A SURVEY OF MEASURABLE SELECTION THEOREMS S. M. ...

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A SURVEY OF
MEASURABLE SELECTION THEOREMS
S.M.SRIVASTAVA
INDIAN STATISTICAL INSTITUTE,KOLKATA
Lectures delivered at University of Dortmund,Germany
April 14,16,2010
2
1.Introduction.
Measurable selection theorems are useful as well as interesting in
their own rights.They have been used widely,so much so that many
important selection theorems have been discovered by their users.The-
oretically they form a very important topic in descriptive set theory.
It won’t be much of an exaggeration to say that they form the heart
of classical descriptive set theory.In these lectures we shall give an
overview of important measurable selection theorems.
Let X and Y be arbitrary sets.A multifunction F:X ￿→Y is a
function with domain X and values non-empty subsets of Y.
The set
{(x,y) ∈ X ×Y:y ∈ F(x)}
is called the graph of F.
For any set A ⊂ Y,we define

A
(F) = {x ∈ X:∃y ∈ F(x)(y ∈ A)}
and

A
(F) = {x ∈ X:∀y ∈ F(x)(y ∈ A)}.
So,

A
(F) = {x ∈ X:F(x) ∩A ￿= ∅}
and

A
(F) = {x ∈ X:F(x) ⊂ A ￿= ∅}.
Now let A be a family of subsets of X and Y a metrizable space.We
say that F is A-measurable if ∃
U
(F) ∈ A for every open set U in Y;
F is strongly A-measurable if ∃
C
(F) ∈ A for every closed set C in
Y.Thus,
F is A-measurable if for every open set U in Y,
{x ∈ X:F(x) ∩U ￿= ∅} ∈ A
;it is strongly A-measurable if for every closed set C in Y,the set
{x ∈ X:F(x) ∩C ￿= ∅} ∈ A.
In particular,a point map f:X → Y is called A-measurable if
f
−1
(U) ∈ A for every open set U in Y.
A function s:X →Y is called a selector for F if s(x) ∈ F(x) for
every x ∈ X.
By axiom of choice,a multifunction always admit a selector.
But when does a measurable multifunction admit a measurable se-
lector?This is a fundamental question and it crops up at several places.
In this series of talks we shall give a survey of major selection theorems.
3
In applications,usually A is a σ-algebra.However,there are inter-
esting cases where it is not so.
If X is a topolgical space and A is the family of all open sets,A-
measurable multifunctions are called lower semi-continuous.So,F
is lower semi-continuous if and only if for every open set U in Y,
{x ∈ X:F(x) ∩U ￿= ∅}
is open in X.
Similarly,F is called upper semi-continuous if for every closed
set C in Y,
{x ∈ X:F(x) ∩C ￿= ∅}
is closed in X.
Note that F is upper semi-continuous if and only if for every open
set U in Y,∀
U
(F) = {x ∈ X:F(x) ⊂ U} is open in X.
Another interesting case is obtained as follows.Let X be a metrizable
space.A countable union of closed sets in X is called an F
σ
-set in X
and a countable intersection of open sets in X is called a G
δ
.It is a
standard fact that every open (closed) set in a metrizable space X is a
F
σ
(G
δ
) set in X.Thus every open and every closed set in a metrizable
space is simultaneously F
σ
and G
δ
.It is also clear that the set Δ
0
2
of all
sets which are simultaneously F
σ
and G
δ
is an algebra on X.Also note
that set of all F
σ
-sets coincides with the family of countable unions
of sets in Δ
0
2
.It is clear that all lower and all upper semi-continuous
multifunctions are F
σ
-measurable.If X and Y are metrizable spaces,
a function f:X →Y is called a class 1 function if for every open set
U in X,f
−1
(U) is an F
σ
set.
2.Preliminaries.
A second countable,completely metrizable space is called a Polish
space.This is the category of topological spaces on which deep and
useful descriptive set theory takes place.
An algebra L on a set X is a non-empty family of subsets of X
which is closed under complementations and finite unions.Note that
if A ∈ L,X = A∪(X\A) ∈ L.So,X and ∅ belong to all algebras on
X.
If an algebra is also closed under countable unions,it called a σ-
algebra.
A measurable space is a pair (X,A) with X a non-empty set and
A a σ-algebra on X.
Lemma 2.1.Let X be a non-empty set,Y a metrizable space and
A be a family of subsets of X which is closed under countable unions.
4
Then every strongly A-measurable multifunction F:X ￿→ Y is A-
measurable.
Proof.Let U be an open set in Y.As Y is metrizable,U = ∪
n
C
n
,
C
n
’s closed in Y.
{x ∈ X:F(x) ∩U ￿= ∅} = ∪
n
{x ∈ X:F(x) ∩C
n
￿= ∅}.
This shows that

U
(F) = ∪
n

C
n
(F).
The result now follows.￿
A measure µ on a measurable space (X,A) is a map µ:A →[0,∞]
such that every sequence {A
n
} of pairwise disjoint sets in A,
µ(∪
n
A
n
) =
￿
n
µ(A
n
).
The measure µ is called finite if µ(X) < ∞;σ-finite if there is a
sequence {A
n
} of sets in A such that X = ∪
n
A
n
and for each n,
µ(A
n
) < ∞.It is called a probability if µ(X) = 1.A probability
space (measure space) is a triple (X,A,µ) such that Ais a σ-algebra
on X and µ a probability (resp.measure) on X.
A measure space (X,A,µ) is called a complete measure space if
whenever µ(N) = 0 (such sets are called µ-null) and M ⊂ N,M ∈ A.
For a metric space X,the smallest σ-algebra containing all open
sets is called the Borel σ-algebra of X,which we shall denote by B
X
.
Unless otherwise stated,a metric space will always be equipped with
its Borel σ-algebra.Sets in B
X
are called Borel sets in X.
Further,if X and Y are metric spaces,a map f:X → Y is called
Borel measurable or simply Borel if
f
−1
(U) ∈ B
X
for every open set U in Y.It follows that
f
−1
(B) ∈ B
X
whenever B is Borel in Y.
A measure µ on B
X
,X metrizable,is called continuous if µ({x}) =
0 for every x ∈ X.
A Borel subset of a Polish space is called a standard Borel space.
Standard Borel spaces satisfy many regularity properties:
Theorem 2.2.Every uncountable standard Borel space is of cardi-
nanility c,the continuum.In fact,they contain homeomorphs of the
Cantor set.
5
Theorem 2.3.If B is a Borel set in a Polish space X and if µ is
a finite measure on X,then for every ￿ > 0,there is a compact K and
an open U in X such that K ⊂ B ⊂ U and µ(U\K) < ￿.
Theorem 2.4.Every standard Borel space is a continuous image of
N
N
.
Sketch of the proof of 2.4.One first shows that every Polish
space is a continuous image of N
N
.Now fix a Polish space X.We
show that the family of subsets of X which is a continuous image of N
N
contains all open sets and is closed under countable intersections and
countable disjoint unions.Finally,one shows that B
X
is the smallest
family of subsets of X that contains all open sets and is closed under
countable intersections and countable disjoint unions.￿
Theorem 2.5.(Borel isomorphism theorem) If X and Y are
two uncountable standard Borel spaces,then there is a bijection f:
X →Y such that both f and f
−1
are Borel.
Borel isomorphism theorem is an important result because it allows
one to choose a standard Borel space at its own convenience.Aparticu-
larly simple proof of it was obtained by B.V.Rao and S.M.Srivastava.
Theorem 2.6.(Measure isomorphism theorem) Let X be an
uncountable standard Borel space and µ a continuous probability on X.
Then there is a Borel isomorphism f:X → [0,1] such that for every
Borel set B in [0,1],λ(B) = µ(f
−1
(B)),where λ denotes the Lebesgue
measure on [0,1].
A metrizable space Y is called an analytic space if there is a stan-
dard Borel space X and a continuous surjection f:X →Y.
Analytic spaces play a fundamental role in measurability.Clearly
every standard Borel space is analytic and analytic spaces are separa-
ble.
It is also known that
every uncountable Polish space contains an analytic non-Borel set.
Theorem 2.7.Let X be a Polish space and A ⊂ X.The following
conditions are equivalent:
(1) A is analytic.
(2) A is a continuous image of N
N
.
(3) There is a standard Borel space Y and a Borel surjection f:
Y →A.
6
(4) There is a standard Borel space Z such that A is the projection
of a Borel set B in X ×Z.So,for every x ∈ X,
x ∈ A ⇔∃z ∈ Z((x,z) ∈ B).
Proof.Since every standard Borel space is a continuous image of
N
N
(2.4),(2) follows from (1) by the definition of analytic sets.Since
the graph of a Borel map is Borel,(3) implies (4).Since projection is
a continuous map,(4) implies (1).￿
Let Z be any set.Asystemof sets in Z is a family {A(n
0
,...,n
k−1
)}
of subsets of Z indexed by the set of all finite sequences (n
0
,...,n
k−1
)
(including the empty sequence e) of natural numbers.The set

α∈N
N ∩
k∈N
A(α|k),
where N denotes the set of all natural numbers and for α ∈ N
N
,α|k =
(α(0),...,α(k−1)),k ∈ N,the restriction of α to k,is called the result
of Souslin operation on {A(n
0
,...,n
k−1
)}.
Theorem 2.8.Let Z be a Polish space and Y ⊂ Z.The following
conditions are equivalent:
(1) Y is analytic.
(2) Y is the result of Souslin operation on a system of closed sets
in Z.
(3) Y is the result of Souslin operation on a system of Borel sets in
Z.
Analytic spaces enjoy many regularity properties.
Theorem 2.9.Let X be a Polish space and Y ⊂ X analytic.
(1) If Y is uncountable,it contains a homeomorph of the Cantor
set.In particular,every uncountable analytic space is of cardi-
nality c.
(2) Y is measurable with respect every σ-finite measure on X.In
other words,every analytic space is universally measurable.
(3) Y has the propery of Baire,i.e.,there is an open set U and
a set N of first category in X such that Y = UΔN,where Δ
denotes the symmetric difference.
The following result is truly a fine and a very useful one.Descriptive
set theory got started with it.It is due to Souslin.
Theorem 2.10.(Souslin) Let Z be a Polish space and A and B two
disjoint analytic subsets of Z.Then there is a Borel set C in Z such
7
that
A ⊂ C & B ∩C = ∅.
In particular,a subset Y of Z is Borel if and only if both Y and Z\Y
are analytic.
The following generalization of this theorem plays key role in selec-
tion theory.
Theorem 2.11.Let Z be a Polish space and {A
n
} a sequence of
analytic sets in X such that ∩
n
A
n
= ∅.Then there exist Borel sets
B
n
⊃ A
n
such that ∩
n
B
n
= ∅.
A particularly elegant proof of 2.11 was obtained by Mokobodzki.
Theorem 2.12.Let Z be a Polish space and {A
n
} a sequence of
pairwise disjoint analytic sets in Z.Then there is a sequence {C
n
} of
pairwise disjoint Borel sets such that ∀n(C
n
⊃ A
n
).
Proof of 2.12.Using Souslin’s theorem (2.10),for each n,get a
Borel set C
n
such that C
n
⊃ A
n
and C
n
∩ ∪
m￿=n
A
m
= ∅.Now take
B
n
= C
n
\∪
m￿=n
C
m
.
￿
Using 2.4 and 2.12,one proves the following important result.
Theorem 2.13 Let X be a standard Borel space,Y a Polish space
and f:X → Y a Borel map which is one-to-one.Then f(X) is
standard Borel.It follows that f:X →f(X) is a Borel isomorphism.
Theorem 2.14.Let X,Y be Polish spaces,A ⊂ X and f:A →Y.
The following conditions are equivalent:
(1) A is Borel and f is Borel measurable.
(2) The graph of f is Borel in X ×Y.
We close this section by stating a corollary of 2.11 in the form that
we shall use.Let X be a Polish space.A subset C of X is called
coanalytic if X\C is analytic.The following simple result is fairly
useful.
Proposition 2.15.Let X be Polish and C ⊂ X.The following are
equivalent:
(1) C is coanalytic.
8
(2) There is a standard Borel space Z and a Borel B ⊂ X×Z such
that for every x ∈ X,
x ∈ C ⇔∀z ∈ Z((x,z) ∈ B).
2.15 is not at all hard to prove.
Theorem 2.16.Let X be a standard Borel space and {C
n
} a
sequence of coanalytic subsets with union X.Then there is a se-
quence {B
n
} of pairwise disjoint Borel sets such that ∀n(B
n
⊂ C
n
)
and ∪
n
B
n
= X.
Proof.Let Y be a Polish space containing X.Set A
n
= X\C
n
.
Then {A
n
} is a sequence of analytic sets with ∩
n
A
n
= ∅.By 2.11,get
Borel sets D
n
such that X ⊃ D
n
⊃ A
n
and ∩
n
D
n
= ∅.Set E
n
= X\D
n
.
Then E
n
is Borel,E
n
⊂ C
n
and ∪
n
E
n
= X.Now take
B
n
= E
n
\∪
m<n
E
m
.
￿
Here is a beautiful consequence of 2.15 that we shall use in selection
theory.
Theorem 2.17.(Dellacherie) Let X,Y be Polish spaces and B ⊂
X ×Y Borel with sections B
x
open in Y.Then we can write
B = ∪
n
(B
n
×U
n
),
B
n
’s Borel and U
n
’s open.
3.Kuratowski—Ryll Nardzewski Selection Theorem.
Kuratowski and Ryll Nardzewski proved a very general selection the-
orem.It encompasses a lot of major selection theorems.Its proof is
simple at the same time beautiful.
Let X be a non-empty set and L an algebra of sets on X.Let L
σ
denote the smallest family of subsets of X containing L and closed
under countable unions.The following basic facts about L
σ
will be
used without proof.
Proposition 3.1.If {A
m
} is a sequence in L
σ
,then there is a
sequence {B
m
} of pairwise disjoint sets in L
σ
such that ∀m(B
m
⊂ A
m
)
and ∪
m
B
m
= ∪
m
A
m
.
Sketch of the proof.Write A
m
= ∪
n
A
mn
,A
mn
∈ L.Enumerate
the double sequence {A
mn
} in a single sequence,say {D
k
}.Set C
k
=
9
D
k
\∪
l<k
D
l
.Now take
B
n
= ∪{C
k
:C
k
⊂ A
n
∧ ∀m< nC
k
￿⊂ A
m
}.
￿
The proof of the following proposition is quite routine.
Proposition 3.2.Let Y be a Polish space and {s
n
} a sequence of
L
σ
-measurable functions from X to Y.If {s
n
} converges uniformly to
s:X →Y,then s is L
σ
-measurable.
Theorem 3.3.(Kuratowski and Ryll Nardzewski) Let X and L
σ
be
as above and Y a Polish space.Then every closed valued L
σ
-measurable
multifunction F:X ￿→Y admits a L
σ
-measurable selector.
Proof.Fix a complete metric d < 1 on Y inducing its topology and
a countable dense set {r
m
} in Y.
We claim that there is a sequence {s
n
} of L
σ
-measurable functions
from X to Y satisfying the following conditions for every n and every
x ∈ X:
(1) d(s
n
(x),F(x)) < 2
−n
.
(2) d(s
n
(x),s
n+1
(x)) < 2
−n
.
Assuming this,we complete the proof first.Since (Y,d) is a complete
metric space and {s
n
} uniformly Cauchy,{s
n
} is uniformly convergent,
say to s.By 3.2,s is L
σ
-measurable.Since F(x) is closed,by (1),
s(x) ∈ F(x).
We construct the sequence {s
n
} satisfying (1) and (2) by induction
on n.Set s
0
≡ r
0
.Note that (1) is satisfied for n = 0.
Suppose s
n
is L
σ
-measurable and satisfies (1).For each m,define
A
m
= {x ∈ X:d(s
n
(x),r
m
) < 2
−n
}∩{x ∈ X:F(x)∩B(r
m
,2
−(n+1)
) ￿= ∅}.
Since s
n
and F are L
σ
-measurable,each A
m
∈ L
σ
:A
m
is the intersec-
tion of following two sets:
A = {x ∈ X:d(s
n
(x),r
m
) < 2
−n
}
and
B = {x ∈ X:F(x) ∩B(r
m
,2
−(n+1)
) ￿= ∅}.
Since {r
m
} is dense in Y,∪
m
A
m
= X.
Now get paiwise disjoint B
m
⊂ A
m
in L
σ
as in 3.1.In particular,

m
B
m
= X.
Define s
n+1
:X →Y by
s
n+1
(x) = r
m
,if x ∈ B
m
.
10
Thus,for each m,s
−1
n+1
(r
m
) = B
m
.This implies that for every U ⊂ Y,
s
−1
n+1
(U) = ∪
{m:r
m
∈U}
s
−1
n+1
(r
m
) ∈ L
σ
.
In particular,s
n+1
is L
σ
-measurable.￿
There are many special cases of this selection theorem that occur in
applications.
Theorem 3.4.Let (X,A) be a measurable space and Y a Polish
space.Then every closed-valued A-measurable multifunction F:X ￿→
Y admits an A-measurable selector.
(Take L = A.)
Theorem 3.5.Let X be a metrizable space,Y a Polish space and
F:X ￿→ Y a closed-valued lower semi-continuous or upper semi-
continuous multifunction.Then F admits a class 1 selector.
Proof.Let L be the family of all subsets of X which are simultane-
ously F
σ
and G
δ
.Then L is an algebra on X.If F is either lower or
upper semi-continuous,F is L
σ
-measurable.Hence,the result follows
from Kuratowski and Ryll Nardzewski selection theorem.￿
Let Π be a partition of a set X.A subset S of X is called a cross-
section of Π if S ∩ C is a singleton for every C ∈ Π.For any A ⊂ X,
set
A

= ∪{C ∈ Π:C ∩A ￿= ∅}.
Thus,A

is the smallest Π-invariant set containing A.
Call A Π-invariant if A = A

.
Now suppose X is Polish.
We call Π Borel measurable if U

is Borel for every open U ∈ X.
Similarly Π is lower semi-continuous if U

is open for every open
in X;
upper semi-continuous if C

is closed for every closed C in X.
Theorem3.6.(Effros) Every Borel measurable partition of a Polish
space into closed sets admits a Borel cross-section.
Proof.Take A to be the set of all Π-invariant Borel sets in X.
Consider the multifunction F:X ￿→X defined by
F(x) = {x}

,x ∈ X.
Then F is closed valued and A-measurable.
11
By Kuratowski and Ryll Nardzewski selection theorem,let s:X →
X be an A-measurable selector for F.Note that s|C is constant on
every C ∈ Π and s(x) ∈ C whenever x ∈ C.
It is easy to see that
S = {x ∈ X:x = s(x)}
is a Borel cross-section for Π.￿
Theorem 3.7.Every lower and every upper semi-continuous parti-
tion of a Polish space into closed sets admits a G
δ
cross-section.
Proof.The proof is similar to 3.6 by taking L to be the set of all
invariant sets which are simultaneously F
σ
and G
δ
.￿
Srivastava and Sreela Gangopadhyay (n`ee Bhattacharya) used a beau-
tiful idea of changing topology and got a serious generalization of Ku-
ratowski and Ryll Nardzewski selection theorem.
Proposition 3.8.Let Y be a Polish space and {C
k
} a sequence
of closed sets in Y.Then there is a finer Polish topology on Y that
generate the same Borel σ-algebra and in which each C
k
is open.
Proof.Let us define f:Y →Y ×2
N
by
f(y) = (y,I
C
0
(y),I
C
1
(y),...),y ∈ Y,
where for any C ⊂ Y,I
C
denotes the indicator function of C.
Then f is one-to-one and Borel.Let Z = f(Y ).
By 2.13,f:Y →Z is a Borel isomorphism.
Since each C
k
is closed,Z is a G
δ
in Y × 2
N
and so,Polish.Also
note that
C
k
= {y ∈ Y:∃α ∈ 2
N
(f(y) = (y,α)) ∧ α(k) = 1}.
Now take the smallest topology on Y making f continuous.This sat-
isfies all the conditions.￿
Motivated by some problems in descriptive set theory,we generalized
the selection theorem of Kuratowski and Ryll Nardzewski as follows.
Theorem 3.9.(Bhattacharya and Srivastava) Let L be an algebra
on a set X,Y a Polish space and F:X ￿→Y a closed valued strongly
L
σ
-measurable multifunction.Supose Z is a second countable space and
g:Y →Z a class 1 function.Then F admits a L
σ
-measurable selector
s:X →Z such that g ◦ s is also L
σ
-measurable.
12
Proof.Let {U
n
} be a countable base for the topology of Z.For
each n,write
g
−1
(U
n
) = ∪
m
C
nm
,
C
nm
’s closed.
Let {C
k
} be an enumeration of the double sequence {C
nm
} into a
single sequence.Take a finer topology on Y as in 3.8.
Now note that still each F(x) is closed,F is L
σ
-measurable and g
is continuous with respect to the new topology.Take a L
σ
-measurable
selector s:X →Y for F with respect to the new topology on Y.
Since g is continuous with respect to the new topology,g ◦ s is L
σ
-
measurable and our proof is complete.￿
We got another generalization of the selection theoremof Kuratowski
and Ryll Nardzewski which we have recently used to show the existence
of a continuous stochastic relation with support contained in a given
set.
Theorem 3.10.(Srivastava) Let X,Y,L
σ
and F:X ￿→ Y be as
in 3.3.Then there is a map f:X ×N
N
→Y such that
(1) for every x ∈ X,α → f(x,α) is a continuous map from N
N
ontoF(x),and
(2) for each α ∈ N
N
,x →f(x,α) is L
σ
-measurable.
4.Selection Theorems for G
δ
-valued multifunctions.
Following a problem raised for some representations of C

-algebras,
a significant generalization of 3.6 was obtained by Srivastava.
Before we state this,let us recall the Vitali partition of the real line
R.
Example.Call two real numbers x and y equivalent if x −y ∈ Q.
Then every member of the induced partition is a translate of Q.Since
they are countable,they are F
σ
.Since they are homeomorphic to Q,
by Baire category theorem,they are not G
δ
in R.It is well-known
that this partition of R does not admit even a Lebesgue measurable
cross-section.It is also pertinent to note that for any open set U in R,
U

= ∪
r∈Q
[U +r]
is open in R.Thus,the Vitali partition is lower semi-continuous.
Using some deep results in descriptive set theory,particularly a sep-
aration theorem of Saint-Raymond for analytic sets with σ-compact
sections,we obtained the following result.The proof of this result is
hard and therefore omitted.
13
Theorem 4.1.(Srivastava) Every Borel measurable partition of a
Polish space into G
δ
-sets admits a Borel cross-section.
This result was obtained by proving the following selection theorem
for G
δ
-valued multifunctions.
Theorem 4.2.(Srivastava) Let X be an analytic space,Y Polish
and F:X ￿→Y a G
δ
-valued Borel measurable multifunction with graph
Borel in X ×Y.Then F admits a Borel selection.
The following result on lower-semicontinuous partitions of Polish
spaces into G
δ
sets is worth mentioning.
Theorem 4.3.(D.Miller) Every lower and every upper semi-
continuous partition of a Polish space into G
δ
-sets admits a G
δ
cross-
section.
5.Von Neumann Selection Theorem.
It is very well-known that there is a Borel set B ⊂ [0,1] ×[0,1] with
all x-sections non-empty,yet there is no Borel map s:[0,1] → [0,1]
whose graph is contained in B.There are several beautiful and useful
(all quite deep) results showing the existence of such s provided the
x-sections satisfy some conditions.In the last section we shall discuss
such results.
Question arises:how good a selector one can have in general?A
very fine and quite useful result was proved by Von Neumann.
Theorem5.1.(Von Neumann) Let X be an analytic space,Y Polish
and B ⊂ X × Y analytic with each x-section B
x
non-empty.Then
there is a map s:X → Y whose graph is contained in B and which
is measurable with respect to the σ-algebra generated by all analytic
subsets of X.
Proof.Since B is analytic,there is a continuous function f from
N
N
onto B.Now define a multifunction F:X ￿→N
N
by
F(x) = f
−1
({x} ×B
x
),x ∈ X.
Then F is a closed valued multifunction whose graph equals
{(x,α) ∈ X ×N
N
:x = π
X
(f(α))},
and so closed in X ×N
N
.
14
Since X is analytic,in particular the graph of F is analytic.It
follows that F is measurable with respect to the σ-algebra generated
by analytic sets.
By Kuratowski—Ryll Nardzewski selection theorem,it has a selector,
say s
￿
,measurable with respect to the σ-algebra generated by analytic
sets.
Now take s = π
Y
◦ f ◦ s
￿
.Since π
Y
and f are continuous,s is
measurable with respect to the σ-algebra generated by analytic sets.￿
Remark 5.2.Since analytic sets are measurable with respect to
every σ-finite measures (2.9),it follows that s obtained in 5.1 is uni-
versally measurable.
We can generlize Von Neumann selection theorem a bit more,
Theorem 5.3.Let (X,A) be a measurable space with A closed
under Souslin operations and Y a Polish space.Suppose B ∈ A⊗B
Y
(the product σ-algebra) with each x-section B
x
￿= ∅.Then there is an
A-measurable s:X →Y whose graph is contained in B.
Proof.Get a sequence {A
n
} in A and a sequence {B
n
} in B
Y
such
that B belongs to the σ-algebra generated by the sequence {A
n
×B
n
}.
Define I:X →2
N
by
I(x) = (I
A
0
(x),I
A
1
(x),I
A
2
(x),...),x ∈ X.
Set Z = I(X).Then a set A ⊂ X is in the σ-algebra generated by
{A
n
} if and only if I(A) is Borel in Z.
It follows that C = (I ×1)(B) is Borel in Z×Y.Now get a Borel set
D in 2
N
×Y such that C = D∩ (Z ×Y ).By Von Neumann selection
theorem,there is a map s
￿

2
N(D) → Y with graph contained in D
and measurable with respect to the σ-algebra generated by analytic
sets.Now define
s(x) = s
￿
(I(x)),x ∈ X.
Since A is closed under Souslin operations,using 2.8,we show easily
that s is A-measurable.Clearly s is a selector of B.￿
6.Borel Uniformizations.
Let X,Y be Polish spaces and B ⊂ X × Y Borel.A Borel uni-
formization of B is a Borel subset G of B which is the graph of a
function,say f,and π
X
(G) = π
X
(B).
By 2.14,π
X
(G) is Borel and f is Borel measurable.Thus,if a Borel
set admits a Borel uniformization,its projection must be Borel.
15
Since there are analytic non-Borel sets,it follows that there exist
Borel sets that do not admit a Borel uniformization.
It is also known that not every Borel B with projection Borel admits
a Borel uniformization.
The study of the existence of Borel uniformizations lies in the heart
of classical descriptive set theory.In this section,we shall present
major uniformization theorems.Since they require deep results from
descriptive set theory,their proofs will essentially be omitted.
6.1.Borel sets with countable sections.
Lusin,the founder of descriptive set theory,proved a highly non-
trivial result on Borel sets with countable sections.If one is prepared
to use recursion theory,one gets a much simpler proof than the classical
proof of Lusin.
Theorem 6.1.(Lusin) Let X,Y be Polish and B ⊂ X ×Y Borel
with sections B
x
,x ∈ X,countable.Then B admits a Borel uni-
formization.
In fact,Lusin proved much more.
Theorem 6.2.(Lusin) Let X,Y be Polish and B ⊂ X ×Y Borel
with sections B
x
,x ∈ X,countable.Then B is a countable union of
Borel graphs.
6.2.Borel sets with compact sections.
The results (but not the proofs) in this section are due to Novikov.
Theorem 6.3.(Novikov) Let X,Y be Polish and B ⊂ X×Y Borel
with sections B
x
,x ∈ X,compact.Then π
X
(B) is Borel.
Proof.(Srivastava) We imbed Y into a compact metric space Z.
Since Y is a G
δ
in Z,B is Borel in X ×Z.Further,its x-sections are
closed in Z.
By Dellacherie’s theorem (2.17),there exist Borel sets B
n
⊂ X and
open sets U
n
⊂ Z such that
(X ×Z)\B = ∪
n
(B
n
×U
n
).
Now observe that
B
x
= ∅ ⇔∃n
0
,...n
k
(Z = ∪
k
i=0
U
n
i
& x ∈ ∩
k
i=0
B
n
i
).
16
So,
X\π
X
(B) = ∪
{(n
0
,...,n
k
):Z=∪
k
i=0
U
n
i
}

k
i=0
B
n
i
.
￿
Corollary 6.4.Let X,Y be Polish and B ⊂ X × Y Borel with
sections B
x
,x ∈ X,compact.Then for every open set U in Y,
{x ∈ X:B
x
∩U ￿= ∅} ∈ B
X
.
Proof.Since every open set in a metrizable space is a F
σ
,there is a
sequence {F
n
} of closed sets in Y such that U = ∪
n
F
n
.Now note that
{x ∈ X:B
x
∩U ￿= ∅} = ∪
n
π
X
(B ∩(X ×F
n
)).
Clearly,x-sections of B∩(X×F
n
) are compact.So,the result follows
from 6.3.￿
6.4 says that the closed valued multifunction x →B
x
from π
X
(B) to
Y is Borel measurable.By 6.3,π
X
(B) is Borel.So,by Kuratowski—
Ryll Nardzewski selection theorem,we get the following theorem.
Theorem 6.5.(Novikov) Let X,Y be Polish and B ⊂ X × Y
Borel with sections B
x
,x ∈ X,compact.Then B admits a Borel
uniformization.
6.3.Borel sets with σ-compact sections.
Theorem 6.6.(Kunugui) Let X,Y be Polish and B ⊂ X × Y
Borel with sections B
x
,x ∈ X,σ-compact.Then B admits a Borel
uniformization.
This is a fairly hard result to prove.We would like to deduce this
from the following very beautiful result of Saint-Raymond.
Theorem6.7.(Saint-Raymond) Let X,Y be Polish and B ⊂ X×Y
Borel with sections B
x
,x ∈ X,σ-compact.Then there is a sequence
{B
n
} of Borel sets in X×Y with compact sections such that B = ∪
n
B
n
.
Saint-Raymond’s result together with 6.3 easily implies that that
B contains a Borel set C with compact sections such that π
X
(B) =
π
X
(C).Then Kunugui’s uniformization theoremfollows fromNovikov’s
result.
6.4.Blackwell—Ryll Nardzewski selection theorem.
17
Let (X,A) be a measurable space and Y a Polish space.A sto-
chastic relation P:X ￿→ Y is a map P:X × B
Y
→ [0,1] such
that
(1) for every x ∈ X,P(x,∙) is a probability on Y,and
(2) for every Borel B in Y,x →P(x,B) is A-measurable.
Theorem 6.8.(Blackwell and Ryll Nardzewski) Let (X,A) be a
measurable space,Y Polish and B ∈ A ⊗ B
Y
.Suppose there is a
stochastic relation P:X ￿→Y such that whenever B
x
￿= ∅,P(x,B
x
) >
0.Then π
X
(B) ∈ A.Further,the multifunction x →B
x
from π
X
(B)
to Y admits an A-measurable selection.
Sketch of the proof.Using familiar good set argument (or π −λ-
theorem),one shows that for every A ∈ A ⊗ B
Y
,x → P(x,A
x
) is
A-measurable.This implies that π
X
(B) ∈ A.
Next one shows that for every ￿,δ > 0,there is a C ⊂ B in A⊗B
Y
with sections C
x
,x ∈ X,compact and of diameter < δ.Further,for
every x ∈ X,P(x,C
x
) ≥ ￿ ∙ P(x,B
x
).Using this repeatedly,one gets
a sequence {C
n
} of subsets of B in A ⊗ B
Y
with section (C
n
)
x
non-
empty,compact and with diameter(C
n
) converging to zero.This yields
the desired selection.￿
Theorem6.9.(Blackwell and Ryll Nardzewski) Let X,Y be Polish
and B ⊂ X×Y Borel.Suppose there is a stochastic relation P:X ￿→Y
such that whenever B
x
￿= ∅,P(x,B
x
) > 0.Then B admits a Borel
uniformization.
6.5.Borel sets with non-meager sections.
The following result was independently proved by Kechris and Sar-
badhikari.Kechris gave a proof using recursion theory whereas Sar-
badhikari gave a classical proof.
Theorem 6.10.(Kechris,Sarbadhikari) Let X,Y be Polish spaces
and B ⊂ X ×Y Borel.Suppose whenever B
x
￿= ∅,it is non-meager in
Y.Then B admits a Borel uniformization.