On the continuity of the vector valued and set valued conditional expectations

VOL. 12 NO. 3 (1989) 477-h86

ON THE CONTINUITY OF

THE

VECTOR

VALUED

AND SET

VALUED

CONDITIONAL EXPECTATIONS

NIKOLAOS S.PAPAGEORGIOU

University ofCalifornia 1015

Department

ofMathematics

Davis, California95616

(Received May 12, 1988 and in revised form January 29, 1989)

ABSTRACT. Inthispaperwestudy the dependence of thevectorvaluedconditional expectation

(for

both single valued andsetvalued randomvariables),onthe a-field and

randomvariablethat determineit.

So

weprove thatit is continuousfor the

L1

(X)

convergenceof the sub-a-fields and of the randomvariables. Wealso presenta sufficient conditionforthe

Ll(X)-convergence

ofthesub-a-fields. Then we extend the work tothe

set valuedconditional expectationusingthe Kuratowski-Mosco

(K-M)

convergenceand

theconvergenceintheA-metric. Wealsoprove a property ofthe setvaluedconditional expectation.

KEY WORDS AND PHRASES. Conditional expectation,

LI(x)

convergenceof sub-a-fields, Hausdorffmetric, Kuratowski-Moscoconvergence, integrable selectors, ldstrom embeddingtheorem.

1980A.M.S.

CLASSIFICATION

CODE: 60Bll

1)

INTRODUCTION

Thepurpose ofthis noteistostudy thedependence ofthe vectorvaluedconditional expectation

(for

singlevaluedandmultivalued randomvariables),onthetwo quantities

thatdefineit. Namelyonthesub-a-fieldand onthe randomvariable.

Let

(f, ,/)

beacomplete probability spaceand X aBanach space. The

followingare wellknownfor X-valuedrandom variables:

(a)

If f

LI(x),

{]n}n>_l

is anincreasing

(decreasing)

sequence of sub-a-fields

and

Thistheorem,knowninthe literature as the

"Neveu-Ionescu

Tulcea theorem",can befoundinNeveu

[8],

propositionv-2-6. Itis aparticular case ofthe "martingale

convergencetheorem"

(see

Metivier

[7],

corollary

11.8).

(b)

If

fn

_s

f in

LI(x)

and

E

₀ isa sub-a-field of

E,

0

_L1

then E

fn

_s

E:E0f

in

(X).

Thisis aconsequence of the fact that the vector valued conditional expectationisa

continuous,linearoperator on

LI(x)

(see

Neveu

[8]).

Combining

(a)

and

(b)

above,it iseasytosee that the followingistrue:

(c)

If

{En,

}n>l

are sub-a-fieldsof

E

asin

(a)

and

fn

f in

LI(x),

En

_L1

then E

fn

EEf

in

(X).

Inthispaper,weexaminewhathappensifthe sequence

{l]n}n_

>

ofsub-a-fieldsof

isnotnecessarilymonotoneincreasing. Ourwork extends thoseof D.N. Nghiem

[9]

and Fetter

[3],

whodealtwithsingle valued randomvariables withvaluesin

.

2)

PRELIMINARIES

Let

(ft,

Z, tt)

be a complete probability space and X aseparable Banach space.

Wewillbe using the followingnotations"

and

Pf(c)(X)

{A

X: nonempty, closed,

(convex)}

P(w)k(c)(X)

{A c.C

X: nonempty,

(w-)

compact,

(convex)}.

Alsoif A

2X\{},

wedefine

IAI

sup

{llxll

x

A} ("norm"

of

A),

a(x*,

A)

sup

{x x)"

xe

A}

x eX

(support

function of

A)

andfor every

zeX,

d(z,A)

inf{llz-xll

xe

A}

(distance

functionfrom

A).

A

multifunction F:2

Pf(X)

issaid tobe measurable,iffor all zeX,

d(z,

F(w))

ismeasurable. Otherequivalentdefinitionsof measurability ofa

multifunctioncanbefoundin

Wagner

[14].

S

we willdenote thesetof integrable selectors of

F(.).

So:

By

S

{f(.)

e

LI(x):

f(w)

e

F(w) #-a.e.}.

\Ve

Itiseasyto show that S isnonempty if and onlyif

inf{llxll:

x e

F(w)}

L+.

function. In

L+-thiscase

S

#

O.

Alsousing

S

wecan define a setvalued integral for

F(.

by setting

Let I;

0 bea sub-a-field of

E

and let F: II

Pf(X)

beameasurable

multifunction s.t.

S

#

I.

Following Hial-Umegaki

[4],

wedefine theset valued

conditionM expectationof

F(.

wit5respect to

E

_0’ tobe the

E0-measurab/e

multifunction

E0F:

f

Pf(X)

forwhichwe have:

the closuretakeninthe

Ll(X)-norm.

If

F(.)

isintegrably bounded

(resp.

convex

valued),

then sois

EOF(.).

On

Pf(X)

wecan define a(generalized) metric, known as the Hausdorffmetric, by

setting:

h(A,B)

max{sup(d(a,B), aeA),

sup(d(b,A),

beB)}.

Recallthat

(Pf(X),

h)

is acompletemetricspace. Similarlyonthespace of all

Pf(X)-valued,

integrably bounded multifunctions, we can define a metric

A(.,.

be

setting

A(F,G)

JI-

h(F(w),

G(w))

d(w).

Asusual, weidentify

FI(.

and

F2(.),

if

Fl(W

F2(w

-a.e..

Thespace of

Pf(X)-valued,

integrably boundedmultifunctions,

togetherwith

A(.,.)

isacompletemetricspace.

A

multifunction M:

E

Pwkc(X)

isa setvalued measure, ifforall x eX

A

a(x

M(A))

is asigned measure.

Alsolet usrecallanotionofconvergenceof setsthat wewillbe usinginthesequel.

Solet

{An, A}n>l

c_g.

2X\{q)}.

Weset:

s-lirnA_n

{xeX:

x

n

_s

x,xn

An,

n

>_

1}

w

n <n

2<...

<n

k<...}-and w-Ii"

A

_n

{x

eX:x

k x,xk e

Ank,

Here s denotes thestrong topologyon X and w the weaktopology. Wesaythat

the

A

n’s

converge

to

A

isthe Kuratowski-Moscosense,denoted by

An

K-M

A,

if

Finally, letusrecall twonotionsofconvergenceofsub-a-fieldsof l]. The first was used

Ll(X)-converges

to

(denoted

by by Nghiem

[9].

So we saythat

EZnf

LI(x)

)

_{ifandonlyiffor every f}

weassociate

El]n(

e

.2’(LI(x)),

thenwesee thatthisconvergenceof the sub-a-fieldsis theconvergenceofthecorrespondingcontinuous, linearoperators inthe strong operator

topologyon

.(LI(x))

(pointwise

convergence).

Thesecondnotionwasused byFetter

[3].

Sowemay say that lim

Zn

if and onlyif liminfZ

n=

v

N

Z

n=limsup

k=l n=k k=l n=k

3)

SINGLE

VALUED CONDITIONAL

EXPECTATION

Let

(fl, l],

)

be acomplete probability space,

{E

n,

}n_>l

X aBanachspace.

LI(X)

;}

THEOREM3.1" Iflim

l]n

’

the___a

n

PROOF: Let K_n

k_V_n

El(

and L_n

kn

El("

Clearly andfor every n>_l, wehave L_n

c_.

l]n

c_

K

n.

Let fe

LI(x).

Wehave:

sub-a-fieldsof

E

and

Kn=

V

Ln=

n=l n=l

ii

E

nf_

EflII

<_

llEnf_

ELnflll

+

E ₁ L

[[EEnEKnf

EZnELnf[[1

+

liE

nf_

Ef[[1"

Recall that the vector valuedconditional expectation isan

Ll(x)--contraction.

So

we have:

K

L_n

[[EEnEKnf

El;nELnf[[1

<

lIE nf

-E

<

lIE

Knf

Ef[[1

+

[[Ef

ELnf[[1

So finallywehave:

l]

f

Lnf

Ef[[

₁

[IEKnf

ii

E

nf_E

II

I<_21IE

+

-Eflll

0 as n-fromthe

"Neveu-Ionescu

Tulceatheorem"

(see

section

1).

Therefore as claimed by the theorem.

THEOREM

3.2: If

Sn

and

fn

-s

f in

LI(x),

En

L1

PROOF: Forevery n>_l, we have:

llEEnf

n

Eflll

<

llEEnfn-

EEnflll

+

llEEnf-

Eflll

_<[If

n-fll

1+lIE

-E

I

Butby hypothesis

Ilfn

fll

0 and

IIEZnf-

Efll

0 as n

.

Thusfinallywe

have that

lie

nf

_n E

II1

0 as m (R).

Q.E.D.

Nextwewill determinea sufficientconditionfor the

Ll(x)--convergence

of a sequence

{En}n_>l

of sub-a-fields of

E.

Forthiswe willneed astronger hypothesisonthe Banachspace X. So assumethat X is strictly convex.

THEOREM3.3: If for every

AtE

and forevery xeX,

{EEn)A

(.)x}n>_

is

Ll(X)-convergent,

the.__an thereexists a sub-a-field of

E

s.t.

E

_n

.

E

n

PROOF: Forevery n>_l, set

Tn(f

E f,fe

LI(x).

Then T

ne

,2’(LI(X))

and

[ITn[

<

forall n>_l. Clearly from our hypothesis, given anysimple function

s(.),

{Tn(S)}n>_l

is

Ll(x)--convergent.

Weclaimthatforevery fe

LI(x),

{Tn(f)}n>_

is

Ll(x)--convergent

too. Tothisendthis

>

0 begiven. Wecan find

s(.

simple function s.t.

[[f--s[[

<

.

Alsothereexists n

o

>_

s.t.for n,m

>_

n_0, we have

IlTn(s

Tm(S)l

<

e/3.

So finally for n,m

>_

n

O wehave:

IlTn(f

Tm(f)[[1

_<

[[Tn(f)

Tn(S)l[1

+

IlTn(S) Tm(S)lll -[ITm(S

Tm(f)lll

_<

2

[If--sll!

+

IlTn(s)- T.,n(s)ll

<

e.

is

Ll(x)-Cauchy,

thusitconvergesin

LI(x)

to

T(f)

and

Therefore

{Tn(f)}n>_l

T

(LI(x))

(see

Kato

[5],

p.

150).

Itiseasytosee that

T(.)

isidempotent and

Ll(x)--contractive.

So invokingtheresult of Landers-Rogge

[6],

wededucethat there

o sta

i

s.t. T

,e.ce

i

the

theorem.

4)

S_ET

VALU..ED CONDITIONAL

EXPECTATION

Inthissection

(fl, E,/)

isacomplete probability space and X aseparable Banach space. Additionalhypotheseswillbe introduced as needed.

Westart with aninterestingobservationconcerningsetvalued conditional

expectations.

,

THEOREM

4.1: If X is separable,

Z

0 isasub-a-field of

E

and F:12

-

Pwkc(X)

is

integrablybounded,

the__n

E0F(w)

e

Pwkc(X)

#---a.e.-PROOF" Fromthe corollarytoproposition3.1of

[10],

wehavethat for all

A

E

_0,

M(A)

I

F(w) d/(w)

{I

f(w) d/(w)"

feS}

e

Pwkc(X).

Using theorem2.2of

A

A

Hiai-Umegaki

[4],

we have:

a(x

M(A))

A

F(w)) d/(w)

==

A-,

a(x

M(A))

_{isasignedmeasure,}

=

M(.

is a setvalued measure.

Applytheorem 3ofCost(!

[1]

toget G:9t

Pwkc(X)

E0-integrably

bounded s.t.

[

G(w)

M(A)

J

A

EEOF

d/t

)

j

--

A

(w)

A

a(x,

f()) d,()

(x, G()) d()

A

A

3₀

==

a(x

E

F(w))= a(x

G(w))

,

_*

for all w e

fi\N(x

), #(N(x ))

0

,

_{* *} *

Let

{Xn}n>

₁ bedensein X and let N= U

N(Xn),/(N)=0.

Let x eX n>_l

* *

pick

{Xk}k>_l

_

{Xn}n>_l

s.t.x

k x Thenfor w

Ct\N,

we have:

and

*

EZOF(w))

*

a(x,

a(x, G())I

*

EZOF(

*

ZOF(w)

*

_(

a(x,

w))

a(x

_k,E

+

(Xk,

G(w))

a(x,

G(w))l

0 as k-.

Sofor all we

\N,/(N)

O, we have:

*

EEOF(

*

a(x

w))

a(x

G

(w))

Nextwe willderiveasetvaluedversionof theorem3.2. Forthis we will need the followingsimplelemmata.

* LEMMA4.1" If

E

₀ isa sub-a-fieldof

E,

f

LI(x)

and v e

L(EO,

X

),

th-n

fl

(f(w),

v(w))

d/(w)

fl

(EE0f(w)’

v(w)) dt(w).

,

PROOF" Let

v(w)=

)A(W)x

with

AcE

₀ and x eX. Then wehave:

I

(E

Z0f(

w),

;IA(w)x) d#(w)

*

(E

E0

f(w),

x

d#(w)

fl A

I

EE0(f(

w

x*)

d/(w)

)A(W)

S

E0

(f(),

x

d(

A

fl

EEO(f(w)

*

Ifl

*

XA(W

x

)d/z(c,.,)

(f(w),

.yA(Ca)x

d#(ca).

Sothe lemma

,

istruefor countably valued

v(.

belongingintheLebesgue-Bochner

space

L(0,

X

).

Butfrom corollary 3, p. 42, ofDiestel-Uhl

[2],

weknow that these functionsare densein

L(R)(ZO

X

).

Soby a simple densityargument,weconclude that the lemma holdsforall v

L(R)(O,

X

).

n>

n

LEMMA

4.2: If

E

₀ isasub-a-fieldof

E,

fe

LI(z

0,

X)

and v

L(X*),

then

[

(f(),

v(w))

d()

[

(f(),

E

y0v())

d().

J

PROOF:

As

inlemma 4.1, we can check that the result holds for

f(.

being a function. Butrecall that simplefunctions aredensein L

I(E0,X).

Sothe lemma simple

follows by density.

FromDiestel-Uhl

[2],

theorem 1,p. 98, weknowthatif X isseparable,then

[LI(x)]

*

L(R)(X

*)

and if feL

I(x),

ve

L(R)(X*),

thentheirduality bracketsare defined

(f(w), v(w)) d/z(w).

by <f,v>

LI(x).

THEOREM4.2: If X is separable,

E

_n

,

F_nF: f

Pwkc(X)

are

integrably boundedmultifunctions s.t.

S

K-M

S

and n

isrelatively w-compactin

LI(x),

then

SIEnFr.,

K-M

$1^

EE

F as n-. n

PROOF: First let g S1" Fromproposition 3.1of

[10],

weknow that S1_F

EEF"

is w-compactin

Li(x).

So S1^

EES

Thus g

EEf,

f

S.

Sinceby hypothesis

S

K-M

S,

wecan find

fn

S

n>l s.t.

fn

f in

LI(x).

n n

Znf

El]f

in

LI(x)

aridfor every n>l E

nf

_n

Then theorema.2 tells us that E _n S l] Therefore we deduce that"

E

nF

_n

i-(1)

SEY,

F

c_

s-lim

SEl]nF

n

Next let ge

w-n

s

ll]nF

"r

_n Bydefinition, wecanfind gk S

lr’nkFr

_n s.t.

k

s,

w

gkg

in

LI(x).

Also

gk=E

kf

k,fk

eS

Sinceby hypothesis U

Fnk

n

k k>_l

w--compactin L

I(x),

by passing to a subsequence if necessary,wemayassume that

fk-w

f in

LI(x).

Since

S

K-M

..,

S,

wegetthat fe

S.

Let

v(.)e

LO(X

*)

n

[LI(x)]

*

and denoteby

<

.,.

>

thedualitybracketsforthe pair

(L1

(X),

LOO(X

*)).

Fromlemmata4.1 and 4.2 we have:

<E

fn,

v>=<E

fn,

El]nv>

<fn’

EZnv>"

Byhypothesis wehave

EZnv

Ev

in

LI(x).

Alsonotethat for all n_>l

l]n

Znllv

(w)

liE

v(w)ll

_<

S

-<

Ilvll(R)

_#-a.e.. Soinvoking lemma4.2of

[12],

we get:

Againthroughlemmata4.1 and 4.2we get:

<f,

El]v>

<El]f,

Er’v>

<EEf,

v>.

Thusfor every v e

L(R)(X

*

ILl(x)]

* wehave:

Enf

Ef

<E

n’V>-<E

,v>

Enf

EZf

in L

I(x)

and fe

S.

So g

EZf

S1A

EE

F.

Therefore wehave:

w-

S1EnFE

_c

S

I’EE

F

(2)

n

From

(1)

and

(2)

above andsincewealwayshave s-lim

S1EZnF

w-I]-S1.

E-nF

n n

we conclude that

SI,ZnF

K-M

$1"

Ey_,

_F as n-.

n

Q.E.D.

REMARKS:

(a)

If for all n>_l,F

n F: fl

Pwkc(X)

andisintegrably bounded,

then the hypotheses of theorem4.2are automatically satisfied. Similarly if forall

n>_

Fn(W

c__

W(w)

-a.e.

with W: f

Pwkc(X)

integrably bounded.

(b)

Conditionsthatguaranteethat

S

K-M

S

canbefoundin

[11]

(theorem

n

Finallyby strengtheningthehypotheseson

Fn(.)

and droppingthe separability

requirement on X wecanproveconvergenceinthe

A(.,.)

metric.

THEOREM4.3: If lim

E

_n and F_n, F:

Pkc(X)

are integrably bounded multifunctions s.t. F_n

_A

F,

the...__n

A(EnFn

E’F)

0 as n-.

PROOF: Using ldstrom’s embedding theorem

[13]

(theorem 2),

wecan embed

Pkc(X)

isometrically as a convex coneinaseparable Banach space

.

Then

Fn(-),

F(.

n>_l canbeviewedas

:-valued

randomvariablesand

EEnF(

),

EF(

are

Pkc(X)-valued,

integrably bounded. Then ifby j(.) wedenote the embeddingof

Pkc(X)

in

X,

theorem3.6of Hial-Umegaki

[4]

and theorem3.1ofthispaper, tell us that

Zn

_E

’F)

llj(EEnFn)-L

()

A(E

Fn,

J(E"F)II

0 asn-.

Q.E.D.

ACKNOWLEDGEMENT: Theauthoris indebtedto thereferee forhis suggestions andremarks thatimprovedthepresentation. This researchwassupported by N.S.F.Grant

2)

3)

4)

)

s)

9)

I0)

11)

12)

13)

14)

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