Nondeterminism of stochastic automata: An etude in measurable selections

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Informatik-Berichte 13 – 02/1981

Nondeterminism of stochastic automata

An etude in measurable selections

L

Introduction

NONDETERMINISM OF STOCHASTIC AUTOMATA - AN ETUDE IN MEASURABLE SELECTIONS

Ernst-Erich Doberkat Fachbereich Mathematik, Lehrgebiet Informatik,.

Fernuniversität Hagen

Measurable selections play an important röle in Mathematical Optimiza- tion, and the development of stochastic dynamic programming is paralled by some theoretical investigations in the field of set valued maps - mainly motivated by the aim to find general and applicable conditions for the existence of well-behaved selectors of those maps.

On the other hand it does not seem tobe too strange to make an attempt of describing some aspects of economic behavior by means of a network of automata - why confine concepts of input, output and changes of internal states to models in Computer Science? And indeed, this has been don.e, in order tobe able to handle the concept of action and

of aggregation, respectively, in a mathematically convenient manner(RO).

Having a look at both the optimization, and the automaton concept, it seems tobe attractive to marry them, and to investigate, say, op- timal actions in a network of automata by means of measurable

selections.

This will not be done here. Rather, a re_lationship between nondeter-

·ministic and stochastic automata is proved, a relationship which may be formulated in the following way: any action which is chosen non- deterministically may thought tobe chosen following some probabilitic law, under some suitable formal assumptions.An equivalent formulation in terms of Computer Science might be that even nondeterministic com- putations, viz., guessing, follows some probabilistic laws, when the models of computations are restricted to acceptors.

This note is organized as follows: Section 2 remembers automata, and some facts from Selection Theory, Section 3 formulates the possibility of representing a stochastic automaton by a deterministic one (whtch has as a byproduct the result that a stochastic automaton frorn a suitably chosen kind may be approximated by a discrete one).

In Section 4 we have a closer look at the nondeterminism of stochastic automa ta.

2. Automata

Let X be the input alphabet and Z the set of states of the automata considered here. (X,Z;R) is said tobe a nande.:te.,'tm,i.n,i.s.U.c. (-~:ta.:te.) o.u,toma.,ton iff QH,R(x,z)c? holds for every xEX, zEZ; z' ER(x,z) is 'a possible new state after input of x EX in state z. No•t-1 let x* be

the free monoid generated by X (with /VI as the le.ngth of vEX*, e as the void word), then R is extended in the usual manner to a relation R* on x*x Z: define

R

*

(e,z) := {z}, and

R

* (

V X , Z ) : :

U{

R ( ^{X , Z I ) ; 2 I E} R

* (

V , Z ) } •

Then, after input of the string vx Ex* in state z E Z, R*(vx,z) is the s et o f a l 1 p o s s i b l e n e w s tat es .

Let X, and Z, be endowed 1-1ith cr-fields A, and C, respectively.

((X,A),(Z,C);K) is said tobe a ,~:tac.ho.-~:t-i.c. (-6:ta.,te.) a.u.-t:oma.,ton iff K is a transition probability from (X xZ, A ®C) to (Z,C). Here AG>C is the sma11est cr-field on X x Z, that contains all measurable

rectangles A ,< C, where A E A, CE C. Then the defining property of K can be rephrased as follows: given CEC, the function

(x,z)-1- K(x,z.)(C) E [0,1] is A@C-measurable, and given x EX, z E Z, the function C i-K(x,z)(C) is a probability measure on C. K is extended to a transition probability on x*xz in the following way:

let A* be the canonical cr-field on x* (cp. 001, p.355 f.) and define inductively (vEX*,xEX,zEZ,CEC)

K*(vx,z)(C) := JK(x,z)(C) K*(v,z)(dz),

then K* is a transition probability from (X*x Z,A*&c) to (Z,C)

z

provided we define

K*(e,z)(C) := e:(z)(C),

----where e:(z) is the Dirac measure on z, i .e. e:(z)(C) = 1 iff z E C, = 0 otherwise. The probabilii;y that after input of vEX* in state z the next state is an element of CEC is K*(v,z)(C).

Let again X and Z be measurable spaces. (X,Z;g) then is said tobe a me.a. .. H1.,'ta.ble. de.te.-'7.m,i.nL~:tJ.c. (.!i:ta.,te,) a.u:toma.,tan iff g:X x Z - is a measurable state transition map. Note that if X and Z are second countable topological spaces and g is an act, i .e. g is jointly. con-

tinuous (see SI), then(X,Z;g) is a measurable deterministic state

a1.itomaton, when X and Z are endowed with their Borel sets, i .e. the smallest o-fie1d that contains the open sets.

The transition map g:X x Z - Z is extended i_n the usual way to a map g*:x* x Z -

z

upon setting

*

g (e,z) := z,

g*(vx,z) := g(x,g*(v,z)).

We need some topological assumptions. Throughout this paper, Xis assumed tobe a measurable space (with the o-fi.eld omitted in n o ta t i o n ) , a n d Z i s a s s um e d t o b e a Po 1 i s h s p a c e , i • e . a s e p a ra b 1 e and completely metrizable topological space. Z is thought tobe endowed with its Borel sets. Denote by F(Z), and C(Z), the set of all non void closed, and compact, subsets of Z, respectively, C(Z) is assumed tobe endowed with the Hausdorff metric, and is thus a separable metric space.

If (Y,Y) is a measurable space, and F:Y F(Z) is a set valued map, Fis said to be me.a..ou.,'ta.ble. iff {y;F(y) 11 U

* ~}

is a measurable subset in Y for any open set U c Z (cp. HI, where this property is referred to as 1-1eak measurability). From Theorem 5.6 in HI it is inferred that F:Y F(Z) is measurable iff there exists a sequence (sn)nElN of measurable selections s :Y Z for F such that

n

F(y) = {sn(:y);n E IN}c

. C )

holds for every yEY, denoting the topological closure; (sn nEIN is sa i d to b e a Ca.,5 :ta.b1.g .1te.p.1tu e.n:ta.:tio n fo r F. U n 1 es s o th e rw i s e

specified, it is assumed for the rest of this paper that a nondeter- ministic state automaton has the property that

R*:x* x

z -

F(Z) i s m ea s u ra b 1 e .

Given a stochastic automaton with extended transition law K*, and VEX, ZEZ, denote by supp K*(v,z) the following subset of Z:

z'Esupp K*(v,z)iff K*(v,z)(U) >0 holds for any open ~eighbourhood U of z'. From PA, Theorem II.2.1 it is seen that supp K*(v,z) is the smallest closed subset C of Z with the property that K*(v,z)(C)=l holds. Hence supp K*:(v,z) 1 - supp K*(v,z) is a closed valued map and is measurable, since for U cZ open supp K*(v,z) n U

*

^{~ iff}

K*(v,z)(U) >0. This implies that a s·tochastic automaton generates (via supp) a nondeterministic one, but it is not clear at al,l

whether or not (supp K)*= supp K* or the measurability property _above holds. This question will be investigated below.

3. Deterministic Representations for Stochastic Automata

Fix for this section a stochastic state automaton S = (X,Z;K). Given v

c:

x*, z

c:

Z, supp K*(v,z) can, by 'construction, be interpreted as the set of all possible ne,~ states after input v in state z; note that supp K*(v,z) = {z' ;K*(v,z)({z'}) > O}, if Z is finite and is endowed with the discrete topology. Now let V= (X,Z;g) be a measurable

deterministic automaton with the property that g*(v,z) E supp K*(v,z) h o 1 d s f o r a n y v E X*, z E Z . Th e n V ca n b e t h o u g h t a s a de t e rm i n i s t i c realization of some of the behavioral possibili'ties of S, and it will be shown now that there is a measurable deterministic automaton which represents in some sense all those possibilities.

3.1. Theorem:

Let S have the property that, given an open set U c Z, v Ex*,

*

{z E Z;K (v,z)(U)

>

^0}

is open. Then the following holds:

a) i f g: X x Z - Z i s mea s u ra b l e, an d g(x,z) E supp K(x,z)

holds for any xEX, zEZ, we have g*(v,z) E supp K*(v,z)

*

f O r a ny V E X , Z E Z,

b) there exists a measurable deterministic state automaton (INx X,Z;g) such that

* * (

lvl c

vvEX vzEZ: supp K (v,z)={g a.,v,z);a.EIN } hol ds.

P ro o f : a ) a n d b ) w i 11 b e dem o n s t ra t e d b y i n du c t i o n o n I v 1 •

1. Assume a) is proved for all v with lvl::. n. Fix then v E Xn, x EX, z E Z. Si nc e

g*(vx,z) = g(x,g*(v,z)) E supp K(x,g*(v,z)) it is obviously sufficient to show that

supp K(x,g*(v,z)) c supp K*(vx,z)

holds. Now letz' Esupp K(x,g*(v,z)), and U be an open neighbourhood of z'. Then

A .- {z;K(x,z)(U)

>

0}

* *

is an open neighbourhood of g (v,z) E supp K (v,z), thus we have

K (v,z)(A) > 0.

*

But

K*(vx,z)(U) ;::_; K(x,z)(U) K*(v,z)(dz) > 0.

A

This implies z' Esupp K*(vx,z).

2. Let (sn)nElN be a Castaing representation for (x,z) i- supp K(x,z).

Then

g(n,x,z):= sn(x,z)

defines a measurable deterministic state automaton (INxX,Z;g) such tha t

g*(a,v,z) E supp K*(v, z)

always holds by part a). In order to prove the claim of part b), define

( ) ·- *( )· INIV/ C S V,Z .-{9 a,V,Z , E } , then it suffices to show that

K*(v,z)(S(v,z)) = 1

holds. Assume this is done for v Ex* with 1v1 s; n, and fix v E xn, XEX, zEZ. Then

H := {z;K(xz)(S(vx,z)) = 1}

is the complement.of the open set {z;K*(x,z)(Z-S(vx,z))>O}, hence H i s c l o s e d • S i nc e

* *

g (ä.l,vx,z) = g(l,x,g {a,v,z)J and

K ( ^{X ,} g * ( Cl, V , Z ) ) ( S ( ^V X, ^{Z ) ) ;?.} K ( X , g * ( ci, V , Z ) ) ( { g ( .e_, X , g * ( a, V , ^{Z ) ) ;} lEiN } ^{C )} = 1 , we see supp K*(v,z) c H by the i.nduction hypothesis. Thus we have

*

f

*

K (vx,z)(S(vx,z)) =

*

K(x,.)(S(vx,z))dK (v,z)=l. 0 supp K (v,z) .

.

In the next Theorem it will be assumed that K*(v,•) is wea.k..e.y

c.on:t.-i..nu..ou.~ for any v Ex*. This means that, if f:Z - IR is a bounded and continuous function, then z 1- ffdK*(v,z) is continuous, too.

This notion of continuity gives rise to the well known weak :t.opology on the set of all probabilities on Z, in which a net (µa)aEI converges to µ iff lim f fdµ = ffdµ holds for every bounded and continuous func-

a EI a

tion f:Z IR. From PA, Theorem II.6.1 it is seen that the·assumption of 3.1 holds in particular if K*(v,.) is weakly continuous for·any VEX*. It is easily verified that this is the case provided K(x,•) is weakly continuous for any v EX, since

- - - - - ~ - -

*

SfdK.(vx,z) concides with

i[;fctK(x, ,)JdK*(v,z).

This formula is an immediate consequence of the definition of K*

above.

From Theorem 3.1 we deduce that a stochastic automaton with continuous state transitions can be approximated by stochastic state automata which have in any situation only a finite set of alternatives for a s ta t e t ran s i t i o n •

3.2 Theorem:

Let the transition law K of S have the property that K(x,•) is weakly continuous for any x EX. Then there exists a stochastic automaton

(JNxX,Z;L) with the following properties:

a) card(supp L(m,x,z))s;m for any mEIN, xE:X, zEZ,

b) the net (L*(a,v,z)) IVI converges to K*(v,z) in the weak topo- a.EIN

*

1 0 g y f O r a ny V E X. , z E

z

~

Proof: O. The idea of the proof is to make use of the deterministic re p re s e n ta t i o n i n 3. 1 i n t h e fo 11 o w i n g wa y: l et (IN x X, Z ; g ) b e a s i n 3..1, and let U (n,x,z) be a neigbourhood of g(n,x,z). L(m,x,z) is

m

going tobe so defined that the masses of Um(n,x,z) with respect to K(x,z) are blurred over Z and adjusted properly.

Let p be a metric on Z which is compatible with the topology;

Br(z) .- {z';p(z,z')s;r} is the closed ball with center z and radius r.

In part 1 a sequence of stochastic automata is constructed for which a) and b) hold i f IVI = 1.

1. Fix for the moment x EX, z E Z, and define form elN the set JN° _m as follows:

l EINO _{m ,}

and i f n ElN~ , such that

. 2

n' := inf{k;p(g(n,x,z),g(k,x,z)) 2:m} EIN, then

Let

lNm := JN~ n{l, ... ,m} ,

and put

am :=-S--- K(x,z)(B

1/m(g(n,x,z))).

~ m

Since {g(n,x,z); nc:lN} is dense in supp K(x,z), we have am - 1, as m - "', hence 1-1ithout lass of generality am can assumed tobe positive for all mE:lN. Now define

L(m,x,z) :=J..

~

K(x,z)(Bl/m(g(n,x,z)))·e:(g(n,x,z)), am ne=]'l

then evidently - m

card(supp L(m,x,z)) scard(INm) s m, and

L(m,x,z)(supp K(x,z)) = 1 hold. Define

Am :=supp K(x,z) -U{Bl/m(g(n,x,z));nEINm}

then the construction of g yields K(x,z)(A ) - O, as m- "". Now let

. m

f : Z - IR b e uni fo rm 1 y c o n t i n u o u s a n d b o und e d, t h e n

1 S fdK(x,z) - jfdl(m,x,z) 1

S ~ f l f - f(g(n,x,z))ldK(x,z) + f lfidK(x,f) n E Il'l n B 1 / m ( g ( n , x , z) ) Am

s s u p p { 1 f ( z ' ) - f ( g ( n, x, z ) ) 1 ; z 'E

s

1 / m ( g ( n, x, z ) ) , n EIN m } + f I f I dK ( x, z.) - 0.

Am

This is so since the first summand converges to O because of the.

uniform continuity of f, the second because of the construction of Am.

From PA, Theorem II,6.1 it is seen now that K_(x,z) is the weak limit of (L(m,x,z))mEIN'

2. Assume that b) is shown for v E Xn. Fix v with IVI = n, x EX, z E Z, a n d 1 e t f : Z - IR b e b o u n de d a n ·d c o n t i n u o u s • G i v e n e:

>

0 , i t i s

inferred from part 1 that there exists n

0 EIN such that

1 ffdK(x,z) -ffdl(n,x,z)l<e:/2

holds for n::::n. Since fand K(x,·) are assu.med tobe continuous,

0

z 1 - j f d K ( x, z ) i s b o u n de d an d c o n t i n u o u s . Th e in du c t i o n hypo t h es i s implies that there exists a.

0 Ell'ln such that

1 f[ S f d K ( x , • ) ] d K

* (

v , z ) -

i [

f d K ( x , • ) J d L

* (

a. , v , z ) 1

<

e: /2 holds for all a.2:a. (where INn has the componentwise order). Thus

0

JjfdK*(vx,z)-SfdL~(:::n,vx,z)I

= lf[fdK(x,•)dK*(v,z)-j[fdL(n,x,·)dL*(a,v,z)I< e:

Theorem 3.2 demonstrates that every stochastic automaton can be approximated by a discrete one, provided the continuity condition

imposed there holds. One might suspect now that given

s

as in 3.2, there exists a sequence {(X,Z;Ln); nEJN}of stochastic automata such tha t

lim L~(v,z) = K*(v,z)

holds in the topology of weak convergence for any v Ex*, i .e. that the extension of the input alphabet by the factorlN might be

u n n e c e s s a r y . Th i s , h o w e v e r, i s fa 1 s e i n g e n e ra 1 , a s t h e fo 1 1 o ~li n g e x a m p 1 e de m o n s t ra t e s . L e t X : = Z : = IR u n de r t h e u s u a 1 t o p o 1 o g y , a n d pu t

K(x,z) := 1

2^{(e:(x+z) + e:(x-z)).}

Si nc e

[fdK(x,z) =½(f(x+z) + f(x-z),

K(x,•) is weakly continuous for all xEX. Define for al' ... ,anE.R, Tc:{l, ... ,n}

AT ( a 1' ... 'a n ) . - f=nl ( -•l ) e: ( i ) ( T) • a i then

K*(x1, ... ,xn,z) = 2-nI{e:(xn + AT(x 1, ... ,xn_ 1 )) ;Tc: {l, ... ,n}}

in easily proved by induction. Let

then

B u t s i nc e

and

g1(x,z) := x + z, g2(x,z) := x - z,

{g1(x,z),g

2(x,z)} = supp K(x,z).

n

= ~ X i ' 1 =0

g;(xl, ... ,xn,xo) = ~ (-l)n-lxi

1 =Ü

hold, it is clear that no sequence (Ln)n ElN as described abovecan exist.

4. Stochastic Representations for Nondeterministic State Automata Let :'-,/ = (X,Z;R) be a state automaton such that R*:x* x Z - F(Z) is

Theorem, it is remembered that a set valued map G:Z + F(Z) is LLµµe.r... (Io(ve..t) ,Hm,i.c.on,ünuoLL..S i ff {z ;G( z) c U} [resp. {z ;G( Z) n U

*

0} 1

is open in Z whenever U c Z is open.

4.1 Theorem:

Assume that R(x,·) is lower semicontinuous for any xEX, and that Z is cr-compact, or that R is compact valued. Then there exists a stochastic automaton (X,Z;K) such that

* *

(*) R (v,z) = supp K (v,z) holds for any vEX*, zEZ.

Proof: 1. Since R:X x Z - F(Z) is measurable, there exists according to D0 2 ,4.2 (if Z is cr-compact)or to DO 3, 4.2 (if R is compac·t

valued) a stochastic automaton (X,Z;K) such that supp K(x,z)=R(x,z) holds for any XEX, zEZ.

2. Now assume the equality (*) has been demonstrated for all v, Jv I s;n. Fix one such v, x EX, z E Z. If U is an open neighbourhood for arbitrary z' E R*(vx,z), there exists according to the construction of R*, z E R*(v,z) such that z' E R(x ;z).

A :=·{z" ;R(x,z") nU*~}

then is an open neighbourhood of z, thus we have K(x,z") (U)>O

for any z" EA. Because of the induction hypothesis K*(v,z)(A)

>

0

hol ds; consequently

K*(vx,z)(U);;::f K(x,z" )(U)K*(v,z)(dz")

>0.

A This implies

R*(vx,z) c supp K*(vx,z).

If z' $Ri:•(vx,z), there exists an open neighbourhood U of z' such that R*(vx, z) n U = 0,

since R* has closed values. Consequently, we have R(x, z) n U = ~

for any

z

E R*(v,z), thus K(x,z)(U) = 0 holds for every zER*(v,z) = supp K*(v,z). But this implies

r----·---·--- ---• --- - .

As a consequence, it is deduced now that the behavior of ,\J can be described approximately, i.e. up to topological closure, by a de t e rm i n i s t i c de v i c e :

4.2 Corollary:

Let N be given as in Theorem 4.1. Then there exists a measurable deterministic state automaton (lJ'lx X,Z;g) such that

R*(v,z) = {g*(a,v,z);aElr'llVl}c

*

holds for every x~X, ZEZ.

Proof: Let K be given as in 4.1. Since K (X, Z) ( U) > 0 i ff R ( x, Z) 17 U

*

^'/J

for any open U c Z, and since R(x, ·) is lower semicontinuous, Theorem 3.1,b) c_an be applied. 0

This Corollary tells us together with Theorem 4.1 that the behavior of a nondeterministic state automaton can be interpreted as stochastic behavior, i .e. the possible behavior of a stochastic state automaton, and can be approximated by the behavior of a deterministic automaton.

Let us assume that the stochastic and nondeterministic automaton con- sidered here are endowed with some informations concerning their initial states. That is, we are given a set F of possible initial states in the nondeterministic, and a probabi1ity p in the stochastic case. Then define

R;(v) :=

U

R*(v,z) zEF

as the set of possible new states after input v EX*, and K* ( v) ( D) : = j K* ( v, z) ( D) p ( dz)

p .

as the distribution of the new states. Now assume we have

* *

R = supp K , and

supp p = F.

Is it then true that

*

*.

RF = supp Kp ?

The next Proposition answers this question.

lov1er semicontinuous for any v ':: x*, If FE F(Z) is a set of initial s ta t e s fo r N , a n d t h e p r o b a b i 1 i t y p i s a n i n i t i a 1 d i s t r i b u t i o n o n Z such that

supp p = F holds, then

supp K* = R* implies supp K;=R;.

Proof: 1. To begin with: given FEF(Z), there exists always a proba- bility p such that supp p = F holds. For since Fis closed and Z is separable, there exists a countable dense subset {z ;n EIN} of F. _n Then

-:;:;-- -n p := L-.- 2 s(z )

nEIN n

is the wanted probability.

2. If v = e, one has

t h u s I v 1

>

O ca R b e a s s um e d i n o r de r t-0 p r o v e R;(v) = supp K;(v).

Si nc e

~ fK*(v,z)(R*(v,z))p(dz) F

= 1, supp K;(v) c R;(v)

hol,ds. In order to demonstrate the reversed inclusion, consider an.·

arbitrary zER;(v) together with an open neighbourhood U of z. Since R*(v, •) is lower semicontinuous,

*

H := {z';R (v,z')nU,t:~l)}

is an open neighbourhood of z such that F n H

* 0.

This impl ies p(H)

>

O, and

z' EH implies K*(v,z')(U)

>0.

Consequently one has K;(v)(U)

>

O, th US z Esupp Kp(v).

*

0

Up to now it is not clear under which conditions the assumptions of Theorem 4.1, and Proposition 4.3, respectively, 1~ork. It turns out

that it is the case if R takes compact values such that R(x,·) is both lower and upper ser.iicontinuous. Sefore discussing this, 1et us introduce Caratheodory maps.

4.4 Definition:

Let T be a measurable space, A,8 be separable metric spaces. A map f:T x A ^• 8 is said to be a Ca.,'ta.-theodotty ma.p i_ff f(t, •) is

c o n t i n u o u s a n d f ( • , a ) i s m e a s u ra b 1 e f o r a ny ( t, a ) E T x A.

Any Caratheodory map is measurable (HI, Theorem 6.1).

Note that C(Z) is a separable metric space under the topology induced by the Hausdorff metric, since the finite subsets of a countable dense subset of Z forma dense set in C(Z). Moreover it is-seen from BE, Theoreme IV.6.1, that R is a Caratheodory map iff R is compact v a 1 u e d, m e a s u ra b 1 e , a n d i f R ( x , • ) i s u p p e r a n d 1 o w e r s e m i c o n t i n u o u s foreveryxEX.

4.5 Lemma:

R:X x z - C(Z) is a Caratheodory map iff R*:x* x z-c(Z) is.

Proof: It is easy to see that R is a Caratheodory map if R* is.

Denote by Rn the ~estriction of R* to xn xz, and assume Rn is shown tobe a Caratheodory map. From BE, Theoreme VI .1.3, it is seen that R 1 has compact values, and that R ₁(vx,•) is lower and upper

n+ n+

semicontinuous. It must be shown Rn+l(•,z) is measurable. Now HI, Theorem 3.2 implies that it is sufficient to demonstrate the following: iff CEC(Z), then

n+ 1 · ) E := {vxEX ;Rn+l(vx,z cC}

is a measurable subset of·Xn+l. Since Rn is measurable, there exists a Castaing representation (Uk)kE IN for it. Since R(x, ·) is continuous with respect to the Hausdorff metric for any x EX, E is seen to·

coincide with

( U

{vxEXn+l;R(x,U,(v,z))cC})c.

kEIN ^K

Because Uk(•,z) is measurabie, vx ! - R(x,Uk(v,z)) is measurab1e as a set valued map. Thus Eisa measurable subset of Z.

•

4.5 yields as an immediate consequence.

4.6 Theorem:

Let N be governed by a Caratheodory map. Then there exists a stochastic automaton S = ( X, Z;K) and a rneasurable deterministic autornaton

(JNxX,Z;g), such

, ( ) • ~ Thl I v 1 , c R

* ( )

K

* ( )

\ g a, V , Z , ~ := ..i·i J

=

V , Z

=

S U p p V , Z

holds for any v Ex*, z ^E Z. If rnoreover FE F(Z) is a set of initial s ta t es f o r ,\/ ,

R;(v) = supp K;(v)

hoids for all vEX, where p is an appropriate distribution of the i n i t i a l s tat es fo r S . 0

One might guess that g in the Theorem above can be chosen in such a

\vay that g* is a Caratheodory map. I conjecture that this is false.

lt can be shovin that, given x EX, there exists a continuous rnap

such that

y . z - z

_x·

Yx(z) E R(x,z)

h o l d s fo r a n y z E:: Z ( MÄ, ( 1 . 1 2 ) ( 6 ) ( e ) , M I , § 9 ) , a n d t h e ext e n s i o n

y* : z _., z

V

*

i s c o n t in u o u s fo r a ny v E X , an d v:(z) E R*(v,z)

holds (cp Theorem 3.1,a)). But the problem is that

V 1- Y~( Z)

is probably not measurable, and it cannot be expected that a countable set of such maps can be found.

Acknowledgements:

The author wants to thank Professors B. Fuchssteiner, 0. Moeschlin, and F. Stetter fo.r some conversations on the subject presented here.

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