Chapter 4 Measurable Section and Selection Theorems with Applications to the Effros Borel Structure

CHAPTER 4 MEASURABLE SECTION AND SELECTION THEOREMS WITH APPLICATIONS TO THE EFFROS BOREL STRUCTURE

W e s t a t e and prove a general a b s t r a c t s e c t i o n theorem

due t o

Hoffmam-Jci?rgensen ( s e e [46],p8*) from which i t seems

t h a t all known measurable s e c t i o n and s e l e c t i o n theorems may be deduced.This theorem i s applied t o t h e E f f r o s s t r u c t u r e and t o some s e l e c t i o n the0rems.A negative r e s u l t concerning existence of measurable i n v e r s e s i s given.

t e r s e c t i o n of these s e t s contains a t most one point ..................................................

MEASURABLE SECTION A N D SELECTION THEOREMS

Proof: We define the Souslin scheme 0

79

by

for all multiindices p a I? .Next the Souslin scheme H

is

defined by induction as follows : n-1

and

H( (n))=A( (n)) \ $G( (k))

-.

1cC I

By induction it is easily seen that

3

cef

H(p)

$' =

for all mdtiindices p e p

By induction we easily show that the Souslin scheme H is monotone decreasi-ng and

P*q

H ( p ) n H(q)=0

if neither

nor q < p . This implies

fi u

S ( H ) = u.1 ( p 8 4 tH(P))

hence

.

S(H)=BE.~'

Let now

m=(ml,..)6h00

x 15 X by

mk=min in I

For each

be arbitrary.We define inductively

CXIM

k > 1

(ml,. ,mk-l ,n)

we then have

[xln

5

A( (m,,.,mk))#O

,hence

using the transitivity of cU we obtain :

3c.

G(

(m17

Since for p 4 P

0

9mk))

0

either Lx] g G ( p )

or [x]

0

G(p)=0

we con-

clude (by induction on k):

1x1 A A( (ml, and

[x]OH(p)=0

,mk))=[XI 0 H( (ml, ,mk) if

and from the assumption

p 4 m .From this it follows : ii) it even follows that Cx] fl B

MEASURABLE SECTION AND SELECTION THEOREMS

80

c o n t a i n s just or,e point.Hence

B

is a section f o r

cy

and

t h e proof i s f i n i s h e d . I n a p p l i . c a t i o n s of t h e p r e c e d i n g theorem t h e S o u s l i n scheme

w i l l o f t e n be a S o u s l i n scheme whose values a r e

A

non empty open s e t s i n a complete s e p a r a b l e m e t r i c space (X,d) such t h a t t h e f o l l o w i n g c o n d i t i o n s are s a t i f i e d :

(nl,snk))

2)

l/k

It f o l l o w s e a s i l y from t h e L i n d e b f p r o p e r t y t h a t a S o u s l i n scheme w i t h t h e s e p r o p e r t i e s always e x i s t i n a complete s e p a r a b l e m e t r i c space.

E f f r o s measurable.

I -

P r o o f : F i r s t we c o n s i d e r t h e case where

i s a P o l i s h space.Let

d*

topology.Let

S=t(x,F)bXS? I x & F i Clearly duct space

X

be a precompact m e t r i c on

d

compatible with t h e topology.We c o n s i d e r now Polish

( X , @)

h

S C_ X% X

.

f

with

the

be t h e s e t

S is c l o s e d and t h e r e f o r e P o l i s h i n t h e proX K P .Let

D

be a complete metric on

S

81

MEASURABLE SECTION AND SELECTION THEOREMS

generating the topology.By induction we now choose a Souslin scheme A

of non empty open sets in S

with the

properties:

ii) D-diam(A( (nl,.,nk)

)5 l/k

We define the equivalence relation (x,A)-(y,B) If

GgS

G=9

A=B

.

on S

Cro

by

is an open set we easily see that

is open in S ; note that this of course depend of the special definition of S .We remark that the equivalence classes are closed in S .It is now clear that all conditions for the applicability of theorem 4.1 are fulfilled If we take the Borel field

3

to be the Borel structure genera-

ted by the topo1ogy.A Borel measurable section for the equi-

valence relation cu is the graph of a choice function,which is measurable since its graph is measurable.This concludes the proof of the

,,Polish,,part of the theorem.

In the general case let

be surjective and

h:??-*X

continuous.We consider the injective mapping h,:? defined by

h,(A)=h-’(A)

.The mapping h,

-+

6

is in general

not Effros measurable but it is measurable from the Borel structure to the Effros Borel structure.To see this we only need to remark that

hF1 ( I F E N - I F/)H#0] )=CAE? I h

and the last set is analytic in X

lytic in

x

AAh(H)#0)

since

( HGN* is an open set ).If

f

h(H)

9

is ana-

is a measu-

MEASURABLE SECTION A N D SELECTION THEOKEMS

82

rable choice function on

2 we put

g(A)=h o f o h,(A)

and

n

it is clear that g is a choice function on X with the desired measurability properties.This concludes the proof.

It can be shown that in the Polish case there exist a sequence fn

that can

fn(A)

of Effros measurable choice functions such

is dense in A

for each closed set A .This

also be shown for a 6-compact metric space.We do not

know whether or not there exist always an Effros measurable choice function in the case of an analytic metrizable space; our conjecture is that this is not the case.

Proof:In the first case we consider the mapping h

h:Y -+ X

defined by

h(y)=f-’(y) .This is indeed a closed

set since the graph of f is closed.The assumption implies that this is an Effros measurable mapping and we obtain a measurable inverse by composing with an Effros measurable choice function.In the last case we obtain by a similar argument an inverse function which is measurable from the

MEASURABLE SECTION A N D SELECTION THEOREMS

83

coarsest Borel structure containing the topology and closed under the Souslin operation to the Borel structure generated by the topology.In particular this function is univer-

sially measurable. Suppose that every continuous surjection from a Polish space onto a compact metrizable space admit a Borel measuC1,C2 C X

rable inverse.Let

be two disjoint coanalytic

.

sets in a standard uncountable measurable space

(X,$))

Let @ be a compact metrizable topology on X

generating

the Borel structure

2

.Let f,g

from Polish spaces (Y,%) and (X\C2)

(Z,e) onto

if S L Y

h(s)=f(s)

now

k:X

sets

S=

Y

+

and h(s)=g(s)

and

Z onto (X,&) if S S Z .Let

-+ S be a Borel measurable inverse of h .The

B,=k-’(Y),

B2=k-’(Z) are Borel measurable in X;

they form a partition of X and we have C2S

( X\ C, )

respectively.We define the continuous surjection h

f r o m the disjoint topological sum

by

be continuous surjections

ClsB2

and

B1 .Hence every pair of disjoint coanalytic sets may

be separated by Borel sets.Now we look over the proof of n

theorem 3.9 .In the argument which shows that X,,\?= analytic we could let a(F)

is

be in an Effros measurable sub-

h

set of Xp .Then we would obtain that the natural imbedding from

%

into

A

X9 maps Effros measurable sets onto

coanalytic sets.The above separation would now show that this imbedding is measurable in contradiction with theorem

3.9

if both

z0

and

4

Xp

are standard and the further

conditions in theorem 3.9 are not fulfilled.This contradiction concludes the proof.

MEASURABLE SECTION AND SELECTION THEOREMS

84

Notes and remarks to chapter 4: The section theorem is as stated due to Hoffmann-Jmgensen and in [f6] he shows how t h e most commonly used section and selection theorems may be deduced from it. The existence of Effros measurable choice functions is due to the author.It seems however that these results was known for compact spaces even before the Effros Borel structure was invented. The negative result concerning non existence in general of a Borel measurable inverse is due to the author and

J.E.Jayne in collaboration.

It is most probable that there does not need to exist Effros measurable choice functions in the case of an analytic metrizable space;we have not been able to decide this question,it may be related to the problem under what conditions the Effros structure is standard. A very important application of the section theorem

is to find a Borel set in a Polish group which intersect

-

every sideclass of a closed subgroup in exactly one point. \

In K=[O,f

the Cantor group we have the seemingly

nice equivalence relation k co h <==+

1-N

1

defined by

k(n)#h(n)]

is finite .It is an

amusing exercise to show that this equivalence relation does not admit a universally measurable section nor does there exist a universially BP-measurable section (indeed no BP-measurable section).This may be,shown using the results of the following chapters.