we set R = R(Q), then

sat (A) = H2(R n (A x x) ), the 2-projection of Rfl(A x x) .

If X and Y are topological spaces and A c X x Y, we say that

A is Q-invariant if A = A^ whenever x and t are related. x t

11

By A_ we mean the x-section (vertical section) of A: X _____________

A - { y e Y: (x,y) e A }. For y e Y, we write A

- { x e X : (x,y) e A } for the y-section (horizontal

section) of A.

/ \

We let A(Q) denote the g-field of Q-saturated Borel sets.

By Q(A) we denote the partition of X induced by the collection

A. Specifically, if x and t are points of X, they are Q(A)-

related if they belong to precisely the same members of A.

The members of 0(A) are called the Q(A)-atoms. We observe / \

that A is Q(A)-saturated if and only if A is a union of Q(A)-

atoms.

We say that (X,A) is a measurable space if X is a set

and A is a a-field of subsets of X. If (X,A) and (Y,8) are

measurable spaces, the product measurable space is

(X x Y, A 0 8) where A 8 8 = 8(A x 8) , the Borel field gen-

erated by rectangles A x B e A x 8. If E c X x Y, we denote

the x-projection of E by n^(E). The y-projection is denoted

by n 2 ( E ) .

The first four lemmas, below, point out some elementary

relationships that will be of later use. These lemmas are

well known. See, for example, Q22^] .

Lemma 2.1. Let X be a Polish Space, A a countably

generated sub-a-field of 8(X), 0 the partition of X induced

12

by A, and R the equivalence relation on X induced by Q. Then

R e A 0 A.

Lemma 2.2. Let X be a topological space, Q a partition

of X by Borel equivalence classes, and A the a-field of Q-

saturated Borel subsets of X. Then Q is the partition of X

induced by A. In mathematical notation, this fact is ex-

pressed as Q = 0 ( A(Q) ).

Lemma 2.3. Let X be a Polish space, A a countably gen-

erated sub-a-field of 8(X), and Q the partition of X induced

by A. Then A is the collection of O-saturated Borel subsets

of X. In mathematical notation, this is written

A = A ( Q(A) ).

The proof of the next lemma, though not difficult, is

not straightforward. The "natural" method to prove that pro-

jections of A 0 B-measurable sets are Borel is to show the

collection of such sets is a a-field containing the A 0 8-

measurable rectangles. Unfortunately, this is false. A

proof of the lemma is included for the sake of completeness.

Lemma 2.4. Let (X,A) and (Y,8) be measurable spaces, Q

the partition of X induced by A, and R the equivalence rela-

tion on X induced by Q. If H e A 0 8 and (x,t) e R, then

H = x = Ht. Consequently, n^'H) is O-saturated.

13

Proof of Lemma 2.4

Suppose (x,t) e R. If H e A x B, H = H . It is then X l!

clear that H = { H c X x Y : H = H } is a a-field contain-2C XZ

ing A x B. Therefore, A ® B c H, proving H = H . — X L

To prove the second assertion, suppose (x,t) e R and

x e IT (H) . By the first assertion, H = 0 if and only if 1 x

H = 0. Therefore t e II^H). This shows that II1 (H) is a

union of Q-atoms, concluding the proof of the lemma.

A multifunction (or set-valued function) from X to Y is

map F : X P(Y), where X and Y are topological spaces. The

graph of F is the subset Gr(F) = { (x,y) : y e F(x) } of

X x Y. For example, if f : X + Y is a (single-valued) func-

tion, then f-1 : Y -*• P(X) is a multifunction. Another ex-

ample is the multifunction R : X P(X) defined by / \

R(x) = { t e X : (x,t) e R(Q) }. For x e X, R(x) is the

equivalence class to which x belongs. We will also use the

symbol R to refer to the graph of the multifunction R; namely, /s

R = R(0). The meaning of R will be clear from the context

and no confusion should occur. When we are referring to R

as a subset of X x X, we will write R rather than R(x). X

14

If F : X + P(Y) is a multifunction, we define

F~(B) = { x e X : F(x) 0 B + 0 }. We say F is Borel meas-

urable if F~(B) e 8(X) for each open set B in Y. In par-

ticular, if F is single valued, this coincides with the usual

definition of Borel measurability. This convention follows

Srivastava [_22j but differs from most other literature |_24j

where this property is termed "weakly measurable" and the

term "measurable" is reserved for multifunctions such that

F~(B) e B(X) for closed sets B in Y. The change is made

merely for notational convenience.

A function f : X -*• Y such that f(x) e F(x) for all x is

called a selector for F. If f is Borel measurable, f is a

Borel selector for F.

Suppose B is a subset of X x Y whose first projection is

X. A function g : X -*• Y is a uniformization for B if Gr(g) c B

If, in addition, Q is a partition for X and g(x) = g(t) when-

ever x and t are Q-related, we say g is a Q-invariant unifor-

mization of B. If R is the equivalence relation determined

by Q, and h : X -> X is an invariant uniformization of R, we

say that h is a cross section for Q. We will often view

uniformizations and cross sections as subsets of the product

space by identifying them with their graphs. For example, if

15

E is a subset of X x y and r is a subset of E such that

|r | = 1 for all x in n (E), then we will call r a uniformi-X J_

zation of E. This terminology is consistent with Auslander

and Moore [_ 4_| . Unfortunately, this terminology diverges

rather badly from that in j 22J where Srivastava's use of

cross section is weaker than what is used here.

A subset S of X that meets each Q-equivalence class in

precisely one point is called a transversal. If, as well,

S is a Borel subset of X, it is a Borel transversal. A Borel

cross section h induces a Borel transversal S; namely S = h(X)

= n2(Gr(h) fl A), where A is the diagonal of X x X. Con-

versely, a Borel transversal induces a cross section. How-

ever, as Examples 4.11 and 4.12 show, the induced cross section

need not be Borel. Thus, existence theorems for Borel cross

sections are more powerful than existence theorems for Borel

transversals.

If X, Y, and Z are topological spaces and g : X x Y Z,

we say g is a Q-invariant map if g(x,y) = g(t,y) whenever x

and t are Q-related. A map g : X x X -> X is called a Borel

parametrization of Q if g is an invariant Borel measurable

map such that for each x g(x,-) : X -*• R(x) is one-to-one and

onto. In particular, notice that for each y, g(*,y) is a

16

Borel cross section of Q, and the graphs of g(*,y) form a

collection of disjoint Borel sets that fill up R.

History of Selection Theory of Partitions

Selection theory can be defined as the body of mathe-

matics devoted to finding selections for multifunctions (se-

lectors, parametrizations), selections of subsets of a product

space (uniformizations, parametrizations), and selections of

partitions (transversals, cross sections, parametrizations).

In this dissertation we are primarily concerned with selec-

tions of partitions, although the three topics are closely

related. Because selection theorems for partitions are often

special cases of more general selection theorems for multi-

functions, reference to the selection theory of multifunctions

is included in the history below, where appropriate. Some

early results are to be found in [_13j and [_18_l •

Closed Equivalence Classes

Perhaps the first significant result in selection theory

is due to von Neumann [^24, p. 8 7 l ] and Yankov | 24, p. 900J .

Theorem 2.5. Let X be an analytic subset of a Polish

space and suppose f : X -> R is a continuous map of X onto R.

-1

Then F = f admits a 8 ( A(X) )-measurable selector, where

8 ( A (X) ) is the a-field generated by the analytic sets.

17

The next major selection result belongs to Dixmier

|_24, p. 883J . Certainly, this is the earliest major result

in the selection theory of partitions.

Theorem 2.6. Let X be a Polish space and Q a measurable

partition of X into closed equivalence classes. Then Q admits

a Borel cross section (and, hence, a Borel transversal).

Both of the preceding results are special cases of the

following theorem of Kuratowski and Ryll-Nardzewski which

Wagner |_24, p. 867j suggests be called the Fundamental

Theorem of Selection Theory.

Theorem 2.7. Let X be Polish, (Y,B) a measurable space,

and F : X -»• P(Y) a measurable closed-valued multifunction.

Then F admits a Borel selector.

Wagner points out that interest in selection theory was

promoted when applications of Theorem 2.7 were found; notably,

in mathematical economics [|3[] and control theory.

F Equivalence Classes

There is no theory to speak of for partitions by F^

equivalence classes. For, let Q be the partition of the

closed unit interval I such that x and y are related if

x - y e Q.. It is a standard example in graduate-level

analysis courses that this partition does not admit a

18

Lebesgue-measurable transversal, let alone a Borel transversal

or Borel cross section. However, the theorem of Lusin [_13j ,

below, shows that the graph of the equivalence relation de-

termined by Q can be filled up by a countable pairwise dis-

joint family of Borel uniformizations. This theorem was

preceded, historically, by the following theorem of Novikov

L17J .

Theorem 2.8. Let X and Y be Polish spaces, and let B be

a Borel subset of X x Y such that each vertical section is

countable. Then B has a Borel uniformization.

Theorem 2.9 (Lusin). Let X and Y be Polish spaces, and

suppose A is an analytic (Borel) subset of X x Y each x-section

of which is countable. Then A can be filled up by a countable

pairwise disjoint family of analytic (Borel) uniformizations.

Gg Equivalence Classes

Kallman and Mauldin [_10j proved a selection theorem for

simultaneous and equivalence classes. The remaining

basic work in this area was done by Srivastava and can be

found in his dissertation |_23j . In particular, he proved

the following result.

Theorem 2.1G. Let Z be a Polish space, X a Borel sub-

space of Z, and Q a measurable partition of X into Gr (in Z) o

equivalence classes. Then

19

(1) Q admits a Borel cross section, and

(2) The equivalence relation induced by Q is a Borel

subset of X x x .

Actually, condition (1) implies condition (2), and in the

presence of a Borel transversal for Q, condition (2) implies

condition (1). An argument for this fact is given in Chapter

IV.

Srivastava's result, Theorem 2.10, is a special case of

his selection theorem for G^-valued multifunctions, below.

Theorem 2.11. Let Z and Y be Polish spaces, X an ana-

lytic subset of Z, A a countably generated sub-c-field of

B(X), and F : X P(Y) an A-measurable G.-valued (relative to ~ o

Z) multifunction whose graph is in A 8 B(X). Then F admits

an A-measurable selector.

We make the observation that if Q is a partition of X

and A is the a-field of saturated Borel sets, then the pre-

ceding theorem asserts that if F is an invariant G„-valued 0

measurable multifunction whose graph is an invariant Borel

set, then F admits an invariant Borel selector.

Under certain restrictive conditions, Srivastava has

obtained a generalization of this theorem.

20

Theorem 2.12. Suppose, in addition to the hypotheses in

Theorem 2.11, F(x) is countably infinite for each x. Then

the graph of F can be filled by a pairwise disjoint countable

collection of invariant Borel selectors.

Srivastava states that he does not know any further re-

sults in case F(x) is uncountable for x e X. Some further

results are provided by Theorem 4.8 of this dissertation.

Mauldin |_15j shows that many of the assumptions of

Theorem 2.11 cannot be dropped, by presenting an example of

a Borel subset B of the closed unit square such that every

vertical section is an uncountable Ggset and, yet, B does

not admit a Borel uniformization.

The fact that A must be countably generated plays a

central role in this development. That this assumption is

not a burden for the -equivalence-class theory can be seen

from the next lemma, by Srivastava. Kallman and Mauldin

|_10j , on the other hand, have an example that shows such an

assumption is not to be taken lightly in the F^ selection

theory. They demonstrate the existence of a measurable par-

tition Q of a Polish space by F^ equivalence classes such

that the a-field of saturated Borel sets is not countably

generated.

21

Theorem 2.13. If Q is a measurable partition of a

Polish space into equivalence classes, then the cr-field,

/ \

A(Q), of saturated Borel sets is countably generated.

The following theorem of Miller explores some additional

resolution of the Borel structure.

Theorem 2.14. Let X be a Polish space and Q a measurable

partition of X into equivalence classes such that the sat-

uration of each basic open set is of ambiguous class a > 0.

Then Q admits a Borel transversal for 0 of class

Y = sup { a + 3 : 3 < a }.

Property-Preserving Maps

One application of the theory of selections of multi-

functions is the preservation of certain descriptive set-

theoretic properties. Consider the following well known

theorem j_12 | .

Theorem 2.15. Let X and Y be complete metric spaces,

A c X, and f : A -»• Y a one-to-one Borel measurable map. If

A e B(X), then f(A) e B(Y).

If the one-to-one assumption is deleted, it is easy to con-

struct examples where the theorem is false. In the following

variations of this theorem, Hausdorff and £oban, respectively,

have found conditions that can be substituted for one-to-one.

22

Theorem 2.16. Let X and Y be complete metric spaces

and f : X -* Y a continuous open map. If A is in X, then

f(A) is G„ in Y. o

Theorem 2.17. Let X and Y be complete metric spaces and

f : X -y Y an open Borel measurable map such that for s ome

fixed metric on X, f_1(y) is complete for all y e Y. If A

is G„ in X, then f(A) is G. in Y. o o

An application of Theorem 2.10 is the following.

Theorem 2.18. Let X and Y be Polish spaces and f : A Y

an open Borel measurable map such that f~ "(v) is G in X for ' 6

each y e Y. If A e B(X), then f(A) e 8(Y).

Proof of Theorem 2.18

Let Q = { f "(y) : y e Y }. Q is a partition of A into

Gg (in X) equivalence classes. To see that Q is measurable,

let U be open in A. Since sat(U) = f - 1 o f(U) e 8(A), it

follows from Theorem 2.10 that there is a Borel transversal

S for Q. Consequently, f(A) = f(S), and by Theorem 2.15,

f(A) is Borel. This concludes the proof of Theorem 2.18.

Parametrization Theorems

Parametrization theorems are, perhaps, the nicest selec-

tion theorems. Srivastava and I independently discovered the

23

parametrization Theorem 5.5, which adds one extra condition

to the multifunction F in Theorem 2.11. A pretty parametri-

zation theorem that has some of the same flavor has recently

been announced by Mauldin and Sarbadhikari. See Chapter V

for the definition of parametrization of a multifunction.

Theorem 2.19. Let X be a Polish space and F : X -»• P(R)

a measurable multifunction such that F(x) is a dense-in-

itself Gr for each x, and Gr(F) is Borel in X x R. Then F 0

admits a Borel parametrization f : X x J -*• Gr(F) such that

f(x,*) is continuous for each x.

Cenzer and Mauldin j 5 | have obtained the following

parametrization theorem which is notable for its very simple

hypotheses.

Theorem 2.20. Let W be a Borel subset of the closed

unit square I x I such that each vertical section is uncount-

able. Then W admits an S(I x I)-measurable parametrization.

S(I x I) is the smallest family containing the Borel

sets that is closed under complements and operation A. Oper-

ation A is described in Chapter IV, and parametrizations of

subsets of X x Y are defined in Chapter V.

A comprehensive additional discussion of measurable se-

lection theorems is to be found in [_24J •

24

Related Theory

The following standard tools of descriptive set theory

are used in this manuscript. The first is a well-known theorem

due to Novikov |_17j .

Theorem 2.21. (Separation Theorem). Let X be a complete

metric space and { } a sequence of analytic sets with empty

intersection. There exists a family { Bn } of Borel sets

such that c for each n, and fl B^ = 0. In particular,

if and A^ are disjoint analytic sets, then there exists

disjoint Borel sets that separate them.

Srivastava |_22_| has presented an invariant version of

this theorem.

Theorem 2.22. (Invariant Separation Theorem). Let X

be a Polish space, Q a partition of X, and suppose that R,

the induced equivalence relation, is analytic in X x x . If

{ } is a sequence of invariant analytic sets with empty

intersection, then there is a family { B } of invariant n

Borel sets such that A c B for each n, and OB = 0. n — n n

From the following separation theorem of Saint-Raymond

j_21 | , Srivastava |_22_j obtained Theorem 2.24, which is used

in Theorem 5.5.

25

Theorem 2.23. Let X and Y be compact metric spaces, and

E and F disjoint analytic subsets of X x Y such that E is x

a-compact for all x. There is a family { B } of Borel sub-n

sets of X x y with compact x-sections such that if B = U B n

then E c B c ( X x Y ) - F.

Theorem 2.24. Let X and Y be Polish spaces, A a count-

ably generated sub-cr-field of B(X) , and G e A 0 B(X) such

that G x is for all x. There is a family

{ G e A ® B(X) } with open x-sections such that G =flG . n ~ n

CHAPTER III

DISJOINT CROSS SECTIONS: A SPECIAL CASE

Let X be a Polish space and Q a measurable partition of

Q by equivalence classes that are Gg sets. In this chapter

we show that Q admits a pair of disjoint Borel cross sections

given the assumption that the Borel transversal induced by

the first cross section is of some finite Borel class. The

problem is solved without this additional restriction in Chap-

ter IV. The reason for presenting the special case is twofold

(1) the general proof has required much circumlocution and an

entire set of new notation, both of which greatly obscure the

fundamental ideas of the proof; and (2) the special case more

closely tracks the author's progress in the solution of this

problem and will give insight into the original obstacles.

Knowledge of the special case is not required in order to

read the general proof in Chapter IV.

We begin with some elementary observations.

Lemma 3.1. Let X be a set and Q a partition of X.

Suppose { An } is a sequence of subsets of X. Then

26

27

(a) sat( UA ) = U sat (A ) and n n

(b) sat(HA ) c flsat(A ). n — n

Proof of Lemma 3.1

Both facts follow easily from the fact that if A c X,

sat(A) = H2(R D (A x X) ).

That equality may not hold in (b) even under some very

stringent conditions is demonstrated in the following example

where X is the real line, Q is a measurable partition by

clopen equivalence classes, and A^ and A a r e Borel sets of

class one.

Example 3.2. Let X = R, Q = { R }, A = Q., and

A2 = R ~ Then sat(A1 fl A2> f sat(A ) fl sat(A ) . For,

satCA ^ fl A2) = 0 and sat(A ) fl sat(A ) = R.

A second observation for the reader to keep in mind is

the following. A partition has been defined to be measurable

when the saturation of every open set is Borel. Does this

imply that the saturation of every Borel set is Borel? As

will be seen below, were this true, it would be very easy to

construct disjoint Borel cross sections for Q. Examples

showing that this is not the case are found in Chapter IV.

Theorem 3.3. Let X be a Polish space and Q a measurable

partition of X by equivalence classes that have at least

28

two elements each. If Q admits a Borel cross section h such

that the induced transversal S is of some finite Borel class,

then Q admits a second Borel cross section, disjoint from h.

Proof of Theorem 3.3

Without loss of generality, we assume S e n„ , Borel's 2p

multiplicative class 2p.

Let Y = X - S and let be the partition of Y induced

by Q. Since each equivalence class has more than one member,

any Borel cross section that is found for Q„„ will also be a Y

Borel cross section for Q; moreover, it will be disjoint from

S. To construct a Borel cross section for Q,., we will apply

iheorein 2.10. Except to show that is measurable, it is

clear that all the hypotheses of Theorem 2.10 are satisfied.

Let U be open in Y. We wish to show that sat_ (U) is a Y

Borel subset of Y. We observe that sat (U) = Y 0 satn(U). vy ^

Were U open in X, we would know that Oy is measurable. If

we knew that the saturation of every Borel set were Borel,

again we would be through. However, neither of these situ-

ations is necessarily the case. So, let W be an open subset

of X such that U = W D Y. The problem will be solved if we

can show that sat^(W 0 Y) is a Borel set in X.

29

We can inductively define a family

{ G : l < k < 2 p } o f subsets of X such that n^.-.i^ -

(1) Y = U G , and G e n„ for all n., O njiij rij 2p-l 1

(2) G = fl G for all n, , and o iij n2rL1n2 1

G e I for all n , niti2 2p -I 2

(1) G = U G for all n., n., and 1 n1n2 n3n1n2n3 1 2

G e n for all n , n1n2n3 2p-3 3

(1), G = U G k n , n ? . . . n , n, n i . • . 11

1 2 2k 2 k+ i 1 for all n^, n , . . . ,n

and G n i n 2 • • • n

e n 2k+i

2p-2k-l

2 k+1

for all n 2k+l'

2k'

(2), G k n i n 2 • • • n 2 k+1

H G for all n,,...,n n , n i • • • n , 1 2k+2 1 2 k+ 2

2k+l'

and G e E for all n , ni . • .n , 2p-2k-2 2k+2 1 2 k+ 2 ^

(2) G = n G for all n , p— 1 ni-.-n n n i • • • n 1

2 p ~ 1 2 p 2 p

G e E for all n„ . n i . . .n o 2p

2 P

Thus, we have

• > n 2 p - l ' a n d

(3) Y = u n u n ••• n g nl n 2 n 3 n 4 n 2

ni n2 ngp

30

We demonstrate that because Y is the complement of a transversal,

the saturation operator commutes with the intersections in

equation (3); specifically, we show that

(4) sat0(W HY) = U H U ••• n satQ(W H G ) . y n n n rij, ^

1 2 3

This is the heart of the solution.

By repeated application of Lemma 3.1, it is clear that

LHS c: RHS. Suppose y e RHS. Then (5) 3nj Vn2 3na Vn4 ••• Vn^

[ 3 x e R(y) n W n G_ _ ] . L »j - ' n, ... iijp J

Let z be the singleton member of R(y) fl S , where R(y) is the

equivalence class of y. Since z I Y, from (3) we deduce that

(6) Vnj 3n2 Vn3 3n4 ••• 3n2p [ z ? Gn% _ ngp] .

We choose an such that (5) holds; for this we choose

an N2 such that (6) holds; for these we choose an

such that (5) holds; etc. In this manner we select a finite

sequence s = (N , N ... , N„ ) such that both (5) and (6) 1 I Zp

are satisfied. So, x e R(y) 0 W fl G and z t G . Therefore, s s s

x 4 z. Hence x £ S; i.e., x e Y. Therefore, s s s

x e R(y) fl W fl Y; i.e. , y e LHS. This proves (4), and shows b

that satQ(WHY) is a Borel subset of X, concluding the proof

of Theorem 3.3.

31

A natural inclination is to simplify the argument for

Theorem 3.3 by an induction proof. It is not clear, however,

that an induction proof can be devised. The lack of a suit-

able inductive hypothesis is the key obstacle remaining and,

thus, merits a few lines of discussion. Although we know

(#) G = n U O ^ G_ _ _for all n, , V ; n, n, n3 n4

nin* " n*p 1

we are not able to inductively "assume"

(##) satQ(w n g ) = n u -• n satQ(w n Gnt_ng) ^ 3 2p

is true for all n^ as a stepping stone to proving (4). The

"trick" used in proving (4) does not work for («*) because

G is not the complement of a transversal. It is possible to n i

extend the technique shown to prove the theorem when the

Borel order of the transversal S is of a fixed infinite Borel

class, and this is demonstrated next. However, the proof is

tailored to the specific ordinal of the class. I do not know

of a generic formula analogous to (4) that will work for

arbitrary countable ordinals.

Theorem 3.4. Let X, Q, h, and S be as in Theorem 3.3

except that S is a transversal for Q of multiplicative class

Then Q admits a second Borel cross section, disjoint

from h.

32

Proof of Theorem 3.4

Define Y = X - S and let W be open in Y. Analogous to

the previous proof, there is a family

* Gpn,n2•••n : 0 - k * o f subsets of X such that k

(1) Y = UG , and G e Z for all p, P P p 2p v'

(2) G = U G for all p, and G e IT ° P ni Pn i r piij 2 p-1

for all n ,

(3) G = fl G for all p, n , and o pnx ii 2 pn1n2 1

G e E for all n„, pn!n2 2 p-2 2'

Gpnr.-n n Gpn1...n f°r a 1 1 P ) ni',',,n2k'

2k 2k+l 1 2k+l

and G e n for all n Pni""n2k+i 2p-2k-l L O r d l i 2k+l'

(3), G fi G for all k pn x • • •n

n i , Pni•••n , , 2k+i 2 k+ 2 2k+2

p, n ,...,n and G e £ 1 2k+l pn x• • .n , , 2p-2k-2

2 k+ 2

f 0 r 3 1 1 n2 k+2'

^ P ~ l Gpn1---n Gpn!...ii f°r P> ni n2p-l'

2 P - 1 2 p 2 p P X

and G e £ for all n. . pn j . . . n o 2p

2 P

33

Thus, we have

(4) Y = u [ u n u ... n g ... ] . p n, ng n3 n , * » *P

Consequently, using the argument in the proof of Theorem 3.3,

we get that

(5) saUW n Y) = u [ u n U ... n satQ(W n G ) ], w p n, n2 n3 nap ^ F l 2p

proving that is measurable and concluding the proof of

Theorem 3.4.

The fact that no generic formula seems to work for all

countable ordinals may possibly be linked with the nonexistence

of a canonical, effective formula for decomposition of count-

able ordinals. P. S. Alexandroff seems to have encountered

this same difficulty in 1917 |_lj while attempting to show

that every Borel set contains a Cantor set. His solution to

the problem, to show (ahead of his time) that every analytic

set contains a Cantor set, seems not to work here. An approach

that does work is a variation on a theme Alexandroff intro-

duced in the early part of that work. This approach is used

in Chapter IV. The basic idea is that, given a Borel set B

in a Polish space X, there exists a well founded subtree T of

Seq and maps H : T -> P(X) and a : T -* cax such that if s e T

then

34

(1) H (0) = B,

(UH(s,n) if | s j is even (2) H(s) =

< flH(s ,n) if |s| is odd,

I Sry CQ") i f Is! is e v e n

(3) H(s) e a U ;

na(s) i f 'S' is odd'

(4) a(s) > a(s,n) for all n if a(s) > 0, and

(5) x e B if and only if 3nt Vn2 3n3 Vn4 •••

[ 3k such that a(nj, - ,n2k) = 0 ,

a(nj, .ngjj.j) > 0, and x e H(nj, •» ,n2k) ] .

The last expression requires explanation to make it precise,

and is also difficult to work with. The notions of "6a-

tree" and "kernel of a 6a-tree" are introduced in Chapter IV

to clarify the idea of an alternating sequence of existential

quantifiers. Intuitively, the map H transforms a 6a-tree

into a 6a-tree of sets with B = H(0) as its vertex. (See

Figure 1.) The saturation operator transforms the second

tree into a 6a-tree of saturated sets. The endpoints of the

6a-tree of sets are open sets, causing the endpoints of the

6a-tree of saturated sets to be Borel. We prove that if all

of the endpoints of a 6a-tree are Borel, then the vertex set

35

6a-tree

6a-tree of sets

II (0)

H(2,l)

H(2,2)

6a-tree of

saturated sets

satH(0)

satH(l)

satH(2)

^satH(3)

satH(2,1)

satH(2,2)

satH(2,3)

Fig. l--6a-tree of saturated sets

36

is Borel. As far as I know, this is a new characterization

Borel sets. The discovery that the vertex of a <5a-tree of

sets is Borel if all of the endpoints are Borel was made by

Professor R. D. Mauldin.

CHAPTER IV

DISJOINT BOREL CROSS SECTIONS OF

A MEASURABLE PARTITION

In this chapter we prove the existence of an infinite

family of pairwise disjoint Borel cross sections for a meas-

urable partition 0. of a Borel subset X of a Polish space Z

under the assumption that the Q-equivalence classes are un-

countable Gg subsets of Z. The proof given in this chapter

is self-contained and does not depend on the special case of

the theorem that was presented in Chapter III.

In the process of proving the selection theorem, Theorem

4.8, we prove a new characterization of Borel sets—Theorem

4.5. This theorem should be of independent interest because

it provides a new avenue in the investigation of properties

of Borel sets. In the last section of this chapter some

examples are given that clarify many relationships among

the properties that are of fundamental import in selection

theorems of this nature. In particular, Example 4.13 shows

that under the conditions assumed for the Selection Theorem

4.8, it is possible that the saturation of a Borel set is

37

38

not Borel. Were this not the case, the somewhat circuitous

route followed in this chapter could have been replaced by

a trivial argument that would have shown even more: that

there exist uncountably many disjoint Borel cross sections

for the partition Q.

The chapter begins with some additional notation. This

is followed by the principal lemmas on properties of trees,

especially Lemmas 4.1 and 4.4. The other half of the selec-

tion theorem, the characterization of Borel sets, is presented

next. The selection theorem is then reduced to the collecting

of previous results. The chapter concludes with some ex-

amples .

Definitions

Some notation has been lifted from Game Theory to form-

alize the sketch presented at the end of Chapter 3. A tree

[_20j is a partially ordered set (T, <_) such that for each

t e T the set of predecessors of t in T is well ordered by

<. If y is an ordinal, the yth level of T is the set of

t e T such that the set of predecessors of t under is

order isomorphic to y. A branch of T is a maximal well-

ordered subset of T. We say that a tree is well founded if

39

each branch is finite. An endpoint of T is an element t of

T that is maximal with respect to <_. The set of endpoints

of T will be denoted by E^. A subset T' of T is said to be

a subtree of T if s e T1 whenever s e T, t e T' , and s _< t.

We are concerned in this dissertation with subsets of

Seq that are trees under the partial ordering defined by

setting s < t if t extends s. A nonempty subset T of Seq is

called a 6c-tree if (1) T is a well-founded tree, (2) the

level of each endpoint of T is an even whole number, and

(3) T is completely regular; i.e., whenever (s, n) e T for

some n, then (s, n) e T for all n.

A rank function on a tree I is a function g : T -»• Ord

that satisfies

I 0 if s e E (Us) = T

' sup { 3(t) + 1 : t > s } if s i E^.

The existence and uniqueness of this function for well-founded

trees is shown in the next section. The rank of a well-

founded tree, r(T), is defined as r(T) = sup { g(s) : s e T }.

If T is a tree, a game G(T) is played as follows. Players

I and II alternately choose natural numbers n_ . A strategy

for player I is a map < : Seq -* W. Given a play (n1, n£, n ,

• • • , n2k^ £ ^ ~ ^T' K determi-nes that next move by player

I; namely <(n;L, n2, n , ... , n2k) . The set of strategies

40

S 00

for player I is denoted = W . An endpoint s e T is

said to be consistent with strategy <_ in game G(T) if

K(S | 2n) = s(2n+l) whenever 2n < |s|. For K e S^ we define

E,J,(k) as the set of endpoints of T consistent with strategy

k in game G(T). We observe that a game G(T) on a 6a-tree T

will terminate after finitely many moves, when an endpoint of

T is reached. The notion of "winner" of G(T) is not required

in this dissertation, and so is not defined. Similarly, it

is not necessary to discuss strategies for player II.

Let T be a 6a-tree. For t e T we define the subtree T

of T by

T = { s e Seq : (t, s) e T }.

It is clear that the set of endpoints of T can be described

by

E t = { s e Seq : (t, s) e ET }.

It is also clear that the rank of T satisfies r(T ) = 3(t). t t

In particular, r(T) = 3(0).

Let X be a set and H : T -> P(X) a map from T into the

power set of X. We define the kernel of H by

ker(H) = U f) H(s). x ' KtSf seErOc)

The restriction of H to T is defined to be

H t : T t h- P(X) : Ht(s) = H(t,s) .

41

6c-Trees

The existence of the rank function on well-founded trees

is well known jj.4] . The first lemma is included for the sake

of completeness, since existence of the rank function plays a

central role in the forthcoming development.

LeTmna 4.1. If T is a well-founded tree, then a rank

function exists and is unique. If T is a subset of Seq, the

rank of T is countable.

Proof of Lemma 4.1

We first show a rank function exists. We set T 0 = E^,

the set of all endpoints of T, and we suppose T is defined

for all ordinals y < a. Define

(1) T = { s e T : t e U T if t > s } ,

a Y<a y

Observe that the sets Ta form a nondecreasing family of sub-

sets of T. We claim that T = UT^, the union taken over all

ordinals a. If not, we can construct an infinite branch in

T as follows. Let s be in T but not in UT . To show there 1 a

is an element of T that is preceded by s^ and that is not

in UTa, we suppose not. For each t > s , then, we can assign

an ordinal ot(t) such that t £ Ta^t^. We then set y =

sup { a (t) : t > s^ } and observe that s^ £ , a contra-

diction. Therefore, there is an element of T such that

42

s < s and s t UT . Continuing by induction we construct 1 2 2 a

an infinite sequence (s^, s2> s^, ...) such that s^ < s2 <

s < ... and s £UT for all n. By Hausdorff's Maximality 3 n a

Principle, there is a branch B of T that contains (s , s2,

s , ... ), and this contradicts that T is well founded. This

proves that T = U T .

We are now in a position to define the rank function.

We set

(2) e : T + Ord : 3(s) = inf { a : s e T } . a

Eecause every collection of ordinals is well ordered, 6 is

well defined. Moreover, if s is an endpoint of T, then $(s)

= 0. It only remains to be shown that

3(s) = sup { 8(t) + 1 : t > s } . Suppose s, t e T,

a e Ord, s e T , and t > s. There is an ordinal y such that ' a

t e T and a > y + 1. Therefore 8(t) = inf {6 : t e T~ } < y - 6

Y; i.e., 3(t) + l £ Y + l £ a . Consequently, sup { 3(t) + 1

: t > s } £ a, and, so, sup { $ ( t ) + l : t > s } _ < inf { a :

s e T a } = B(s), which proves inequality in one direction.

To show the other direction, let a = sup { 3(t) + 1 : t > s }

If t > s then B(t) + 1 £ a; i.e. , t e UT , the union taken

over y < a- This shows s e Ta and, consequently, 3(s) £ a =

43

sup { 3(t) + 1 : t > s }, completing the proof of equality.

This concludes the proof of the existence of a rank function

on T.

To compute the rank of T, let K denote the cardinality

of T, « + the successor cardinal of K , and a the least ordinal

with predecessors. We show the rank of T is less than a.

For suppose there is an element s of T such that 3(s) >_ a.

Then there is an element s that is maximal with respect to

this property; viz., 3(s) >_ a, and if t > s, then 3(t) < a.

But 3(s) = sup { 6(t) + 1 : t > s } < a, because |{ t £ T :

t > s }| < |T| < N+. This contradiction proves r(T) < a. In

case T c Seq, this means r(T) < w .

To see that a rank function on a well-founded tree is

unique, suppose 3 and 3* are rank functions and 3 ¥ 3*. There

is an element s of T such that 3 ^ ) f 3* (s^ . Hence, there

is an element s2 of T such that s2 > s^ and 3(s2) ^ 3* (s2) .

Continuing, we generate an infinite, decreasing sequence of

ordinals, an impossibility. Therefore a rank function is

unique. This concludes the proof of Lemma 4.1.

We now begin a study of <5cr-trees. If T is a 6a-tree, X

is a set, and H : T P(X) is a map, we shall refer to { H(s)

: s e T } as a 6a-tree of sets. If Q is a partition of X and

W is a subset of X, we shall call

44

{ sat^(W H H(s)) : s e T } the induced 6a-tree of saturated

sets. The next two lemmas follow the outline of the previous

chapter. Lemma 4.2 establishes that the kernel of a <5a-tree

of Borel sets is itself a Borel set, and Lemma 4.3 shows that

the kernel is in fact the vertex set of the tree.

Lemma 4.2. Let T be a 6a-tree, X a topological space,

and H : T ->• P(X) a map. The following are then true.

(i) If (n,p) e T for every (n,p), then

ker(H) = U fl ker(H ) , and n, p

n p ' (ii) If H(s) e 8(X) for every s E E^, then

ker(H) £ B(X)

Proof of Lemma 4.2

We first prove (i). If x £ ker(H), then there is a

strategy k such that for every endpoint s of T consistent

with k, x £ H(s). Set n1 = k(0). For n2 £ W, define

KHj ,n2 : T

ni,n2 * N : K

n i >n 2( s ) " K(V n2 • s ) ' So' f o r a 1 1

n and s £ E (K ) we have x £ H(n , n , s) = H (s) Z -V-I « I > N2 J- ^ ,n2 ni ,n2

since (n^, , s) is an endpoint of T. This shows that there

is an n^ such that for every n2 there is a strategy K ni ,n2

on the tree T such that for every endpoint s of T ni,n2 n1,n2

consistent with K we have x e H (s); and this shows nx,n2 n!,n2

that x £ RHS.

45

Suppose x e RHS. Then, there is an n^ such that, for every

n 0, x e ker (H ). That is, there is a strategy K on I n 1 ? n 2

b J n l,n 2

the tree T such that for every endpoint t of T con-ni,n2 •/ r n x,n 2

sistent with K we have x e H (t). Fix such an n,. ni,n2 n!,n2 1

Define < : T •> W such that (1) K(0) = n^, (2) for each ri ,

K(n n t) = k (t) if t is in T and (3) 1 2 n x,n 2 n x,n 2

K(S) = 1, otherwise. Then, K is a strategy on T, and if s is

an endpoint of T consistent with K we can find an n^ e N and

an endpoint t of T such that s = (n,, n„, t). Moreover, n x,n 2 1 2

t is consistent with strategy K on the tree T ni ,n2 m ,n2

Therefore x e H (t) = H(n,, n., t) = H(s). This shows

nj,n2 1 2

x e LHS, and shows that assertion (i) is true for all 6c-

trees T.

The proof of (ii) is by induction on the rank of T.

Assertion (ii) is easily seen to be true for trees of rank

zero. Suppose T is a So-tree of rank greater than unity, and

assume that the lemma holds for all trees of lesser rank.

For all pairs (n n ), we have r(T ) = 3(n , n ) < 3(0) _ L Z nx ,n2 I z

= r(T). By hypothesis, ker(H ) is Borel for all pairs n l > n 2

( ni, n 2). From (i) we get that ker(H) is Borel, completing

the proof of Lemma 4.2.

46

Lemma 4.3. Let X be a topological space and T a 60-

tree. Suppose H : T P(X) satisfies

(i) If s e T, |s| e E, and s t E„,

then H(s) =UH(s,n), and n

(ii) If s e T and |s| e 0, then

H(s) = nH(s,n).

Then H(0) = ker(H).

Proof of Lemma 4.3

The proof is by induction on the rank of T. We assume

the lemma is true whenever we have a tree of lesser rank than

the rank of T and a map that satisfies (i) and (ii). We may

assume 0 is not an endpoint of T so that, by (i) and (ii),

H(0f) = U n H(n,p) . n p

From Lemma 4.2(i),

ker(H) = U n ker(Hnp) . n p **

Since the rank of T is less than the rank of T and H n,p n,p

satisfies hypotheses (i) and (ii), H(n, p) = H (0) = ker(H ., n, p n , p /

which proves the lemma.

The preceding lemma showed that the kernel of a 6a-tree

of sets is the vertex set of that tree. The next lemma shows

that this property is shared by the kernel of the induced <5a-

tree of saturated sets. This is a surprising result because

47

the hypotheses of the preceding lemma are not satisfied by

the tree of saturated sets. Specifically, let Q be a parti-

tion of a topological space X, W a subset of X, and

H : T -* P(X) a map. The set W induces a map

H w : T ?(X) : Hw(s) = sat[ W U H(s) ] .

In general, we do not expect that satisfies hypothesis

(ii) of Lemma 4.3. (See Example 4.13). Nevertheless, the

next lemma states that H^(0) = ker(H^) if H(0) is the com-

plement of a (not necessarily Borel) transversal for the

partition Q.

Lemma 4.4. Let X, T, and H satisfy the hypotheses of

Lemma 4.3. Let W c X, Q be a partition of X, and S a (not

necessarily Borel) transversal for Q. If H(0) = X - S then

(*) satQ(W H H(0O) = ^ ^ k ) satq(W n H(s)) .

Proof of Lemma 4.4

From Lemma 4.3 we have

(1) H(0O = U n H(s) . /ccSf seET(*;)

Therefore,

(2) W n H(0) = u n W n H(s) V ' KESt seEfOc)

and, by Lemma 3.3, LHS c RHS.

48

Next, suppose y e RHS. Let R(y) denote the equivalence

class containing y. We have

(3) 3K e ST Vs e ET( /c) 3 x k s

[ XKS £ R(y) n w n H(s) ] .

Let z be the singleton member of S fl R(y) . From (1) ,

(4 ) Vac c ST 3s E ET( /c) [ z 4 W n H(s) ] .

Choose K satisfying (3), and for this k choose s satisfying

(4). Then, x e R(y) Pi W fl H(s) and z i H(s) . K S

Therefore x f z, which implies x i. S. So, KS ICS

x e R(y) O w n H(0) K S

and, consequently, y e RHS, concluding the proof of the lemma.

Borel Schemes

In this section we introduce a new characterization of

Borel sets. The main result, Theorem 4.5, shows that in very

general topological spaces a Borel set is the vertex set of

a 6a-tree of Borel sets that satisfies all the hypotheses of

Lemmas 4.2 and 4.3. The remainder of the section, a digres-

sion from the main body of this chapter, is devoted to

comparison of the new characterization of Borel sets with

schemes for generating analytic sets.

One new definition is required for Theorem 4.5. A

space X is a G^-topological space if each closed set a set.

49

Theorem 4.5. Let X be a Gg-topological space and

Y e 8(X) . There exists a 6a-tree T and maps H : T P(X) and

a : T -* w that satisfy

(1) H(0) = Y,

(2) If s e T and |s| e E

(a) If s t E then H(s) = U H(s,n) , 1 n

(b) II(s) e £ . v , v v a(s)

(c) a(s) is an even ordinal,

(3) If s £ T and |s| e 0

(a) H(s) = PI H(s ,n) , n

(b) H(s) £ n f v , v a (s)

(c) a(s) is an odd ordinal,

(4) If s, t £ T and s properly extends t

then a(s) < a(t), and

(5) If s is an endpoint of T, a(s) = 0.

Proof of Theorem 4.5

There is an even ordinal y, y < , such that Y e ,

Borel's additive class y. Define T = { 0 } , H (0) = Y, and o o

a (0) = y. T is a 6a-tree and (1-4) are satisfied by o o

(T , Hq, aQ). Suppose m is a whole number and (Tm, Hm> am>

has been defined such that T is a 6a-tree, (1-4) are satis-m

fied, and if 1 < k < m then T, is a subtree of T and H. and ' — — k m k

50

a, are extended by H and a , respectively. Consider an end-k m m

point s of the tree T . The length of s is even because T m

is a 5a-tree. By (2b) Hm(s) e ^. If a m(s) = 0 we do

m

not extend s in defining the tree Tm+^- a

m(s) > ^

(2c) a (s) >2. So, for all natural numbers n and natural 9 m —

number pairs (n, p) there are sets and A and ordinals

Y and y that satisfy (6-12): 1 n np

(6) H (s) = U A , m n n

(7) A = n A for all n , n p np

(8) am(

s) > Y n > Y

n p f o r a 1 1 ^n' '

(9) y is an odd ordinal for all n, 'n

(10) y is an even ordinal for all (n, p) , np

(11) A e II for all n, and n y

n

(12) A e E for all (n, p). np y

r np

We define, for all (n, p) , Hm+1

CO 3 II

i

Hm+1 (s, n, p) = A ,

np'

am+l CO II

2 >

am+l (s, n, p) = Ynp'

and

Tm+1 = T U { (s

m , n) : s e T and a(s) > C } m

U { (s, n, P) : s e and a(s) > 0 }

51

In addition, for all s in T we set H , ,(s) = H (s) and m m+1 m

°Wl(s:) = V ( s ) ' T h e n Tm+1 l s a < Tm + r

Hr t . <Vl>

satisfies (1-4), and if 1 £ k £ m then is a subtree of

Tm+1 a n d Hk a n d ak a r e e x t e n d e d by H m + 1 and am+1, respectively.

This completes the inductive definitions of T , H , and a . m m m

We can, now, define T = UT . Because of (4) and the fact that m

Y is well ordered, T is well founded and, hence, a 6a-tree.

Define H and a to be the unique maps that extend H and a , m m

respectively, for all m. Then (1-4) are clearly satisfied by

(T, H, a), and (5) is satisfied by construction since, if

a(s) > 0, then s would have been extended at some finite

stage. The completes the proof of Theorem 4.5.

The implications of Theorem 4.5 lead in several directions

The obvious analogy with Operation A is one offshoot and will

be briefly explored shortly. Corollary 4.6 lies in a rather

different direction and has an immediate bearing on Theorem

4.8, the major theorem of this dissertation.

Corollary 4.6. Let X be a topological space, Q a

measurable partition of X, and S a Borel transversal for Q.

Then Qy, the partition of Y = X - S induced by Q, is meas-

urable.

52

Proof of Corollary 4.6

Let T be a 6o-tree and H and a maps, as guaranteed by

Theorem 4.5. We desire to show that sat (W CI Y) is Borel in Y

Y for every open subset W of X. Let W be an open subset of

X. By Lemma 4.4, sat^(W 0 Y) = ^(0) = kerCH^. By Theorem

4.5, T, X, and satisfy the hypotheses of Lemma 4.2 (ii).

Thus, sat (W fl Y) e S(X). Therefore, satn (W fl Y) = X Hy

Y fl sat^(W fl Y) is Borel in X, proving that is measurable.

We digress, now, to compare the results of Theorem 4.5

to Operation A. We say that a set H is the result of Operation

A applied to a family of sets

A = { H(s) : s e Seq } if H = ^ ^ H C ^ n ) .

The following theorem is well known; see, for example, j_12J .

Theorem. Let X be a Polish space. A subset A is an

analytic subset of X if and only if A is the result of Oper-

ation A applied to a regular system of closed sets.

By analogy, we make the following definition. Let X be

a topological space and T a 6a-tree. A set H is the result

of Operation 8 applied to a family A = { H(s) : s e T } if

H = U n H(s) . ksS-j seE'j(K)

The result of Operation 8 is simply the kernel of the map H.

53

Theorem 4.7. A set B is a Borel subset of a Polish

space if and only if B is the result of Operation 6 applied

to a regular system of closed (open) sets.

Proof of Theorem 4.7

Lemma 4.2 (ii) can be invoked to show that the result

of Operation B applied to a system of either closed or open

sets is Borel. In the other direction, suppose B is a Borel

set. By Theorem 4.5 there is a <5c-tree T and a collection

of sets { H(s) : s e T } such that B = H(0). By Lemma 4.3,

B = ker(H); i.e., B is the result of Operation 8 applied to

{ H(s) : s e T }. Moreover, H(s) is open whenever s is an

endpoint of T. The family { H(s) } is not usually regular.

Because a result of Operation B depends only on the endpoints

of T, this set can be easily modified to form a regular

family by defining a new map H* : T -> P(X) by H*(s) = H(s)

if s e , and H*(s) = X, otherwise. B is then the result

of Operation B applied to the regular system of open sets

{ H*(s) : s e T }. Because every open subset of a metric

space is an F , it is easy to induce a So-tree T** and a

regular system of closed sets { H**(s) : s e T** } such that

B = ker (H'v,v) , concluding the proof of Theorem 4.7.

54

Borel Cross Sections For Measurable Partitions

The main theorem is now an easy consequence of the pre-

ceding results.

Theorem 4.8. Let X be a Borel subset of a Polish space

Z, 1 < n < n , and Q a measurable partition of X by sub-

sets of Z of cardinality > n. Then Q admits n disjoint Borel

cross sections. In particular, if each Gg set is uncountable,

then Q admits infinitely many disjoint Borel cross sections.

Proof of Theorem 4.8

The existence of a Borel cross section h for Q is guar-

anteed by Theorem 2.10. Set Y = X - S, where S = h(X) is the

Borel transversal induced by h; let R be the equivalence re-

lation induced by Q; let Qy be the partition on Y induced by

Q; and let be the equivalence relation induced by Qy- If

each Q-equivalence class has at least two members, it is clear

that any cross section for will also be a cross section for

Q. If n = 1, the proof is complete. Assume n 2. Y is a

Borel subset of the Polish space Z, is a partition of Y by

G.subsets of Z, and, by Corollary 4.6, Qv is measurable. So, 0 1

another application of Theorem 2.10 yields a Borel cross sec-

tion h2 for Qy which, by the previous remarks, is a Borel

55

cross section, also, for Q and is clearly disjoint from h.

It is clear that this process may be repeated (n-1) times to

produce the required n disjoint Borel cross sections for Q,

concluding the proof of Theorem 4.8.

Problem. A major unsolved problem remains. Are there,

in general, uncountably many disjoint Borel cross sections

for Q? Or, even, a Borel parametrization? In case each Gr o

equivalence class is dense-in-itself, it is not difficult

to continue the above process to produce H^disjoint Borel

cross sections. Actually, more is possible in this case --

there is a Borel parametrization. This fact is proved in

Theorem 5.6.

This chapter will be concluded with a few examples il-

lustrating some of the key relationships involved in selection

theorems for partitions. Let X be a Polish space, Q a meas-

urable partition of X by Borel equivalence classes, R the

equivalence relation on X x x induced by Q, S a Borel trans-

versal for 0, and fg : X -* S the induced cross section. We

ask the following questions.

1. Under what conditions is f Borel measurable?

2. If B e B(X), must sat(B) e 8(X)?

3 . Mus t R e B (X x X) ?

We also ask what relationships exist among 1-3.

56

We first observe that 3 implies 1. Indeed, suppose R

is analytic in X x X. If B is a Borel subset of X, then

f s ~1 (B) = sat(B n s ) = n 2 [R n ([B n s ] X x ) ]

is an analytic subset of X. Similarly, fc ^(X - B) e A(X).

By Theorem 2.21, f ^(B) e 8(X).

We next observe that 1 implies 3. Suppose f^ is Borel

measurable. Define

V = { (x, y, u, v) e X x X x X x X : u = f ( x ) and v = f(y) }

and W = { (x, y, u, v) e X x X x X x X : u = v } . Then V is

homeomorphic to Gr(f_) x Gr(f_,) and, hence, is a Borel set;

W is, clearly, a closed set; and R = n12(V 0 W), the (1, 2)-

projection. Since I (V fl W) is one-to-one, R is Borel.

In |_22j > Srivastava proved that R e 8(X x X) if each

equivalence class is G „. Therefore, is always Borel meas-0 O

urable in this case. If, however, each equivalence class is

merely F , Example 4.10, below, demonstrates that R need not

be Borel. Example 4.12 gives an example of a partition by

F^ equivalence classes such that f^ is not Borel measurable.

Example 4.10 shows, as well, the existence of a closed

set whose saturation is not Borel. In example 4.13, below,

we demonstrate that even if each equivalence class is closed,

2 need not be satisfied.

57

Examples 4.10-4.12 are special cases of the following

construction. Let Qq be a measurable partition of the closed

unit interval I such that Qq contains at least two equivalence

classes (say, Eq and E^) and some equivalence class (say, Eq)

contains at least two points (say, yQ and y). Let A be a

non-Lebesgue-measurable subset of I. We proceed to construct

2

a partition Q of I = 1 x 1 . We begin by defining a second

partition Q of I. Q is defined to be the partition that

differs from Qq only in that y belongs to E^ rather than Eo<

We, then, say that points (u , v ) and (u^, v^) of I x I are

Q-related if (1) u = u, and (2) if u e A then v and v, are o 1 o o 1 Q -related, and (3) if u t A then v and v., are Q-related. xo o o 1 1

Some properties of the partition Q are established in the

next Lemma. To facilitate proof of the lemma and examples, we

define L = I x { y } , L = I x { y }, and L. = I x { y }. J o o 1 1

We say that disjoint sets E and F are dense-in-each-

other if each point of each set is a limit point of the other

set.

Lemma 4.9. (1) Q is measurable if and only if (*) E and E, are dense-in-each-other, o 1 '

(2) If (*) does not hold, there is an open set whose Q-

saturation is not even analytic, (3) If (*) holds and there

53

is an ordinal a 0 such that satq (V) is in I for all o

open sets V, then satq(W) is G^ in I x I for all open sets

W, and (4) sat^C!^) is a nonanalytic subset of I * I.

Proof of Lemma 4.9

Suppose (*) is true. For U, V, c I,

sat (U x V) = jjU 0 A) x sat (V)J U [~(U - A) x sat (V)J ^ o '1

= U x sat_ (V). o

If U and V are open then sat^ (U x V) is Borel (and of class

G in case (3) ). Since I x I is separable, the same is true a

for sat^CW) if W is open in I x I, proving (3) and half of (1)

Suppose (*) is not true. If y is not a limit point of E ,

there is an open set V in I such that y e V and V 0 E = 0.

So, L1 n satg( I x V) = A' x { y^ } ^ A(I x I), which implies

sat^Cl x V) t A(I x I). The procedure is similar if yQ is

not a limit point of E or y1 is not a limit point of Eq .

This concludes the proof of (2) and the second half of (1).

Finally, L fl sat^I^) = A x { y } implies that

satQ(LQ) t A(I x I).

Example 4.10. There is a measurable partition Q of

I x I by F^ equivalence classes such that satp(W) is open for

open W and yet (a) there is a closed set L whose saturation

59

is not analytic and (b) the induced equivalence relation R

is not analytic. For, let Qq be the partition of I defined

by u ~ v if u - v e let E = Q. fl I, E = (^2~ + Q) fl I,

yo = .75, y = .5, and y^ = 2~- 1. Since sat^ (V) = I for "O

open V, by Lemma 4.9 (3) sat^(W) is open for open W. The

proof of (a) is Lemma 4.9 (4). To see (b), observe that

N 2 ( (Lq x I*") fl R ) = sat^(L O) i A ( I 2 ) , which shows that

R t A(I4).

Example 4.11. There is a measurable partition Q of

I x I by equivalence classes of Borel ambiguous class 2 such

that sat^CW) is open if W is open and yet there is a closed

transversal S for Q such that fg is not Borel measurable. For,

let E , E^, y y, and y be as in Example 4.10. Set

E2 = (Eo U a n d Qo = ^ Eo' El' E2 L e t y2 £ E2 a n d

define L2 = I x { y2 }. By Lemma 4.9 (3), sat^(W) is open

for open W. Set S = L U L, U L„. S is a closed transversal r o 1 2

C1(Ln-o

for Q and, since L f l f c , ^ ( L ) = A x { y } (

fg1(LQ) t A (I x I) .

Example 4.12. There is a measurable partition Q of

I x I by F equivalence classes such that sat^(W) is F for a Q a

open W and yet there is a G. transversal S for 0 such that f0 is o b

60

not Borel measurable. For, let E , E,, y , y, and y, be as o i yo 1

in Example 4.10. Define Qq by setting u ~ v if (u, v) e Eq

or (u, v) e or u = v. Then

sat (U x V) = U x satQ (V) = U x (V U Eq U E ^ e F^CI x I) ^ vo

implies sat^CW) e x I) for open W. Define

S = L U L, U (I x (E U E J ' ) . o 1 o 1

Then S is a G. transversal for Q. As in Example 4.11, f is not o o

Borel measurable, concluding the demonstration of Example 4.12.

A careful examination of the next example yields a nice

bonus. The dividends from this example are summarized in the

corollaries that follow.

Example 4.13. There is a measurable partition Q of the

closed unit interval I by closed uncountable equivalence classes

such that the saturation of every open set is F , the satura-

tion of every closed set is closed, and, yet, there is a Borel

set H such that sat(H) is non-Borel. For, let g be a contin-

uous map of the closed unit interval onto the closed unit

square. If Q = { g-1(t) : t e I x I }, then Q is a partition

of I by uncountable closed sets. (The construction of Q is

due to R. D. Mauldin.) If M is a closed subset of I, g(M) is

a closed subset of the square and, consequently,

sat(M) = g 1(I x n2 ( g(M) ) ) is closed. If U is open, U

61

is an F and, so, sat(U) is an set. By the proof of

Srivastava's theorem [Theorem 2.10J, there is a Borel trans-

versal S for Q and a Borel cross section f : I •+ S.

Purves [_19j has proven that if X and Y are Polish spaces ,

A e 8(X) , B e B(Y) , and h : A -> B is a Borel measurable map

from A onto B, then the image of each Borel subset of A is

Borel in B if and only if

| { y e B : | h _ 1(y) ] > } | < .

Applying this theorem to the function f, we discover the

existence of a Borel set H. in the closed unit interval whose

image under f is non-Borel. Since sat(H) fl S = f(H), sat(H)

is non-Borel, concluding the example.

Corollary 4.14. Suppose 0 (not necessarily measurable)

is a partition of a Polish space that admits a cross section

that is Borel measurable. Then the Q-saturation of every

Borel set is Borel if and only if Q has only countably many

uncountable equivalence classes.

It is interesting to note that if Q has only countably

many uncountable equivalence classes, then Q is a measurable

partition.

62

Corollary 4.15. If Q is a measurable partition of a

Polish space by uncountably many uncountable equivalence

classes, then there is a Borel set whose saturation is non-

Borel.

CHAPTER V

BOREL PARAMETRIZATIONS

In this chapter we deal with two kinds of parametrization

theorems. Theorems 5.5 and 5.6 enumerate conditions that are

equivalent to the existence of a Borel parametrization.

Theorems 5.2 and 5.3 identify sufficient conditions for the

existence of a Borel parametrization.

Throughout this chapter we assume X and Y are Polish

spaces. A Borel parametrization for a subset G of X * Y is

a Borel isomorphism g : X x Y -*• G that maps { x } x Y onto

{ x } x g„. A Borel uniformization of G is a Borel measurable X ' """"

function f : X •+ Y such that Gr(f) c G. We note that a Borel

parametrization of G can be decomposed into a continuum of

disjoint Borel uniformizations that fill up G.

If Q is a partition of X, we say that a subset G of X x Y is

Q-invariant if G = G whenever x and t are Q-related. When X u

there is no ambiguity as to the partition 0, we say, simply,

that G is invariant. A Borel parametrization g of G is_ said

to be invariant if, in addition, g(x,y) = g(t,y) for all y

whenever x and t are related.

63

64

If F : X -*•' P(Y) is a multifunction, we say that a func-

tion g is an (invariant) Borel parametrization for F if g is

an (invariant) Borel parametrization of G = Gr(f); that is,

i f g : X x Y -> G is a Borel isomorphism that maps { x } x Y

onto { x } x F(x) (and such that g(x,y) = g(t.y) for all y

if x and t are related).

A conditional measure distribution on X x Y [_15j is a

map y : X x 8(Y) -> R such that for each x, y(x,») is a meas-

ure on 8(Y) and for each E in 8(Y) , y(»,E) is a Borel meas-

urable function on X. If, in addition, u(x,») = y(t,»)

whenever x and t are related, we say \i is invariant. If v

is a measure on Y and E is a subset of Y such that

(1) v(E) > 0 and (2) if F is a subset of E then either v(F) = 0 or

v(F) = v(E), then E is said to be an atom of the measure v.

A measure v is atomless if it has no atoms. In case v is a

measure on the Borel subsets of Y, v is atomless if the v-

measure of each point is zero.

In |_15j > R- D- Kauldin proved the following theorem.

Theorem 5.1. Let X and Y be uncountable Polish spaces

and G a Borel subset of X x Y such that each vertical section

of G is uncountable. Then the following are equivalent.

65

(1) G admits a Borel parametrization,

(2) There is a conditional probability distribution on

X x y such that for each x, y(x,G ) = 1 and y(x,*) X

is atomless,

(3) G contains a Borel set M such that each vertical

section of M is a nonempty compact perfect set.

The next theorem is an invariant version of Theorem 5.1.

This theorem is used in the proof of Theorem 5.5. No proof

will be presented for Theorem 5.2 because it can be verified

by retracing the steps of the proof of Theorem 5.1 and showing,

at every stage, that the invariant properties are preserved.

Theorem 5.2. Let X and Y be uncountable Polish spaces,

Q a partition of X, and G a Borel subset of X x Y such that

each x-section of G is uncountable. The following are equiv-

alent .

(1) G admits an invariant Borel parametrization,

(2) There is an invariant conditional measure distri-

bution u on X x y such that y(x,G ) = 1 for all x X

and y(x,*) is atomless for each x,

(3) There is an invariant Borel subset T of G such that

each x-section is a nonempty compact perfect set.

66

Of course, if any of the conditions of Theorem 5.2 ob-

tain, then G is an invariant subset of X x Y.

The next theorem, if easy to prove, is somewhat of a

surprise. It says that certain partitions admit a parametri-

zation if and only if they admit an invariant parametrization.

Theorem 5.3. Let X be a Borel subset of a Polish space,

Q a measurable partition of X by uncountable equivalence

classes, and R the equivalence relation induced by Q. The

following are equivalent.

(i) Q has a Borel parametrization,

(ii) R, as a subset of X x x , has an invariant Borel

parametrization,

(iii) R has a Borel parametrization,

(iv) There is an invariant conditional probability dis-

tribution y on X x x such that for each x,

u(x,R) = 1 and u(x,*) is atomless, X.

(v) There is a conditional probability distribution v

on X x X such that for each x, v(x,R ) = 1 and X

v(x,«) is atomless,

(vi) There is an invariant Borel subset T of R such that

each vertical section of T is a nonempty compact

perfect set,

67

(vii) There is a Borel subset M of R such that each

vertical section of M is a nonempty compact per-

fect set,

(viii) There is a Borel subset H of X that meets each

member of Q in a nonempty compact perfect set.

Proof of Theorem 5.3

We observe, first of all, that (ii) implies (iii), (iv)

implies (v), and (vi) implies (vii). Moreover, by Theorem

5.1, statements (iii), (v), and (vii) are equivalent, since,

by Theorem 2.10, R is a Borel subset of X x X.

Even though we know from Theorem 5.2 that statements

(ii), (iv), and (vi) are equivalent, we do not use this fact

in this proof for three reasons. First, no proof was offered

for Theorem 5.2; second, the arguments concerning these state-

ments may provide the reader with some insight into the inter-

relationships involved; and third, the proof is not difficult,

even without the aid of Theorem 5.2.

We point out at this time that the existence of a Borel

measurable transversal S for Q is guaranteed by Theorem 2.10.

The transversal S will be referenced in several parts of

this proof.

68

We show (i) implies (ii). Suppose F : X x X + X i s a

Borel parametrization of Q. Define

G : X x X X x X : G(x,y) = (x,F(x,y)). G is easily seen to

be a one-to-one invariant Borel measurable map because it can

be written as a combination of one-to-one Borel measurable

maps. Namely, let h : (x,y) -* (x,x,y) and i : x -> x. Then,

G = (i x F) o h. Because F(x,*) maps X onto R(x), G(x,«)

maps X onto { x } x r Therefore, G is an invariant Borel

parametrization of the subset R of X x X. This proves that

condition (ii) is satisfied.

We next show that (vii) implies (vi). Let M be a subset

of X x X satisfying condition (vii). For each x e X, let

s : X X be the Borel measurable cross section that is in-

duced by S. We define

T = { (x,y) e X x X : ( s (x) , y ) e M } .

T is an invariant subset of R and T = M , N is a nonempty X S \ X y

compact perfect set for each x. T is a Borel subset of X x X

since T = (s x i) "'"(M), where i : X -* X is the identity map.

We show next that (vi) implies (viii). Let T be a subset

of X x x that satisfies the condition (vi). We set

H = II (T) , the 2-projection of T. To see that H is a Borel

69

subset of X, we first observe that H = n ( (S x X) 0 T ).

This is because H is invariant. We next observe that the

restriction of the projection map to the set (S x x) fl T is

one-to-one. For, suppose (x,y) and (x',y^) are points of

(S x X) 0 T and II (x,y) = H (x',y'). Then y = y \ Since T

is a subset of R, this means (x,x') e R. But x and x' belong

to S. Therefore, x = x^, proving that n | (S x X) fl T is

one-to-one and that H is Borel. To see that H meets each

equivalence class in a nonempty compact perfect set, fix

x e X and let R(x) denote the equivalence class containing

x. Then it is clear that H fl R(x) = T . x

We now show that (viii) implies (vi). Let H be a set

that satisfies (viii) . Define T = R fl (X x H) . T is a

Borel subset of R. If x e X, T = R(x) fl H is, by assumption, X

a nonempty compact perfect set. To see that T is invariant,

suppose (x,t) e R. Then, T = R(x) fl H = R(t) fl H = T . X L

To show that (v) implies (iv), suppose v satisfies (v).

Define y : X x S(Y) : y(x,E) = v(s(x),E), where s is the

cross section defined earlier in this proof. The map y

is an invariant conditional probability distribution that is

easily seen to satisfy the required conditions.

70

We next show that (iii) implies (ii). Suppose G is a

Borel parametrization for R. Define

F : X x X - * X x X : F = i x (JI o G o (s x i)) , where

i : X -> X is the identity map. F is an invariant Borel

measurable map and F(x,y) e { x } x R for all (x,y). F maps

X x X onto R; for suppose (x,y) e R. Then (s(x),y) e R and

there is some element t such that (s(x),t) e R and

G(s (x) , t) = (s (x) , y) . Thus, F(x,t) = (x, n 2 o G(s(x),t) ) = (x,y)

To see that F is one-to-one, suppose F(x,y) = F(x',y'); i.e.,

( x, n2 o G (s (x) , y) ) = ( x', n 2 o G(s(x'),y') )• Then

x = x", n2 o G(s(x),y) = n2 o G(s(x),y'), and, finally, y = y'.

This shows F is an invariant Borel parametrization of R.

Finally, we show (ii) implies (i). Suppose F is an in-

variant Borel parametrization of R. Then n o F is a Borel

parametrization of Q. This concludes the proof of Theorem

5.3.

The final goal of this chapter is to prove Theorem 5.5.

This theorem was first proven by S. M. Srivastava in his

dissertation Q23J . I discovered this theorem independently,

and because my proof is an application of Theorem 5.2, and

also because it utilizes a rather interesting indexing scheme,

it is presented here. This indexing scheme should be applicable

to other unsolved parametrization problems.

71

A subtree T of Seq will be called a regular finite

binary tree if each branch is finite and of the same length,

and if whenever s e T is not an endpoint of T, then there are

precisely two natural numbers m and p such that (s,m) e T

and (s,p) e T. We observe that the rank of T satisfies

r(T) = n if the nth level of T is nonempty and the (n + l)st

level is empty. The empty set is regarded as the zero

level; for example, if T = { 0 }, then r(T) = 0.

Suppose T is a regular binary tree of rank k. We re-

quire some notation to represent the regular binary trees of

1c rank (k + 1) that contain T. T has 2 endpoints. Every

regular binary tree of rank (k + 1) that contains T can be

k+1

constructed from T by adding elements of N until each end-

point of T has exactly two extensions. To represent this

fact mathematically, we fix a one-to-one onto map

(*) h : W + (W x M) - A, ET

where A is the diagonal of M x N. For K e W , h o < maps

each endpoint s of T into an ordered pair

(m,p) = ( (h o <)1(s), (h o K)2(S) ) such that m f p. This

induces a tree T(K) of rank (k + 1) that contains T. It is

described by

(**) T(K) = T U { (s, (ho K)j (s) ) : s e and j = 1 or 2 },

72

where (s, (h o K)^(S) ) denotes the element (s,m) of Seq that

extends s, where m = (h o K)J(S). Using this notation, we

have T = U T(K) . T

KCM

The following lemma due to Arsenin [_2^ and Kunugui | 1 lj

is also needed in the proof of Theorem 5.5.

Lemma 5.4. Let X and Y be Polish spaces and G a Borel

subset of X x Y such that each vertical section is a compact

subset of Y. Then the x-projection of G is Borel in X.

Theorem 5.5. Let X and Y be Polish spaces, A a countably

generated sub-a-field of B(X), F : X P(Y) an A-measurable

function such that Gr(F) e A x 8(Y), and, for all x, F(x) is a

dense-in-itself G^ subset of Y. Then F admits an A-measurable

parametrization.

Proof of Theorem 5.5

As in |_23_j, Theorem 4.1 and |_15_J, Theorem 1.1, the

underlying structure of this proof is the effective procedure

[_12, page 418] for selecting a point from a nonempty G^ set.

Let d be a metric on Y such that the diameter of Y satisfies

6(Y) <1. We can find a system (see Theorem 4.1, referenced

above) { V(s) : s e Seq } of nonempty subsets of Y such that

73

(a) V(0) = Y,

(b) 6(V(s) ) < 2~k for s e Nk,

(c) { V(s,m) : m e N } is an open basis of V(s) for

s e Seq, and

(d) CI ( V(s,m) ) c V(s) for s e Seq and m e N.

Let G = Gr(F). If we consider Y to be a subset of the Hilbert

cube H, then Y is a G. subset of H since Y supports a complete 0

metric. So, G e A x g(H) and each x-section of G is a Gg

subset of H. By Theorem 2.24 there is a collection of sets

{ Gn e A x 8(H) } such that each x-section of Gn is open in

H and G = H Gn. Set G° = X x H, and let Q = Q(A) be the par-

tition of X induced by the collection A. Let 7"n be the set

of regular binary trees in Seq of rank n and T = U the col-

lection of all regular, finite binary trees in Seq. We show

there is a collection { B(T) : T e T } of subsets of X such

that if T is a regular binary tree of rank n, then

(i) B(T) = U B(S), the union taken over all regular

binary trees S of rank (n + 1) that contain T,

(ii) If S is a binary tree of rank n distinct from T,

then B(T) fl B(S) = 0,

(iii) If T possesses distinct endpoints s and t such that

cl ( V(s) ) n cl ( V(t) ) + 0, then B(T) = 0,

74

(iv) If s is an endpoint of T and x belongs to B(T),

then G fl V(s) j1 0 and cl ( V(s) ) c (Gn) , X x

/V

(v) B(T) e A(Q), the a-field of Q-saturated Borel sets,

and

(vi) B( { 0 } ) = X.

We define B ( { 0 } ) = X and suppose that B(T) has been

defined for all binary trees of rank k or less in a manner

consistent with (i) - (vi). Let T be a binary tree of rank

2

k and s an endpoint of T. Let h : W -> W - A be the map de-

fined in (*), and let i e W. Suppose (m,p) = h(i). Define

(1) Z(s,i) = { x e B(T) : G fl V(s,m) * 0 + G D V(s,p), X X

cl ( v(s,m) ) fl cl ( v(s,p) ) = 0, and

cl ( v(s,m) ) U cl ( v(s,p) ) c (Gk+1) }. X

We claim the set Z(s,i) has the following properties:

(2) Z(s,i) = B (T) 0 F~ ( V (s , m) ) fl F" ( V(s,p) )

n ( x - nx ( (X x H - Gk + 1 )

n ( X X (cl ( v(s,m) ) U cl ( V(s,p) ) ) ) ) )

if cl ( V(s,m) ) 0 cl ( V(s,p) ) = 0,

(3) Z (s, i) = 0 if cl ( V(s,m) ) fl cl ( V(s,p) ) f 0,

(4) Z(s,i) e 8(X), and

(5) Z(s,i) is Q-saturated.

75

Property (2) follows from the fact that

(6) { x e X : cl ( V(s ,m) ) U cl ( V(s,p) ) c (Gk+1) }

X

k + 1

= { x e X :x£ n1( ( X X H - G ) H ( X x (cl (V(s,m) )

U cl ( V(s,p) ) ) ) ) } .

Property (3) is a trivial consequence of (1). To see that

Z(s,i) is a Borel set, we observe that the set in (6) is the

projection of a Borel set each x-section of which is compact,

and we apply Lemma 5.4. This proves (4). Property (5) fol-

lows from (1) since both G and G are invariant subsets of

X x Y.

We now show that

(7) B(t) =UZ(s,i). l

Suppose x e B(T). By (iv), G H V(s) f 0 and, since G is X X

dense-in-itself, we can choose distinct points xq and x^ in

G H V(s). Let j e { 0,1 }. There are open sets W. of Y x J k + 1

such that W. c V(s), x. e W. c cl(W.) c (G ) , and J - J J ~ J - x

cl(W ) fl cl(W^) = 0. By (c) , we can find natural numbers m

and p such that xq e V(s,m) c W and x^ e V(s,p) c W^. So

x e Z(s,i), where i = h "'"(m.p), and this proves B(T) cUZ(s,i). i

The reverse inclusion is obvious.

76

It is, now, evident that there is a pairwise disjoint

family { D(s,i) } of saturated Borel subsets of X such that

D(s,i) c Z(s,i) for all i and

(8) B(T) = U D(s,i). i

We have shown, thus far, that for a fixed endpoint s of

T, equation (8) holds; that is, for each s e E^ there is an

i e M such that x e D(s,i). We express this by

(9) B(T) = sn E T .UND(s,i).

This can be rewritten as

(10) B(T) = U E n D( s, K(S) ). KE/V T seE

T

We define B(T(K)) = H D( s, K(S) ), where T(K) is defined seEm

by (**) . From (10), B(T) = (J B(T(K)), and (i) is satisfied, ET

k eW

If K ^ K", there is an s E E^ such that K(S) F K^(S). Since

{ D(s,i) : i e W } is a pairwise disjoint family,

D(S,K(S)) F| D(S,K'(S)) = 0. This shows that for any pair,

T(K) and T(K^), of distinct binary trees of rank (k + 1) that

extend T, B(T(K)) fl B(T(K')) = 0. This fact, coupled with

the inductive hypothesis (ii), proves (ii).

77

To show (iii), suppose (s,m) and (t,p) are distinct

endpoints of T(K) such that cl ( V(s,m) ) f"| cl ( V(t,p) ) f 0,

where s and t are endpoints of T. Suppose s # t. Since

B(T(K) ) c B(T), if cl ( V(s) ) D cl ( V(t) ) + 0, then

B(T(ic) ) = 0 by the inductive hypothesis (iii). If, on the

other hand, cl (V(s) ) fl cl (V(t) ) = 0, then

cl (V(s,m) ) 0 cl ( V(t,p) ) ccl ( V(s) ) H cl ( V(t) ) = 0,

contrary to our assumption. Suppose, then, s = t. By (3),

we have that B(T(K) ) c D ( s,h "'"(m.p) ) c Z ( s,h ^(m,p) ) = 0

This argument shows that (iii) holds.

Suppose j = 1 or 2, (s,(ho K)^ (s) ) is an endpoint of

T(K), and x e B(T(K) ). Then

x e B(T(K) ) c D(S,K(S) ) c Z(S,K(S) ). By (1),

G fl V ( s, (h o ic) . (s) ) 5s 0 and x J k+1

cl ( V(s,(h o K).(s) ) c (G ) . This demonstrates property J x

(iv)

Since E , is a finite set, property (v) follows from the

fact D(s,i) is a saturated Borel set for every s and i. This

completes the inductive definition of the sets { B(T) : T e 7 }

We next construct a family { M(t) : t e 2* } of subsets

of X x Y such that for t e 2n

78

(I) M(t) is an invariant Borel set,

(II) Each x-section of M(t) is a nonempty closed subset

of Y with diameter less than 2 n and is contained

in (Gn) , x

(III) If i = 0 or 1, M(t,i) c M(t), and

(IV) If t" e 2n and t' f t, then M(t) fl M(t') = 0.

Set M(0) = X x Y and suppose M(t) has been defined for

r k'

t e 2* of length k or less. Fix T e T, . Since T has 2 end-

points , we can index E^ by

E,p = { s(t,T) : t £ 2 " }. Define, for j e { 0,1 },

M(t,j) = U U E ( B(T(k) ) x c l ( V(s(t,T),(h o k) TET K-EW T J

k

(s(t,T) ) ) ) ).

Condition (I) is satisfied because the union is countable

and, by statement (v), B(T(k) ) is an invariant Borel set.

To prove (II), we fix x £ X. By (i), (ii), and (vi) ,

E there is a unique T in T and k in W such that x £ B(T(k) ) .

K. Therefore, for j e { 0,1 },

(11) M(t,j)x = cl ( V(s(t,T), ( h o k) (s(t,T) ) ) ),

and, by (11) and (b), M(t,j) is a nonempty closed subset of

Y of diameter less than 2 From (iv) we see that

M(t,j)x c (Gk+1)x.

79

Statement (III) is seen to be true from (d) and (i).

k+1

Suppose (t,j) and (t',j") are distinct elements of 2

If t ^ t', M(t,j) n M(t',j') c M(t) fl M(t') = 0 by the induc-

tive assumptions (III) and (IV). Suppose t = t". Without

loss of generality we can assume j = 0 and j' = 1. To prove

(IV) it suffices to show M(t,0) f l M(t,l) = 0 for all x. X X

So, fix x e X, and let T and k be the unique elements such

that x e B(T(k) ). Let s = s(t,T). Then, by (11) and (iii)

we see that M(t,0) 0 M(t,l)

X X = c l ( V(s, (h o k ) 1 ( s ) ) ) n c l ( V(s, (h o k ) 2 ( s ) ) )

= 0.

This proves (IV) and concludes construction of the family

{ M(t) }.

For n e W define M = 0 U M(t). M is an invariant Borel o11

n te2

subset of X x Y because of (I), and each x-section of M is a

Cantor set by (II), (III), and (IV). By Theorem 5.2, G ad-

mits an invariant Borel parametrization. This concludes the

proof of Theorem 5.5.

80

Theorem 5.6. Let X be a Polish space and Q a measurable

partition of X by equivalence classes that are dense-in-

themselves Gr sets. Then 0 admits a Borel parametrization. 0

Proof of Theorem 5.6

Define F : X -> P(X) by F(x) = sat(x). F(x) is a dense-

in-itself Gr for each x. Let A be the a-field of saturated 0

Borel sets. If U is open in X, F~(U) = sat(U) e A , proving

that F is A-measurable.

Let R be the equivalence relation induced by Q. By

Theorem 2.10, Gr(F) = R is Borel in X x X. Since Gr(F) is

invariant, Gr(F) e A x 8(X). Thus, by Theorem 5.5, F admits

an A-measurable parametrization; i.e., R admits an invariant

Borel parametrization. From Theorem 5.3, Q admits a Borel

parametrization, concluding the proof of the theorem.

CHAPTER VI

UNSOLVED PROBLEMS

In Chapter IV, I proved the existence of infinitely

many disjoint Borel cross sections for certain measurable

partitions of Polish spaces, extending the existence theorem

of a single cross section by Srivastava. In extending

Srivastava's theorem, I have raised several new questions.

1. Can his theorem be extended even further, to include

(a) uncountably many disjoint Borel cross sections,

(b) continuumly many disjoint Borel cross sections,

or (c) a Borel parametrization? I have no counter-

example to any of these conjectures.

2. Can his theorem be generalized further to show the

existence of infinitely many disjoint invariant

Borel selectors for an invariant multifunction?

Specifically, I make the following conjecture. Let

X and Y be Polish spaces, A a countably generated

sub-a-field of the Borel sets of X, and F a G.-valued o

81

82

A-measurable multifunction from X to Y whose graph

is in A 0 B(X). Then F admits an infinite family

of pairwise disjoint A-measurable selectors.

Several new directions are opened up by Srivastava's

theorem. Two of them are listed below.

3. Can Srivastava's theorem be generalized to non-

separable metric spaces?

4. Conjecture: Let X and Y be Polish spaces, A a

countably generated sub-o-field of B ( X ) , and F an

A-measurable multifunction from X to Y such that

Gr(F) e A 0 B(X) and such that there is a fixed

metric on Y with respect to which F(x) is complete

for all x. Then F admits an A-measurable Borel

cross section of class 1 whose image is a trans-

versal .

This conjecture is a natural generalization of Theorem 2.17.

I conclude this chapter with the outline of an approach

to question 1. I use the notation in |_12J . Suppose X is

Polish and Q is a measurable partition of X by uncountable

( ot)

dense-in-themselves equivalence classes. For E e Q, let E

be the derived set of E of order a, and let E^W*^ = PIE^01^ be

83

the dense-in-itself kernel of the set E. Let

( a X = U { E : E e Q } for a < 10. . Let Q be the partition a — 1 a r

of X induced by Q. Since, for E e 0, E^Wl^ = E^a^ for some a J ^

a < to , Q is a partition of X into perfect equivalence 1 w x w i

classes. A positive answer to the following two questions,

in conjunction with Theorem 5.6, would imply an affirmative

answer to question 1(a).

5. Is a Borel subset of X?

6. Is Qx a measurable partition of X^'

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