119 ^9/1 A/0. / 5 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
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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
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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