Selection theorems and their applications

Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and B. Eckmann, Z0rich

263 T. Parthasarathy Indian Statistical Institute, Calcutta/India

Selection Theorems and their Applications

Springer-Verlag Berlin. Heidelberg. New York 1972

A M S S u b j e c t Classifications (1970): 49 A 35, 54 C 65, 90 D 15

I S B N 3-540-05818-4 S p r i n g e r - V e r l a g B e r l i n • H e i d e l b e r g . N e w Y o r k I S B N 0-387-05818-4 S p r i n g e r - V e r l a g N e w Y o r k • H e i d e l b e r g • B e r l i n This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin • Heidelberg 1972. Library of Congress Catalog Card Number 72-78192. Printed in Germany. Offsetdruck: Julius Beltz, Hemsbach/Bergstr.

PREFACE

This volume of lecture notes contains results on selection theorems and their applications.

Some of the material of this volume had been

given as seminar talks at Case Western Reserve University during ]969 - 1970. This volume contains nine sections each of them followed by selected references.

We hope this volume will be profitable to

specialists in game theory, dynamic programming,

control theory, mathe-

matical economics as well as to all mathematicians

interested in this

area of Mathematics. I wish to express my sincere thanks to the following Professors: Henry Hermes, Marc Jacobs, Ashok Maitra and Sam Nadler Jr., for several useful suggestions. A particular measure of gratitude is due to Mr. Arun Das who patiently and accurately prepared the final typescript of this volume. My wife Ranjani proof-read the manuscript.

To her I owe my heart-felt

thanks. Finally I wish to express my gratitude to the Indian Statistical Institute for providing the excellent research facilities and to Springer - Verlag for undertaking the publication of these notes.

T. Parthasarathy November 20, 1971

Indian Statistical Institute

CONTENTS

Introduction

. . . . . . . . . . . . . . . . . . . . . . . . . .

I. C o n t i n u o u s

selections

. . . . . . . . . . . . . . . . . . . .

2. C o n t i n u o u s

selections

on m e t r i c

continuum

. . . . . . . . . .

I

3

14

3. C o n t i n u o u s s e l e c t i o n s and s o l u t i o n s of g e n e r a l i s e d differential equations . . . . . . . . . . . . . . . . . . . .

24

4. M e a s u r a b l e

35

5. G e n e r a l

selections

theorems

6. T w o a p p l i c a t i o n s

7. v o n - N e u m a n n ' s

and an a p p l i c a t i o n

on s e l e c t o r s

selections

choice

theorem

8. On the u n i f o r m i z a t i o n

of sets

9.

on s e l e c t i o n

Supplementary

remarks

stochastic

games.

. . . . . . . . . . . . . . . .

of m e a s u r a b l e

measurable

to

49

. . . . . . . . . .

58

. . . . . . . . . . .

65

in t o p o l o g i c a l

theorems

spaces

.....

. . . . . . . . .

76

85

This volume

is dedicated

to my parents.

I 2~rRODU3 t'lo ~ The purpose of these notes is to prove a few selection theorems and to mention some applications of these theorems. briefly outline the contents of these notes.

We will now

These notes are divided

into nine sections. We start with a selection theorem due to Michael,

in section

one, which yields a characterlsation of paracompactness. 2 we are concerned with the following question. exists a continuous selection on subsets of some topological

Suppose there

2 )[ (= space of all nonempty closed

space X), then what can you say about X ?

In other words for what space section on 2 X

In section

X

does there exist a continuous

?

The aim of section three is to show how the existence of continuous

selection for certain set-valued ~aps leads to the

existence of classical solution of some generallsed differential equations.

In section four, we prove a selection theorem due to

Dubins and Savage and then we apply this theorem to prove the existence of optimal

stationary strategies for the two players in

zero-sum two-person stochastic games. In section five we establish the following result due to Kuratowskl

and Ryll-Nardzewskl.

metric space.

Let

F : X --> 2 Y

x " F(x) ~ G # @ ~ e S

Let

Y

be a complete separable

be a measurable map [That is,

whenever G is open in

Y

and

S

countably additive family induced by a field of subsets of Then there is a selector G is open in Y.

Y

such that

X].

f-l(G) e S whenever

We also deduce a few more selection theorems with

the help of this theorem. hypothesis,

f - X->

is the

In section six, assuming the continuum

%~ present an example (due to Orkin) of a non-analytic

subset of [0, I] which is a Blackwell space.

This example depends

on a measurable selection theorem proved in the previous section. In section seven we prove essentially the following measurable choice theorem due to von-Neumann.

Let

T = [0, I] and

arbitrary complete separable metric space. set-valued function from

t.

F

be an

is an analytic

T to X, then there is a Lebesgue measur-

able point-valued function almost all

If

X

f

: T -- X

such that

f(t) • F(t) for

We use this result while characterising extreme

points of sets of vector functions. Given a set E in the cartesian product and Y, a set and

I[X U

of

U

is said to uniformlse E, if the projections E

and

U

through

the set

Y

for each

x ~ ]IX E

above

consists of a single point.

x

X x y of two spaces X

onto

(~x) ,~Y)/] U

X

7[X E

coincide, and if,

of points of

Y

lying

In section eight, we are

concerned with the question of the existence of such uniformising sets. In the last section we mention further results on selection theory with some remarks.

I. CO~21NUOUS SELZCTIONS One of the most interesting and important problems in topology is the extention problem. T~o topolgical spaces given,

and

Y

together with a closed subset A of X, and we ~ u l d

know whether every continuous function to a continuous function U

X

A) into

Y.

f

from

g , A -~ Y

are llke to

can be extended

X (or at least from some open

Sometimes there are additional requirements on

which frequently take the following form ". for every must be an element of a preassigned

subset of

which we call the selection problem, the extention problem,

and presents

Y.

x ¢ X,

f,

f(x)

This new problem,

is clearly more general than a challenge even when

A

is

the null set or a one-point set (where the extention proble= is trivial). Let

X

and

Y

denote topological spaces and

family of non-empty subsets of Y. for

~

every

is a continuous

f : X--

If Y

$ : X -- 2 Y such that

2 Y denote the then a selection

f(x) z $(x) for

x ¢ X.

Example I.I: Let @(x) = u-l(x).

u

: Y -- X

Then

is continuous and

f

be onto.

is a selection for

f(x) ~ u-l(x) for every

Example 1.2: Let ~u." X -> 2 Y, let ~A.

Define

~ : X -" 2 Y by

@(x) = V ( x )

if

if and only if

x ~ X-A. f

Define

A < X

if and only if

and

f ". ~ -> Y

is a selection for ~

$

by f

x ¢ X.

~(x) = ~g(x)~

Then

$ ". X --~ 2Y

g if

be a selection for x s A

and

is a selection for

which extends

g.

The selection problem of the first paragraph can now be rephrased as follows : Under what conditions on " X-> ion for

2Y

can every selection for

@, or at least for

@IU

@iA

X,

A : X, Y and

be extended to a select-

for some open set

U ~ A

The purpose of this section is to give a solution to the above problem. Definition" Call ~ : X -~ 2Y lower semicontinuous l.s.c.) if

{x : $(x)~U~@}

is open in

X

(abbreviated as

for every open set U c Y .

The first step in answering the question, is provided by the following elementary but important necessary condition. Proposition I.I'.

If

~ " X -~ 2 Y has the property that, for every

Xoe X, there exists a selection for $~U (U a neighbourhood of which has a preassigned value

yo~ @(xO) at

xo)

xo, then ~ is lower

semlcontinuo us. Exsmple I.I*'. ~

is

I.s.c. if and only if

Example 1.2"" If ~ i s

u

is open.

l.s.c., A is closed in X and g

is continuous,

then ~ is l.s.c. Proof of proposition i.I: Let G = ~x : $(x) ~ V ~ ~}

V

be open in

is open in ~.

Y ~ we must show that

For each

Xo~ G, pick a

Yo ¢ #(Xo): then by assumption,

there exists a selection

for some neighbourhood

xo such that

U*o " Uo ~ x

Uo of

". fo(X) ~ V), then

is ~ontalned in

G.

Hence

f(Xo) = Yo"

fo for #~Uo, Now if

Uo* is a neighbourhood of

G

x0

which

is open and the proof is complete.

One can easily prove the following facts : (i) If

# : X -> 2 Y is l.s.c,

= ~

for every

x

where ~

of V(x) and #(x) respectively, (ii) If

# : X -~ 2Y is

with the property that the function (iii) If

Y

and if

l.s.c, #(x)NU

: X -~ 2 Y is such that and

then

(~x)

~ is l.s.c.

and if

U

is an open subset of

@(x) = U ~ $(x) is

is a topological linear space, and if ~

Y

is non-empty for every x ~ X, then

@ : X -~ 2 Y defined by

l.s.c., then the function

denote the closure

: X-~

l.s.c.

~ " X-,

~Y is

Z Y, defined by ~(x)--eonvex-hull

Now we are set to prove the following important result due to Michael which gives a characterisation of paracompact spaces. X

Recall

is paracompact if it is Hausdorff and if every open covering of

X

has an open locally finite refinement. Theorem l.l: The following properties of a

Tl-space

X

are equivalent

[i]. (a) X

is paracompact.

(b) If

Y

is a Banach space then every l.s.c, function ~'X -~ ~ ( Y )

(family of all closed convex subsets of Y) admits a selection. First we shall establish the follo~ing lemma. ~

~

If

X

semi-continuous and

is paracompact,

function from

X

Y

a normed linear space,

to the ~on-empty convex subsets of Y

r ~ O, then there exists a continuous

f(x) ~ S r ( T ( x ) ) Proof" Let

for every

~ a lower

x ~ X,

where

Uy = ~x : y ~ Sr(~Y(x)) )

f : X -~ Y Sr(Y(x))

for

y s Y.

such that

= ~y~d(y, ~y(x~)qr). Then

Uy = ~x ~ W~(x) C~ Sr(Y ) # ~}. Since y is l.s.c.,

U

is open for every

the open sphere of radius is a covering for

X

r with centre

and hence, since

open locally finite refinement if

~ # ~].

~W~ : ~ ~ A)

y]. X

W~ C V~ .

Clearly

Sr(Y) stands for ~Uy : y ~ Y) it has an

[ We assu~e V~ # V

there exists an open covering

Now we shall construct a family

~Poo : ~ ~ A) of continuous functions from interval with the following properties and

there

is paracompact,

~V~ " ~ ~ A).

By a theorem of Dieudonue, such that

y.

X

to the closed unit

" (i) p~ vanishes outside

~ p~ ~ ( Y )

( = closed convex subsets of

Then there

l.s.c, Y)

function

for which there

is no selection. Proof: Let

Z

be the set of rationals in

ordered as a sequence

Zl, z2, . . . .

y(x) # 0

for only finitely many

y(x) _~ 0

for all

/

x ¢ Z }.

C

if

X, and suppose

Let

Y = {y : y E

x ¢ Z ~ .

Finally

x ~ X-

Let

Z

is

~(Z),

C =~y'y ¢ Y,

let

Z

¢(x) C ~{

y " y ~ Y, y(z n) _> I/n }

It is easy to check that no selection for Suppose each

zn z Z

¢ is

l.s.o.

if

x -- z n.

Let us show that there is

¢.

f

were a selection for

has a neighbourhood

kf(x)](z n) • I/2n

X

f

is continuous,

such that

By induction pick a sequence k of distinct integers such that znk+l ~ ~ U for all k. i=l n i

{nk} Then

whenever

U n in

$ , since

x ~ ~n"

{ ~nk } is a sequence of closed subsets of

intersection property, then

and hence there is an

[f(Xo)](znk ) • 0

for all

X

with co finite

xo ~

~ k=-I

Unk.

But

k, which is impossible.

As applications of the above theorem we cite the following two propositions " Proposition 1.2 " Let and

X

be a paracompact space,

$ : X -~ F~(y), a l.s.c, function.

re(x) = inf { ~ l.s.c,

with

m(x) > O.

y ~L : y ~ ~(x) }

p(x) Z 0

for all

Y

Let

and suppose that x,

and

Then there exists a selection

a Banach space

p : X -~ R

p(x) > re(x) whenever f

for

$

such that

is

I0 llf(x) il_~ p(x) for every

x ¢ X.

~¢(x) N{

\ {o}

(x)

Let

y : Y ¢ Y, ~ Y

--> ~ ( Y )

• X

by

is

l.s.c.

@(x) -- y(x----~then

@

Hence if we define is also

l.s.c.

by the above theorem, there now exists a selection and

f

p(x)>0

i: p(x)--0.

It is not hard to check that ~ @

~ ~ p(x)} if

f

Hence

for

@

satisfies all our requirements.

Proposition 1.3 :

A topological space

X

is paracompact if and

only if it is dominated by a collection of paracompact subsets. Before proving proposition

1.3, we ~dll start with a

definition. Def~nltion :

Let

X

closed subsets of

be a topological space and d5 a collection of

X.

Then _~(Bn) continuously,

B n is as in the introduction.

of continuous (n norm, and

CLO, T] will denote the space

vector valued) functions on

Cb[O,T]

the compact subset of

[O,T] with the uniform

C[O,T] defined by

Cb[O,T] = { x : x ~ C~O,T], x(O) = x°,Ix(t)-x(t')l~ blt-t'l} . If

Q ~ C(B n)

and

y a En

we use the notation

p(y, Q) = inf { iy-qL " q ~ R } ,

while for

Ql' Q2 ~ -~(Bn),

h(Ql, Q2) denotes the Hausdorff distance between these sets. For

Q " [0,T] -> _9(Bn), define the variation of

subinterval

it - ~

t], ~ > O, denoted by

Vt

•

Let

P

denote a partition of

of points

(Q)

t-~

Property A " lira a- ~0

IT vt~ 0

=

For each

sup PeO

as follows.

it-a, t], i.e. a finite collection

For the partition

P,

k vt't-~ (Q • P) = ~=OZ h(Q(t ~+i ), Q(ty ))

vt (Q) t-~

on the

'

t-~ = to • t I < ... < tk+ 1 = t, and let

set of all such partitions.

Q

~

denote the

define

and

vt (Q, P). t-q

x ¢ Cb[O'T],

(R(x(.))) dt -- 0

vtt-~ (R(x(.))) ~ LI[O,T]

uniformly for

x s

Cb[O , T].

and

26

(Assume

x(t) = x(t) = x(O)

is defined for

Vt (R(x( • ))) t-~ This convention will be assumed throughout,

t-~ < O.

for

t < ~

so that

when necessary). At this Juncture we would like to make the following observations (i)

"

It is clearly possible that the variation

unbounded for some the special case

t, Q

yet

vt~

(Q) ~ L I.

t (Q) Vt_ ~

For example,

is

consider

a point valued function in the interval

[O,1]

defined by J(l-t)

Sin(l/l-t)

~en

t # I

when

t = I.

q(t) 0

(il)

If

R

is Lipschitzian,

say

h(R(x), R(x')) _~ Klx-x'l,

clearly property A is satisfied. Indeed vtt.~ (R(x(.)) ,P) -k k ~=07 h(R(x(t +i)), R(x(t ))) _< K b ~=OZ (t +l-t ) = K b ~ for x a Cb[O,T] . (iii)

Let

R " E n --> ~(B n) continuously and let

We define the variation of Let

R

in

S, d e ~ t e d

~y:

ly-x°l_~ b T}.

V(R,S) as follows.

k

be any finite collection of points yl ,...,yk+l in S such k that Z ly +i _ y ~ _~ b T and let /k denote the set of all such ~=i k collections. Let V(R,S,k) = Z h(R(yi+l), R(yi)) and V(RpS) is i=l defined as sup V(R, Stk) . If V(R,$) < oc we say R has bounded k~A variation in S. Clearly if R has bounded variation in S, then for any

x ~ Cb[O,T] ,

is finite for all In this case,

t

V~.~ (R(x(.)))

En).

Another way in which equation (I) appears is in the theory of control

33 systems having equations of the motion of the form ~ = f(t,x,u), 0 x(O) = x , where the control function u may be chosen as any measurable

r

vector valued function with value at time

a preassigned set

U(t) I , and, in this case, ~

is

fco. Strategies and stationary strategies for Player I!

are defined analogously. A pair (~, r) of strategies for Players I and II associates with each initial state

s

an

nth-day expected gain

rn(= , C)

(s)

for Player I and a total expected discounted gain for Player I c I(~', F ~) (s) F

We shall say that the stochastic game has a

sup inf I(~, F)(s) = inf sup I(~, F)(s) v

for all

F

for every

s ~ S.

42

In case the stochastic game has a value, the quantity sup inf I(~, F )(s), F ?r function.

as a function on

S, is called the value

The stochastic game problem was first formulated by Shapley [6] who took

S, 4

B

to be finite, assumed that play would terminate

in a finite number of stages with probability one, and considered only what we have called stationary strategies.

Shapley was able to

prove under these conditions that the stochastic game has a value and that both players have optimal strategies.

See [8] or [61.

We

shall prove a generalisation of Shapley's result under suitable assumptions on

S, A, B, q, r.

Specifically, we shall assume that

(1) S, A, B

are compact metric spaces, (ii) r

function on

S ~

and

b n -~ bo,

A ~B,

and (ili) whenever

is a continuous sn J

So,

q(-LSn, an, bn) converges weakly to

an -~ ao,

q(. ISo, ao, bo).

These conditions will remain operative throughout the rest of this section. In order to prove the main theorem, we need some lemmas. X

is a compact metric space, we denote by

probability measures on the Borel sets of

PX X.

If

the space of all It is well-known that

PX, endowed with the weak topology, is a compact metric space. Lemma 4.6 " S ×A

xB.

Let

i

be a continuous, real-valued function on

Then, i(s, ~, k) = IJ i(s,a,b) d ~(a)dk(b), s ~ S,

~ PA' k s PB, is a continuous function on

S x PA ~(PB"

For a proof see [4]. Lemma 4.7 " where

X

Let

u

be a bounded, continuous function on

is a Borel subset of a Polish space and

metric space.

Then

u* . X -~ R

defined by

Y

X ~

Y

is a compact

u*(x) -- max u(x,y) y~ Y

45

is continuous.

Moreover

u, " X --> R

defined by

u. (x) = rain u(x,y) is continuous. y~Y Proof of this lemma is easy and hence omitted. Lemma 4.8 : where

X

u

be a bounded,

continuous function

is a Borel subset of a Polish space and

metric space. into

Let

Y

Then, there exist Borel maps

such that

U(X,g(x)) =

f

Y

y

is a compact

and

u(x, f(x)) = max u(x,y), x z X y~Y

X x

g

from

X

and

rain u(x,y), x ~ X. y~Y

This lemma is an immediate consequence of theorem 4.1. If

X

is a topological space, denote by

all bounded,

continuous functions on

X.

C(X) the family of

For each

w ~ C(X), define

Kw(S,~,k) = r(s,~,k) + ~ I w(-) dq(. {s,@,k), s ¢ S, ~ ¢ PA' k ¢ PB

where

r(s, S, k) = ~I r(s, a, b) d @(a) d k(b)

and

q(.~s, S, k) = 5S q(.is, a, b) d iz(a) d k(b). It follo~s from lemma 4.6, S ~PA

XPB"

Kw

is a continuous function on

From Sion's minimax theorem Is] we have,

sup inf ~PAXSPB

Since

Kw

Kw(S,~,k) =

is continuous on

Inf XSPB

S ~

sup Kw(S,~,k) , s e S. SaPA

PA "( PB

it follows from lemma 4.7 that

sup

by max

Thus, we have

and mln, respectively.

max ~¢PA

mln XSPB

Kw(S,U,X) =

mln XSPB

and

and inf

PA' PB

are compact,

can be replaced above

max Kw(S,~,X) , s ~ S. ~ePA

44 Lemma 4.9 : from

S

For each

into

PA

max

and

w ~ C(S), there exist Borel maps PB, respectively,

f

and

g

such that

rain ke PB

Kw(s , ~, X) = rain k~ PB

Kw(S , f(s), X)

~ e PA

min XeP B

max ~eP A

Kw(S , ~, k) = max 6eP A

Kw(s , u, g(s)), s ~ S.

and

Proof "

We prove the first assertion.

S ~ S, ~ ¢ PA" S XPA.

By lemma 4.7,

~

Let

¢(s,s) = rain Kw(S,~,k) , Xe PB is a continuous function on

Hence, by virtue of lemma 4.8, there exists a Borel map

f " S -~ PA max

rain

~eP A

kaP B

such that Kw(S , ~, X)

~(s,f(s)) = max ~(s,N) = ~zP A for all

s s S.

On the other hand,

¢(s, f(s)) =

min Kw(S , f(s), X) for all X,P B the proof of the first assertion. For each Tw(s) =

w e C(S) define

max ~ePA

min XePB

Len~na 4.10 :

The operator

P_~

Wl, w 2 e C(S).

if

Let

w ~ C(S),

check,

T

T T

~ w Ii denotes

is monotone, we get

This completes

as follows •

Kw(S , @, X) = rain XePB

It is clear from lemma 4.7,

:

Tw

s $ S.

maps

max Kw(S , @, k). ~¢PA

C(S) into

C(S).

is a contraction mapping o n Plainly,

w 1 _< w 2 + LLWl-W211,

sup ~w(s) ~. ssS

C(S). where,

Since, as is easy to

Tw 1 _< T(w 2 + li Wl-W2i~ )

-- Tw 2 + ~liwl-w 2il

45 Consequently and

w 2)

Hence,

Tw~-Tw I _< 8 ~l-W21i , T

Similarly,

Twl-TW 2 --~ 8 ~tWl-W21L

which shows

is a contraction mapping as

(interchanging

wI

II Twl-TW2il ~ ~ ~ Wl-W21L

~ < I.

This completes the

proof of the lemma. Since

C(S), when equipped with the supremum norm, is a

complete metric space,

T

has a unique fixed point in

virtue of the Banach fixed point theorem. fixed point of

T.

Let

w~

C(S), by

be the unique

Then it follows from the definition of

Kw,(S , ~, k) and lemma 4.9 that there exist Borel maps from

S to

PA

and

PB, respectively,

f* and

such that, for every

g*

s ~ S,

w*(s) = min [r(s,f'(s),k) + 8 I w*(. ) dq (.Is,f*(s),k)] kcP B = max [r(s,~,g*(s)) + ~ I w~(.) dq(.Is,N,g*(s))] ~ E PA = r(s,f*(s),

...(*)

g*(s)) + 8 I w*(.) dq(.Is, f*(s), g*(s)).

We shall show after one lemma that

w* is the value function of the

stochastic game. Our next task is to solve the above functional equation (*). To accomplish this, denote by functions on function from to

PB,

S.

With each ordered pair (f,g), where

S to

S to PA

and

we associate an operator

(L(f,~)w)(s) = r(s,f(s),g(s)) We may interpret Player I,

M(S) the f~nily of all bounded Borel

g

f

is a Borel

is a Borel function from

L(f,g) - M(S) --> M(S)

S

defined by

+ ~ S w(.) dq(.Is, f(s),~(s)),

s ¢ S.

(L(f,g)w)(s) as the expected amount Player II pays

when the initial state of the system is

and II take actions according to

f(s) and

g(s),

s,

Players

I

and the game is

46

terminated at the beginning of the second day with Player II paying Player I

w(t) units of money, where

t

is the state of the system

on the second day. Lemma 4,11 " M(S)

and

The operator

L(f,g) is a contraction mapping on

l(fOo gO~ is its unique fixed point in

The proof is straight-forward and omit it.

M(S). Now we are ready

to state our theorem [4]. Theorem 4.2 :

Let

S, A, B

be compact metric spaces, let

continuous, real-valued function on moreover, that, whenever S ,~ A '~ B,

S ~

A

,( B,

converges weakly to

be a

and assume

(Sn, an, b n) -~ (So, ao, b o)

q(. ISn, an, b n)

r

in

q(. !So, ao, be) .

Then, the stochastic game has a value, the value function is continuous, and

Players

I

and

II

have optimal stationary

strategies. Proof:

Observe that the above functional equation (*) can be

rewritten as

L(f*, g~) w* = w ~.

w ~ = l(f.(oo), g,(Oo)).

It follows from lemma 4.11

that

In view of this we have,

I(f*(°°), g*(°°))(s) -- max [rCs,~, g*(s)) + asP A 8 I l(f *(°°),g*(°°))(.)dq(. Is,~,g*(s))]

= min [rCs, f*Cs),k) + ~ I ICf *(9°),g*(e('))( )dq(-Is, f*(s),k)] ke PB It follows from Blackwell

[Theorem 6, pp ~$2, [I]] or from

[Theorem 3.1 in [4]], that,

I(f *(~),g*(~))(s) = sup I (~, g*(~)) (s) 7r

=

inf l(f*(°°) C)(s) F

for

s ¢ S.

47 Consequently, l(f*(°°),g *(°°)) = sup I (~,g,(Oo)) Z inf sup I(~, C ). ?r v ?r On the other hand,

l(f*(°°),g *(°°)) = inf l(f "(°°) c ) < F

sup inf I( ~, V) ~ C

Hence, inf sup I( =, c) = sup Inf I(~, v). C ~ = f This proves that the stochastic game has a value, that the value function is I(f,(oo), g,(OO)) = w~ and so continuous, and that f, (co) g, (oo) , are optimal stationary strategies for Players I and II respectively. Remark 4.1 "

This completes the proof of theorem 4.2.

If in theorem 4.2 we allow

S

to be merely a Borel

subset of a Polish space, our proof breaks down because lemma 4.6 fails.

However, we can eliminate this difficulty by imposing

somewhat stronger conditions on

q

and

r, namely, that

should be a continuous function on

S ~

PA x p B

I w(.) dq (. Is, ~, k) for every when q

and

A

w ~ C(S). and

r,

B with

are finite]. A, B

and that

should also be continuous on [ ~te

r(s,s,k)

S ~

PA ~

that these conditions will be satisfied Then under these conditions on

compact metric spaces and

S

a Borel subset

of a Polish space, the conclusions and the proof of theorem 4.2 remain valid.

PB

48

REFERENCES

[1]

D. Blackwell, Stat

S6 [1968],

[2]

K. Kuratowski,

[3]

A. Maitra,

A. Maitra

226-235.

Topology

Vol I, Acad. Press, P.W.N. [ 1 9 6 6 ] .

Sankhya Series A. 30 [1968], 211-216.

and

T. Parthasarathy,

Jour. optimi, theory

[5]

Ann. Math.

Discounted dynamic programming on compact metric

spaces,

[4]

Discounted dynamic programming,

T. Parthasarathy

and

t~o-person games,

On stochastic games,

And its Appl , 8 [1970], 289-~00.

T.E.S. Raghavan,

Some topics in

American Elsevier Publishing Company,

New York [1971].

[6]

L. S. Shapley,

Stochastic games,

Proc. National. Acad. Sci

U.S.A., 39 [1983], i095-II00.

[7i

L.E. Dubins and L.J. Savage, Fow to gamble if you must, Xcgraw-~Ki!, New Yor~ [1965].

5.

GFz~ERAL THEORE~S ON SELECTORS

In this section we shall first prove a general theorem on selectors

due to Kuratowski

theorems on selectors Jacobs

and

[2].

Ryll-Nardze%,ski

and deduce a few

Lastly we shall state a theorem of

[3]. Let

space.

X

be a set o f arbitrary elements

Let

S

be a countably

additive

and

Y

a metric

family of subsets of

X

cO

[that is, if

An a S

for

the following statement Lemma 5.1 "

Let

f(x) = lim fn(X)

fn " X -- Y

Let

assumption Ifm

for

whenever

whenever

K = ~

c Y.

G

I/n

for

Then

G

Let

and let

is uniform.

is open in

is open in

Suppose that

Y

(I) n.

Y.

K n = ~ y " p(y, K) _
Y.

follows

is open in

f'l(G) e S

whenever

f(x) e F(x) according to

G

(2) n.

Y, this sequence By lemma 5.i, it Y.

Finally

Thus the proof of theorem S.l is

complete. Remark 5.1 : [namely by

The theorem remains true by replacing the condition

~ x " F(x) C~G # @ } E S

~ x "

F(x)~

K # ~ } e L

can be seen as follows. F

set

:

G = K IU

Y

K2•

whenever

whenever

G

is open in

K c Y

is closed.

being metric, every open ...,

where

Y ]

Kn = ~n"

This

G


PA ~

PB

where

rwCs) = ~[ (~',x') : max [r(s,s,~,') + ~ f w(-) dq(. Is,~,X')] = rain [r(s,t~',X) + ~ I w(.) dq(. Is,#',X)} k

Further-

63 and

w

is a bounded real-valued Borel measurable

These set valued functions Olech [2]. theorem 6.2.

rw

function on

S.

are measurable by a result of

Now one can use (the selection)

theorem 5.3 to prove

The rest of the proof follows along similar lines

to that of theorem 4.2, -with some minor ,~difications the proof of theorem 6.2 is omitted.

and hence

64 REFERENCES

[1]

D. Blackwell,

On a class of probability spaces, Proceedings

of the third Berkeley Symposium on Mathematical Statistics and Probability Vol 2, p 1-6, edited by J. Neyman, University of California Press [1956].

[2]

C. Olech,

A note concerning set valued measurable functions,

Bulletin De L'Academie Polonaise Des Sciences, S _ ~ ~e~ S i e ~ ,

[3]

M. Orkin,

~.ath, ~s~ro et Pl~vs. 13 [1965], 317-321.

A Blackwell space which is not analytic,

To appear.

[4]

T. Parthasarathy, Discounted and positive stochastic games, Bull. Amer, Math. Soc

77 [1971], iS4-136.

7.

VON-NEUMANN'S

MEASURABLE CHOICE THEOREM

The purpose of this section is to prove a measurable theorem due to von-Neumann.

choice

After giving the proof, we shall

mention a few consequences of this theorem. Choice Theorem 7.1 [4] ". Let

S

be a complete separable metric

space,

S

and

A

an analytic set in

defined and continuous for all values.

Let

assumes.

K

Let

A

and has real numerical

be the set of all values, which ~ (k)

is a monotone,

a ~

F(a) a function which is

be an arbitrary

nondecreasing,

F(a), a c A

N-function,

rlght-continuous,

that is,

T (k)

bounded function.

Under these assumptions we have (i)

K

is a

T (k)

measurable set.

(ii)

there exists a function

X ~ K

and assumes values in (a)

k ~ K

If

with (b)

Proof :

0

S

such that

is an open subset of

f(k) c 0 For every

is

S, then the set of all

"[ (X)-measurable.

k ~ K,

F(f(k)) = ~.

We shall prove the theorem when

and nowhere constant

~-i~anction,

a topological mapping of a ~ ~ • b

f(k), which is defined for all

where

a

=

T (k)

therefore

is a continuous

k -~ % (k) = ~

is

- co ~ k ~ oo on the finite interval T~(- co),

b

= p~(+ co).

The general

situation can be reduced to this case [4]. Let us now replace and then write again

X

F(a), f(k) by for

our assertions unaffected, ~.

The domain of

k

~.

(F(a)),

f("C-I(~)),

This leaves the structure of

but there are these changes regarding

is now

a

~ k • b

and

not

- oc ~ k • + co.

66 In this new domain we have to use the notion of common (Legesgue)measurability and not that one of T(k)-measurability. -9 I~(k)

[Our mapping

transformed the latter into the former].

We now introduce the '0 space of Baire', So, which is the set of all sequences

4 = (n!,n 2 .... ),

nl, n 2 ... = 1,2, . . . .

It is metrized by the definition Dist [(ml,m2,...) , (nl,n2,...)] = u

if

mv = nv

= _I_ if

vo

vo

for every

is the smallest

integer for which

Thus

SO

is a complete, separable metric space.

set of all those

~ = (nl, n2, ...), for which

a finite number of

vo

exists such that

mvo • %

establishes an ordering of all

~ ~ 4

we have sphere in 41, 42) . then

is open.

~ < 4 So

[The countable nv W i

occurs for

If

So.

(ml, m2,...) • (hi,n2,...) and If

p ~ 4, and

in the entire sphere is a set of the form

mv ~

= nv

for

P -- (nl, n2,...)

if a

is fixed, then the set of vo

is defined as above, then

Dist (~, B) • I/v o. ~l -< ~ ~ 42

Every

(with fixed ¢ ~ O,

belongs to it, if and only if,

Dist ~ (ml,m2,...) ' (nl, n2,... ) } < ~, where

So] .

v < vo ,

If its center is (ml, m2, ...) and its radius

v = 1,2, ..., v I

m v W nv.

v's only is obviously everywhere dense in

On the other hand the definition,

v.

vI

may be written in the form

that is

m v = nv

is the greatest integer

for

_< i/~.

This

~I -< p ~ c(2 if we put

~I = (ml'''',mVl-l, mvl, I, I, ...), ~2 =(ml,"',mVl-l, mvl+l,l,l, --~-

67 As

A

Is an analytic set In a complete, separable metric

space, therefore there exists a function and continuous for every

~ ~ So

and ~Lich maps

[The mapping may be many to one]. the set of all

There As

F(~(~)), ~ s So.

continuous for every

~ z So

~(~) which is defined

K

So

onto

A.

Is clearly

F(@(~)) also Is defined and

and the real numbers form a complete,

separable metric space, we can conclude that

K

is an analytic

set and consequently measurable. Consider now a wlth

F(@(~)) = X

k ~ K.

by

Denote by the set of all

T(A).

as

F(@(~)) Is continuous,

nI

with

As T(X)

(n I, n2, ...) ~ T(A)

the smallest

n2

(us, n4, ...)

by

with ~

k ¢ K,

Is closed. for auy

etc.

Let

for every