conditional expectation of integrands and random sets

2. Theory

Annals of Operations Research 30 (1991) 117-156 117

CONDITIONAL EXPECTATION OF INTEGRANDS AND RANDOM SETS

A. T R U F F E R T

A VA MA C, Perpignan Cedex 66025, France

The conditional expectation of integrands and random sets is the main tool of stochastic optimization. This work wishes to make up for the lack of real synthesis about this subject. We improve the existing hypothesis and simplify the corresponding proofs. In the convex case we especially study the problem of the exchange of conditional expectation and subdifferential operators.

Keywords: Integrand, random set, conditional expectation, selection, indicator function, support function, polar function, tangent cone, normal cone. directional derivative, subdifferential.

1. Introduction

Integral optimization originated with the work of Rockafellar on convex polarity of integral functionals [37,38] (see also the synthesis in [39]). Stochastic optimization followed as the concept of conditional expectation of an integrand was cleared up (always in the convex case). The first results have been obtained in 1972 by Bismut [3] and Valadier [9, chapter 8] the first, by using the existence of linear positive lifting (see Ionescu-Tulcea) and the second by applying existence results on multivaried conditional expectation to an epigraph. One also obtains, in this way, the results of Daures [10], Castaing and Clauzure [8].

The decisive step was taken by Hiai [22,23] (see also Hiai and Umegaki [24]) who proved the existence of conditional integrand without convexity in the framework of integral representation theory. By a completely different method, Thibault [43] extended this result by establishing the existence of the conditional expectation of a vector-valued Lipschitzian integrand. The evolution of representation theory for globally defined non-convex integral functionals (Klei [28], Bourass and Truffert [5], Foug+res and Truffert [19], and Castaing [7]) allowed the discovery of existence conditions for conditional integrands (Klei [28], E1 Bouamri and Foug~res [13] and Truffert [44]). One of the drawbacks of the preceding works is the diversity of definitions of a conditional integrand and the various hypotheses of existence which are not obviously comparable. Recent results of Foug4res and Truffert [20], on local integral representation, will lead us to a unified approach of conditional integrands under almost optimal existence hypotheses.

�9 J.C. Baltzer A.G. Scientific Publishing Company

118 A. Truffert / Conditional expectation of integrands

Following Hiriart Urruty's work [25] on the mean value functional (1.1), in the case of the coarse o-field, here we present a survey of the concept of conditional integrand with the aim of improving the existing hypotheses and simplifying the corresponding proofs. We introduce the conditional core linked to an indicator integrand and point out the often unnoticed role it plays. In the convex case we make explicit, through support functions, the relationship between the conditional expectation of a random set and the conditional integrand.

In the first section we gather many previously unknown properties of non-convex conditional integrands, then in the second section, among other things, we treat in the convex case two well known major problems (Bismut [3], Valadier [9, chapter 8], Rockafellar and Wets [41] and Truffert [45]), the exchange of conditional expectation and subdifferential operators, and the closure of the image of the set of selections of the exact and approximate subdifferentials under the conditional expectation operator.

1. Non-convex conditional integrand

PRELIMINARIES Let us consider (~2, d , P), a probability space, ~ a sub-o-field of ~r X,

some Banach space with norm l" ], and a Borel o-field ~'x. We denote by A~176 ') (resp. ~oo(~)) the set of the ~r (resp.

~-measurable) functions from ~2 to X and by L ~ r and L ~ their quotient spaces for the equality a.e.

Any function f : ~2 x X ~ - ~ which is ,~r174 ~x-measurable is called a ~-integrand. If f(to, �9 ) is also lower semicontinuous (1.s.c.) (resp. continuous, convex, proper . . . . ) on X, for P-a.e. to ~ ~2, then f is said to be lower semicontinuous (resp. continuous, convex proper . . . . ).

The l.s.c, regularization f of an integrand f is nothing other than the pointwise regularization, that is, for each to ~ ~2,

f ( to , x) = f - ( - ~ = lim inff ( to , y ) . y " ~ X

This is a .~-integrand, where ~ ' denotes the P-completion of ~ ' . To each ~r f , we associate:

- The integrand operator (also denoted by f ) :

I: u - o f ( u ) = f ( . , u(.))

- The integral functional associating with each u ~A~176 Ij(u) equal to the superior integral of f ( u ) (also denoted by f d f (u ) d P).

- The epigraph:

epi f = ((to, x, t) ~ ~2 x X X R: f(to, x) <~ t} ~zar174 ~ x | ~a .

A. Truffert / Conditional expectation of integrands 119

For any sub-a-field N of zar we denote by A~+ (resp. A~-) the set of the functions u of &~176 for which the positive part f ( u ) + (resp. the negative part f (u ) - ) of f ( u ) is P-integrable. These two sets are N-decomposable; we recall that a subset D of .SPx~ ') is N-decomposable if for any u, v ~ D and for any N-measurable set A, we have:

u l A + vl A, ~ D.

We denote by A T the union of A% and A~-.

1.1. CONDITIONAL INTEGRAND

Let f: ~ ? • R be a zae-integrand. For any u~&~176 f (u) + (resp. f (u) - ) admits a unique ~-conditional expectation E~e(f(u) + ) (resp. Ee( f (u ) - )) (see for example [34; I-2.9]); in particular if u ~ A~, then E~(f (u) + ) or Ee( f (u) - ) is integrable and so

E~(f (u) ) = E~( f (u ) +) - E~( f (u ) - )

is the unique (for P-a.s. equality) ~-conditional expectation of f (u ) . We call a conditional ~-integrand of f any ~-integrand g: f2 • X --* R satisfy-

ing the following property:

Vu~A-~, g ( u ) - - E ~ ( f ( u ) ) P-a.s. (1)

Remarks g is sometimes referred to as the conditional expectation of f but the same

terminology is also used for the integrand whose epigraph is the conditional expectation of the epigraph of f ; we will see in section 2 that the two notions differ. So we prefer to call g the "conditional integrand of f " ; this also avoids any possible confusion with the conditional expectation of the function f (u ) .

If f admits g as a conditional ,~-integrand, then: (i) The integral functionals associated with f and g, coincide on Aft. (ii) A~.c A~g~ and A ~ c ag-. ~

The introduced notion is "stable", that is, a ~-integrand is its own conditional ~-integrand.

In the case ~ = (t~, ~} and f(~o, x)-integrable for each x ~ X, the "mean value function" Ef of f , defined on X by:

El(x) = fs~ f(~o, x) dP(~o)

is a conditional ~-integrand of f .


1.2. P-DISCRETELY DENSE SET

1.2.1. Foug4res introduced the topology of P-discrete conoergence: when X is equipped with the discrete topology, it is the topology induced by convergence in probability.

Ps u,, ">u ,=, P(u,, e: u) ---, 0.

A subset H is dense in s for the topology of P-discrete convergence if for any u of 5~176 there exist a sequence (u,,) in H and a sequence (B,,) in of P-covering of ~ such that:

uaB, ,=u,,18' ' for any n ~ N .

Next we will see that it is sufficient to have property (1) on a subset H of A'y which is P-discretely dense in Z#~

Recall that (B,,) ~ ~N is a sequence of P-covering of ~. if it is increasing and if P ( I2 \ u , ,B, , ) = 0 .

Before giving some examples, let us remark that a subset of 5 ~ 1 7 6 is P-discretely dense in s176 if and only if it is rich in the sense of Valadier (see [6, chapter 1.2]).

Examples (1) Any subset containing a P-discretely dense subset of ~ . o ( ~ ) is itself

P-discretely dense in ~'~ (2) The intersection of two ~-decomposable P-discretely dense subsets of

ZP]?(N) is P-discretely dense, as soon as it is non-empty. (3) The space ,._9~ ') of ~-measurable and essentially bounded functions is

P-discretely dense in s More generally, for any ~-random radius fl ( that is B: ~2 ~ R* ~-measurable), the space of density fl

is P-discretely dense in 5~176 as u 0 ~5~ as soon as u 0 is .~-measurable.

Another important example is given by the following result, (see the end of section 1.2 for the proof):

LEMMA

Let f be a ~'-integrand on f2 x X, such that A y is non-empty. Then A T- is P-discretely dense in zpo(..~) if and only if f takes P-almost surely its values in

To determine a conditional integrand, we can limit our study to a P-discretely dense subset of A T in ~oo(~):


PROPOSITION

Let f : ~ x X ~ N be a Jae-integrand. If there exists some P-discretely dense subset H of A~ in $a~ and a ~-integrand g from ~2 • X to It~, such that for any u ~ H, g(u) is a ~-condit ional expectation of f (u) , then g is a conditional N-integrand of f .

Proof Let u ~ A ~ . There exist a sequence ( B , , ) ~ N of P-covering of ~2 and a

sequence (u,,) in H, such that:

ulB, ' = U , , 1 B , ' for any n ~ N.

So for each fixed integer n, we deduce

f(U)IB~=U(u,,)I~, ' and g(u) le ,=g(u , , ) lB , ' P-a.s.

Combining these equalities and [34, I-2.9] (see lemma 2), we obtain

Eee(f(u))lB,, = Ee~(f(u,,))lB,, = g(u,,)lB, ' P-a.s.

and thus

Eee ( f (u ) ) lB ,=g(u ) lB , P-a.s.

So, g(u) and E e ( f ( u ) ) coincide P-almost surely as P-a.s. limits of the sequences ( E ~ ( f ( u ) ) l e , ) and (g(u)lB,,).

1.2.2. As an example, we are going to determine the conditional integrand of an affine integrand.

We suppose that the dual X* of X is separable; we denote by (. I ' ) the bilinear form of duality. If c~ is a sCrandom radius, we denote by ~,},~(sr the space of functions v such that av ~.,wX.(sr

THEOREM

(i) Let /3 be a N-random radius, v ~ S ~ and u 0 ~ f f ' ~ The affine integrand:

( U o - - I v ) : (,o, x ) ~ ~ x x - - , ( .0(~0) - . ~ I v ( , o ) ) E ~

admits as conditional N-integrand the following function:

( . 0 - �9 q e T v): (,o, x ) ~ ~ x x - ~ ( . 0 ( , o ) - x i e ~ ( / 3 v ) ( ~ ) / / 3 ( , o ) )

(ii) If X = N, then (u o - �9 ) �9 E % is a conditional N'-integrand of (u o - - ) . v as soon as v is quasi-integrable.

Proof Since (u 0 - u[ v) is quasi-integrable for any u ~ u 0 + ~ o , ~ ( , ~ ) , which is P-dis-

cretely dense in Sa~ it is sufficient by 1.2.1 to show that for any u ~ ~ ' / 3 ( 5 ~ ) , (u[Ege( f lv ) / f l ) is a conditional expectation of (u I v). Since u/ f l ~ . o ( ~ ) ,


fly ~.s162 and (u I ES~(flv)/fl) is equal to (u / f l lEa( f lv) ) . We deduce (i) and (ii) from the following results:

LEMMA 1 [9, CHAPTER 8, 3.3] The map E~: L~-.(..~') 1 ---, L x . ( ~ ) and the embedding map, from L~(N') in

L~?(s~'), are mutually transpose.

LEMMA 2 [33, CHAPTER 1, 2.9] For any quasi-integrable function v of .L,a~ and any u Es176 u. E~

is a conditional expectation of u. o.

Proof of lemma 1.2.1 Let us suppose that A'~- is P-discretely dense in ~ ~ and the domain ~

of the ~'-random set,

too = {(~0, x ) ~ ~2x x: f ( w , x ) = - o e } c ~ ' | ~ x ,

admits a non-null probability; from [9, chapter 2.2], there exists a ~-measurable function ho~ such that:

V,o ~ boo f( ,o , ~ ( , o ) ) = - oo.

So we deduce the existence of some ~'-measurable set f2o~ with a non-null probability and a ~-measurable function uo~ such that:

By hypothesis, there exist a sequence f2,,) of a P-covering of f2 and a sequence (u,,) in A y such that:

u j ~ , , - u,,l~,, Vn ~ N.

From the choice of the ~,,, there exists an integer p for which ~ o c~ ~2. still admits non-null probability, and we have

G = - o o

This contradicts the fact that up belongs to Ay-. Conversely, let us consider u e&~176 and the following sets

~,, = (o: ~ f2: f ( w , u ( ~ ) ) >/ - n } .

They are measurable and form a sequence of a P-covering of g. Let u 0 be an arbitrary point of Ay . We put

u,, = ul~,, + u01~; ~ A~-

The sequences (a2,,) and (u,) answer the question.


1.3. UNICITY OF CONDITIONAL INTEGRAND

THEOREM Let f: f2 • X ~ IR be a ~-integrand satisfying the following conditions:

(i) f admits a conditional ~-integrand, (ii) A'~ is P-discretely dense in Z~~176

If g and k are two conditional ~-integrands of f , then they coincide P-almost surely: that is, there exists a P-negligible set N such that g and h coincide on (~2\N) • X.

We can immediately deduce this theorem from the following result:

LEMMA Let g and h: ~2 x X ~ R be some ~'-integrands. If there exists a P-discretely

dense subset H of ,LP~ such that

VB~,VuEH s h(u) dP~ fBg(u)dP,

then there exists a P-negligible set N such that:

h~<g on (~2\N) • X.

Remark The lamina still holds if we replace P by a a-finite measure. Before proving it

in this more general case, we give a very important result showing that the operator of conditional expectation respects the order of the integrands; many "boundedness" properties for the integral functionals result in growth conditions for the associated integrands [18], and the conditional integrands acquire similar properties.

COROLLARY Let fl, f2: ~?x X ~ be two sr admitting Eefl

conditional ~-integrands. If Aei, N A e;: is P-discretely dense in s176176 ), then

fl ~<f2 P-a.s. ~ E~fa <~ E~f2 P - a.s.

and Eeef2 as

Proof of the lamina Let (~?, ~ , P) be a o-finite measure space, H a P-discretely dense subset of

o~~ and g and h two ~-integrands satisfying the hypothesis of the lemma. Let ~ be the P-completion of ~ ; H is P-discretely dense in ~ x ~ and h and g considered as ~-integrands still satisfy


"h(u) f * V/}~ ~ , Vu ~ H f~ dP~< g (u ) d P .

so we can suppose without restriction that the o-field ~ is P-complete. Let us suppose that the domain of the ~- random set

r = {(,~, x ) ~ S ~ x X: h(,~, . ,-)> ~(,~. x)}

adn~ts a positive measure. From [9; chapter 2.2] we can find a ~-measurable function u. such that h(~, u(w)) > g(w, u(w)) for each ~ ~ dora F. Since H is P-discretely dense in 5 ~ 1 7 6 there exists a sequence (u,,) in H and a sequence (B,,) of a P-covering of ~ such that: u18, ' = u,,1R, ,, Vn ~ ~ . Let us consider, for each integer n, the following set:

,Q,, = {~0 ~ B,, n dom F: g(w. u(w)) ~< n i l (w)} ,

where fl is a positive integrable function. If ( B, ~ is a o-finiteness sequence, there exists an integer n 0 for which ~2 ~ ~,,,, n B, ~ admits a finite positive measure, since (~,,) is a sequence of a P-covering of dom F. From the choice of g2,,, we have for each w ~ ~2~

Since f,,*,,g(u,,,,) d P < + oo, we o b t a i n :

h(u,,,,) de > f,, g(u,,,,)dP.

This contradicts our hypothesis, since u .... belongs to H.

1.4. EXISTENCE OF CONDITIONAL INTEGRAND

After the works of Hiai [22,23], several authors developed the integral representation theory of non-convex functionals [28.20,7] and its repercussions on the existence of the conditional integrand of a l.s.c, integrand [28,13,44].

Combining a local integral representation theorem, obtained by Foug6res and Truffert [20] with proposition 1.2.1, we give a general existence result under regularity assumptions less restrictive than the one given in the quoted works.

Given a .~-random radius a, a subset D of ..,qo~.~(.~) is called a o-decomposable if for any sequence (u,,) in D and any ..~'-measurable partition (A,,) of $2, the function u = E,,u,,1A, ' belongs to D as soon as it is in s

A function F defined on a o-decomposable D of &~ containing the origin, is called a d-functional measure if

Vu~ D A ~ d ~ F(ulA) is a measure.

A. Truffert / Conditional expectation of mtegrands 125

From this definition, we deduce that an integral functional Ii associated with a J- integrand f on f2 • X is a measure functional on D if f is null at 0 and if D is contained in A~.

F is said .Y-ff-l.s.c. if for any u ~ D and for any sequence (u , ) in D, we have:

lim,, II( u,, - u ) /c~ lu = 0 ~ F( u ) ~< lim,, F( u,, ).

Before stating the main result of existence, recall the local integral representation theorem obtained by Foug~res and Truffert.

LEMMA I20, SECTION 3.3, THEOREM B] Let cr be a aCrandom radius, D a o-decomposable of L-a~'"(:,~') containing the

origin, and F: D ~ IRa .,~-measure functional Jo~-l.s.c, There exists a l.s.c. (null at the origin) ~'-integrand jr on [2 • X whose

associated integral functional I/ coincides with F on D.

Remark The integrand f given by the lemma is the normalization of the Radon-Niko-

dym field associated with F [20, section 2,2]; so it is especially a l.s.c, aM-d-integrand, that is, for any positive real r, the function ( ~ o , x ) ~ • infl.,_.,, I <r,~,~lf(o0, y ) is a ,,~'-integrand [17, section 3.2].

THEOREM Let f : ~2 • X ~ IR be a aC-integrand, null at the origin. If there exist a

aM-random radius ~ and a o-decomposable D of ~.~./l(aM) containing the origin, such that: (i) D is P-discretely dense in Z,'~',~.(aM), ( i i ) D c A~, (iii) 1i is 3-~-l.s.c., then f admits a unique l.s.c, conditional aM-d-integrand.

COROLLARY: [42. SECTION 2.2] Let f be a l.s.c, proper .~-integrand on (2 x X. If there exist a aM-random

radius/~ and u 0 ~~ ), satisfying (i) f(uo) is integrable, (ii) V r > O, infl.~_,,,a.)l <-,nf(', x) is integrable, then f admits a unique l.s.c, conditional aM-d-integrand.

Remarks (1) When f is minorized by an integrable function or more generally, when f

admits as affine continuous lower bound ( - i v ) + a, with v ~..L,~ and a integrable, hypothesis (ii) of the corollary is satisfied. So we rediscover, in the first case, results of E1 Bouamri and Fougdres [13] and [28] and, in the second case, the result of Hiai [22].


(2) When /3 is identically equal to 1, (ii) corresponds to the quasi-integrability assumption of Thibault [43, section 2].

(3) If P is atomless by [18], (ii) is equivalent to (ii') ~.~.~'~(.~') c a-~.

(4) We will see (1.6.l) that the conditional integrand given by the corollary is proper. Conditions (i) and (ii) are thus sufficient to have the existence of a proper 1.s.c. conditional integrand; conversely, if f = -q0 where cp is a convex continuous positive radial ,~-integrand. These conditions are necessary from [44, section 2.7].

P~vof of the theorem The restriction of 1"/, to D is a ~-measure functional fff-l.s.c.; from the

preceding lemma, there exists a l.s.c. ~-integrand g such that I/ and I~ coincide on D. In particular, for any ~'-measurable set B and any u ~ D, ul 8 ~ D, and we have:

f /I . ) dP= fBgl l de, since f and g are null at 0. g(u) is a conditional expectation of f (u) and the theorem therefore results from 1.2.1 and 1.3.1, since D is P-discretely dense in

Proof of corollary Let D =SCaV'"(..~); D is obviously a a-decomposable of 5 ~ ') which is

P-discretely dense. Let us consider the variation of f at u 0,

fo: (w, x) ~ ~2 • X--*f(w, x + u0(r - f ( w , Uo(W)).

By (ii), D is contained in A~,. and I/,, is ~ by [4, section 3.1]. So fo admits an unique 1.s.c. conditional ~-d-integrand go. Let us consider

g: x ) s2 • g0( , x - +

where a o is a conditional expectation of f(u0); g is still a 1.s.c..~-d-integrand and it is the unique conditional ~-integrand of f .

1.5. STABILITY BY OPERATIONS

Combining the general existence theorems of the preceding section with some results of "stability by operations", we obtain a new existence theorem without regularity of the considered integrand. We have seen that the general notion of conditional integrand, introduced in 1.1, is "stable". That is, any ~-integrand is its own conditional ~'-integrand. An integrand obtained as the sum, product or composition of a ~-integrand and a regular ~-integrand is generally not regular. So it is important to know when these operations are stable by transfer to the conditional integrand.


The first result, concerning the conditional integrand of a sum, is easily obtained from 1.2.1.

1.5.1. Conditional integrand of a sum

PROPOSITION

Let f, g: ~ x X---, R be some :aC-integrand admitting respectively E~f and E~g as conditional ~-integrands. If A~n Ag c~ Af+g is P-discretely dense in &oo(~), then E~f+ Eag is a conditional ~-integrand of f + g.

1.5.2. Conditional integrand of a product

PROPOSITION

Let f: ~• be a .~r admitting E~f as conditional ~-integrand and g: ~2 • X - , R a ~'-integrand. If A~c~ A~nA~.g is P-discretely dense in , ~ o ( ~ ) then (E~f ) .g is a conditional ~-integrand of f . g.

Proof Let w ~ a-~ C~ A~ C~ AT.g, the second assertion of 1.2.2 applied to .f(w) ensures

that g(w). E~(f(w)) = g (w) - (E~f (w) ) is a conditional expectation of f (w) . /~C~A~ is supposed P-discretely dense in &oo(~), the g(w). Since A'~N,~g ,~/.g

proposition results from 1.2.1.

1.5.3. Conditional integrand of a composition

Let us consider X and Y, separable Banach spaces, and ~ x and ~ y , their Borel o-fields. If f is a ~r on ~2 x Y, and g: ~2 x X ~ Y is a ( ~ | ~x , N'r)-measurable function, we denote by f . g the function defined on ~2 x X by:

f.g(~0, x ) = f ( ~ o , g(o:, x ) ) ~ •.

It is still a ~-integrand satisfying the following equality:

AT.g= { U ~ ~ g(u)~A~) ,

in particular A~.g. (resp. AT.g-) is P-discretely dense in s as soon as A% (resp. A%) is P-discretely dense in ~ o ( ~ ) .

Consequently, we immediately deduce the following result of the definition of the conditional integrand.

PROPOSITION Let f: ~2 x Y ~-~ be a sr admitting ETf as conditional ~-in-

tegrand and g: ~2 x X--, Y a ( ~ @ ~ x , ~y)-measurable function. Then Eeef. g is a conditional ~-integrand of f . g.


Example Let us consider u ~s and B ~.5~,o(,~,). The integrand

(~. x ) ~ n • x--,f(,~,/3(,o)..- + .( ,~))

admits as conditional integrand

(,0, x) ~ f2 • X--, Eef(co, /3(co)x + u(~o)).

1.5.4. Combining the properties of the conditional expectation of a function [34, section 12-10] with 1.2.1, we easily obtain:

PROPOSITION Let ( f , ) be an increasing sequence of d-integrands on f2 • X and (E~f,,) a

sequence of associated conditional .~-integrands. If X'~,,_ is P-discretely dense in ~ o ( . ~ ) then sup,,E'-'~f, is the unique condi-

tional ~-integrand of sup.f , .

1.6. REGULARITY OF THE CONDITIONAL INTEGRAND

The aim of this section is to bring out the main properties of regularity preserved by the conditional integrand. For most of them, we will use the fact that the conditional integrand operator respects the order of the integrands.

1.6.1. Domain properties

For any ~-integrand f on ~2 • X, we call the domain of f , dora f = {(~0, x ) ~ ~ x X: f(~o, x ) < +oo} .

It is a ~-random set of X (that is, dom f ~ s r 1 7 4 ~'x). Its closure,

d o r a / = {(~o, x ) e ~ X X : x e c l , ( d o m / ( w , - ) ) } .

is a closed ~-random set of X. We denote by L~ the set of the ( P-classes of) ~r selections u of dora f :

VwP-a.s. (~o, u ( ~ 0 ) ) ~ d o m f .

PROPOSITION Let f be a ~r on ~2 x X, admitting E ~ f as conditional integrand. If

the following conditions, (i) A~. is non-empty, (ii) A~- is P-discretely dense in .~,ao(~), are satisfied, then E~ is proper.

If, moreover, there exists a ~-random radius/3, such that (i ') Xy, n.s176176 is non-empty, (ii') s is contained in X~,


then dom E~f is contained P-almost surely in d o m f .

Proof The condition (i) ensures that E~f is not identically equal to + oo. On the

other hand A~ V- contains A~ and so it is P-discretely dense in ~9~176 from (ii). The properness of E~f therefore results from 1.2.1.

If (ii') is satisfied, I Z and IE, / coincide on ,L*~ ') and we deduce from [5, section 2.5] the following inclusion:

0 L~'#(~) N L~ L~ '~ (~ ') n L~-~E,,/( ~ ) c

By [19, section 1.4.4], there exists a sequence (v,,) of selections of dora E~f belonging to L~ '~ (~ ) such that the countable random set U ,,Gr v,, is dense in dom E~f. From the preceding, the v,, are some selections of d o m f and we obtain: dora E~f = U ,,Gr u,,c d o m f P-a.s.

1.6. 2. Some elementaop properties

P R O P O S I T I O N

Let f : ~2 ~ be a d-integrand, such that _A~- is P-discretely dense in ~o.o(~). We assume that f admits a conditional integrand E~f.

If f is convex (resp. even, odd, positively homogeneous, sub-linear) so is E~f.

Pt-oof Let us show the result for convexity. (The proof is similar for the other

properties.) If f is convex, then we have "q~0P-a.e., 'qa ~ [0, 1], Vx, y ~ X:

f ( ~ , c~x+ (1 - ~ ) y ) ~ < ~f (w, x) + (1 - ~)f(~o, y) . (2)

Using the functions g and h defined on ~2 • [0, 1] • X z X by

g( ,~ , ~, x , y ) = ~ x + (1 - ~ ) y e x .

h(~o, u, x, y )= af(w, x )+ ( 1 - a ) f ( w , y ) ~ ] - o o , +oo] ,

relation (2) becomes:

f.g(~o, a, x, y) <,%h(o3, a, x, y). (3)

From the results of section 1.5, f . g and h admit respectively E~f .g and (~, a, x, y) ~ aE~ x) + (1 - a)E~f(~, y) as conditional integrand.

- 0 Since AT.g is P-discretely dense in ,LPt0,11•215 ) we deduce from section 1.3.3 and (3): VwP-a.s., Va ~ [0, 1], Vx, y ~ X:

E ~ / ( ~ . ~ x + (1 - . ) y ) ~< ~ , E ~ / ( . , , x ) + (1 - ~ ) E ~ U ( ~ , y ) .

1.6.3. Upper sernicontinuity

Let f be a ~-integrand and u 0 ~ ~ 1 7 6 such that f(uo) is integrable. As shown by the following example, f(~o, �9 ) could be upper semicontinuous (even


continuous) at u0(co ) for P-a.s. co ~/2 , although its conditional integrand is not:

f : (to, x ) ~ ] 0 , l] X R - - + 0 i f x ~ < 0 and l x if x > 0 . d=.~' lO.1 l Ca)

and ~ ' = { q~, /2 } ; E e f = 6=- is not upper semicontinuous at the origin.

In [18], Foug~res shows that if f(co, �9 ) is P-a.s. upper semicontinuous at Uo(W ) then there exists a .saC-random radius a such that the (..~-measurable) function co ~/2--+ sup i.,._,,,,oo)1 <,~(,,)f(w, x ) i s P-integrable. a is called an u.s.c, radius o f f ag u o.

The converse property is obviously satisfied, when f is convex and lower semicontinuous; in the preceding example a ( ~ ) = ~0 is an u.s.c, radius of f at 0. On the other hand, we easily verify that f does not admit some constant u.s.c radius at 0. This implies the non-continuity of ES~f at the origin.

P R O P O S I T I O N

Let f be a .~'-integrand on /2 • X, admitting E ~ f as conditional integrand; we assume A~ P-discretely dense in .L,('~ Let u 0 ~ s

If f admits at u 0 an u.s.c .~-random radius, then so does Egef. If, moreover, f is convex, lower semicontinuous and satisfies the assumptions

of corollary 1.4, then Eeef(w, .) is continuous at Uo(OO ) for P-almost surely oa ~/2.

Remark In the case where X is the (separable) dual of a Banach space (for example, X

reflexive) the preceding result extends the one given by Daur~s [10].

Proof of the proposition If f admits at u 0 an u.s.c. ,~-random radius /3o, there exists an integrable

function a, satisfying

Vco P-a.s., V x ~ X, f ( to , x) <~ SB{ ..... l~,,)(to, x) + a(oa),

where B(u 0, /3o) is the following .~-random set:

{(,o, x)~/2• Ix-Uo(<O)I--</30('~)}. From the corollary in section 1.3, Ee~f satisfies the growth condition

Voa P-a.s., V x ~ X, Eef( to, X) <~ 6B(uo.&,(oa , x) + Eea(co),

and /3o is therefore an u.s.c, radius of ES~f at u o. Under the additional hypothesis, E~f is proper and 1.s.c by 1.4 and convex by

1.6.2. From the preceding, it is continuous at u o.

1.6.4. Coercivi(v and in f-compactness

A ~-integrand f on /2 x X is coercive [16, section 1.3] if it satisfies

'v'~o P-a.s. inf f(o0, x ) > - oo .x-~ X


and

lim f(02, x) > 0. I , l ~ Ixl

When f is minorized by a P-integrable function, the coercivity of f is also equivalent to the existence of some zC-measurable, positive and P-integrable functions c~ and ~r, for which f satisfies the following growth condition [18] (see also [15, section 2.4] and [16, section 1.3])

Vo~ P-a.s., Vx ~ X: f(02, x) >/a(02)Ix I - ~r(02). (4)

As a consequence of the corollary in section 1.3, we establish that the conditional integrand of f is also coercive:

PROPOSITION 1

Let f be a sC-integrand on ~2 x X admitting a conditional integrand. If f is coercive and minorized by an integrable function, so is its conditional integrand.

Proof Let g(~, x): = a(02)Ix I-7r(02), where a and 7r are given by (4) from 1.5.1

and 1.5.2,

E~g(02, x) = E ~ ( 0 2 ) t x 1 - e~"(02)

is a conditional ~'-integrand of g. Since the conditional expectations Eea , Ee~r of a, 7r are still positive, the

coercivity of Eel results from the corollary in section 1.3.

If, moreover, f is assumed convex, f is coercive if and only if for any (P-a.s.) co, f(o2, �9 ) is minorized and inf-bounded [16, section 1.4]; so we can deduce an inf-compactness result when X is reflexive:

COROLLARY

We assume that X is a reflexive separable Banach space. Let f be a convex 1.s.c. ~aC-integrand, minorized by an integrable function. When f is weakly inf-compact, so is its conditional integrand.

Proof Under the assumptions of the corollary, by 1.4 f admits a unique conditional

.~-integrand E~f which is still convex by 1.6.2, lower semi-continuous, coercive and minorized by an integrable function. So Ee~f is inf-compact since X is reflexive.

When X is finite dimensional and ~ = {~, ~2}, Hiriart Urruty [25, chapter II-3] (see also [53]) established inf-compactness of Ef (see remark 1.1) without


supposing f minorized by an integrable function; following the same approach, we next show that the result still holds for the conditional integrand.

Given a eaC-integrand f on ~2 x X, let us first characterize the conditional integrand of the asymptotic function of f , defined on a2 x X by (see [32, section 6.8.31):

f~(~0 x) = sup f(~o, ;kx + u(w)) - f ( , 0 , u(,0)) , ~ ,

x>0

where u is an arbitrary selection of the domain of f .

LEMMA Let f be a convex 1.s.c. d-integrand on $2 X X, satisfying:

(i) there exists u o ~Z-a~ for which f (uo) is integrable; (ii) X~ is P-discretely dense in ,~o (~ ) .

If f admits a 1.s.c. conditional integrand E~f , so does its asymptotic function and we have:

E~(foo) = (ES~f)o~ P-a.s.

Proof Let (2,,,) be an increasing sequence of positive reals, such that lim,,2t,, = + oo.

Let us consider

f(~o, 2t,X + Uo(W))-f(~o, u0(~o)) f,,(,o, x ) = x,,

From 1.5.3, f , admits

E~f(~o, 2t,x + Uo(~O)) - E~f(co, Uo(~O)) E~f,,(w, x )= X,,

as conditional integrand, and so we obtain

f~ = sup f,, and (E~f)oo = sup E~L,. t l n

Since A~ is P-discretely dense in .E~176 so is A~ and the lemma therefore results from 1.5.4.

PROPOSITION 2 On top of the assumptions of the preceding lemma, we suppose X finite

dimensional and the existence of u~ ~.L~~ for which f(uoo) is P-integrable. Let v 1 ~ x l . ( d ) .

If f is inf-compact for the slope vl, then Eeaf is inf-compact for the slope E~v].


Proof The functional f(w, .) is inf-compact for the slope vl(oa ) if the function

defined by h(w, x)=f (~0 , x ) - (xlvl(w)) is inf-compact, in other terms, if for any X ~ JR, the recession cone of the set

r~h(,o, . ) = ( x ~ X : h(,o, x ) ~ X }

is reduced to (0} (since X is finite dimensional). From [32, section 6.8.4], ToH~(w, .) is the recession cone of each of the

non-empty sets Txh(~o, �9 ). So inf-compactness of f is equivalent to the positivity of h,= on/2 x(X\{O}).

By construction of /7 we have

h,o(w, x) = f~(~ , x) - (x I v,(<.<,)). Since the intersection of the .~-decomposable A~- and .s176 ) is non-empty,

it is P-discretely dense in Z,O~ So we deduce, from 1.2.2 and 1.5.1, that h~ admits

e~(/,~)(<,,, .,-)= E~(/~)(<o, x ) - (x I E%,(<,,)) as conditional integrand. By the preceding lemma we obtain

Vco P-a.s., Vx :# 0 (Eef)oo(o:, x) - (x I E%l(,o)) > 0,

which ensures the inf-compactness of E e f for the slope E~av,.

1.7. CONNECTION BETWEEN CONDITIONAL INTEGRAND AND CONDITIONAL LAW

Let us say that a transition probability (w, A)--+ P(A I~) from (/2, ..@) to (/2, .sac) is a family of conditional laws if

VA E d , V B E ~ , P(A r iB )= foe (A I ,o )e (do , ) . (5)

This formula extends to ,7< /2 • ---, [0, + oo] .~ |

Let f be a !.s.c. zae-integrand on /2 x X, satisfying assertions (i) and (ii) of corollary 1.4. The following function

Ef: (.,, x)~/2xX--.ff(~, x ) P ( d g l , o ) ~ ] - m , + ~ ]

is called the ~-conditional expectation o f f (relative to conditional laws P(-l" ))-

THEOREM

Let f be a 1.s.c ~r on /2 • X satisfying conditions (i) and (ii) of corollary 1.4. The ~-conditional expectation of f is a conditional ~-integrand of f ; in particular, Ef is lower semicontinuous.


Proof (a) Let us first suppose that f is non-negative: the .~| ~x-measurability of

the conditional expectation Ef results from [33; chapter 3, 2.1] by considering the transition probability defined on (Q • X) •162 and independent of x, Q • X is equipped with the product o-field ~ | ~'x.

Let u ~ L ~ and B E ~ ; by applying formula (6) to q~,.B(~, ~ ) = f(~, u(o~)) �9 1s(w ) we obtain:

SiI<" So for any ~-measurable function u, Ef(u) is a conditional expectation of f(u) and Ef is therefore a conditional ~-integrand of f.

(b) When f only satisfies (i) and (ii), by [44, section 1.4] there exist a Young's ,~r q0 and a P-integrable function a, such that:

'q'w P-a.s., Vx E X f( , . , , x) >~ - cp (~ , x) + a ( ~ ) .

Let g(w, x) = f ( w , x) + qo(~, x) - a(w); q0 and g are non-negative so by (a) Eq~ and Eg are their conditional integrands. By 1.4, f admits a unique conditional integrand E ~ f and we deduce from 1.5.1

Eg = E~ f + Eep - Ea,

where Ea(~) = faa(~)P(d~]~) is a conditional expectation of a (see [11, section 9.1]. On the other hand, we have

Eg = Ef + Eep - Ea,

and we deduce that Ef and Ea~f coincide almost surely.

1.8. CONDITIONAL INTEGRAND OF INDICATOR FUNCTION: ~-RANDOM CORE

When ~ = (~, ~2} Hiriart Urruty [25, chapter 20] introduced the notion of continuous intersection of a random set /':

Nr w ~ $2 \N

where JV" denotes the family of the P-negligible sets of ~r He notably showed that its indicator function is nothing other than the mean values function (1.1) of the indicator function of F. He also expressed its support function and established its connection with the expectation of a random set.

The aim of this section is to extend this concept when ~ is some o-field of ~r in order to express the conditional integrand of an indicator integrand. We will also study in section 2 (see section 2.1.1, corollary 3) the connections with the conditional expectation of a random set.


Recall that a subset l` of s • X is a ~-random set of X, if l` belongs to ~ | N x. If for P-a.s. to, the set

r(to) = (x x: (to, x) r ) is closed (resp. convex, non-empty . . . . ), l` is said to be a closed (resp. convex non-empty . . . . ) ~ - random set of X.

A function u: f2 ~ X, is a selection of l`, if:

VtoP-a.s. u(to) ~/'(to). We denote by L ~ r (resp. L ~ the set of the P-classes of ~r (resp.

N-)measurable selections of l`.

THEOREM Let F be a closed ~r set of X.

(i) There exists a largest (for P-a.s. inclusion) closed N-random set Feel` of X contained P-almost surely in l`.

(ii) FeeF is the unique closed N-random set of X such that 8F~r = Eee(6r) and therefore L ~ = L % r ( N ) .

Feel" is called the N-random core of l`. It is non-empty as soon as l` admits a N-measurable selection. Using the results of 1.6 and (ii) we can see that FeeF generally inherits the properties of F: if l` is a convex (resp. c o n e . . . ) set, so is Feel`.

Proof L ~ is o (L~ (see [19, section 3.1]): for any sequence

(u . ) of N-measurable selections of F and any N-partitions (B.) of s E,,u.IB. is still a N-measurable selection of F.

On the other hand, L ~ is closed for the essential convergence if u. ~ L ~ and ql u , , - u[ l~ ~--y:~ 0 then u is a N-measurable selection of F, since F is closed.

According to [19, section 3.2] there exists a unique closed N-random set Fe~F such that L ~ L % r ( N ), this is the essential supremum of the family of graphs (Gr u),~L%ee) (in the sense of Valadier [46, section 1.4]). From [46, section 14], there exists a sequence (u , ) of N-measurable selections of F, such that the countable random set A = U ,Gr u, is dense in F e E and FeeI` is thus P-almost surely contained in l` which is closed.

2. Conditional integrand and convex analysis

Let us consider (~2, ~ , P ) a probability space, N a sub-o-field of F, X some Banach space with separable dual X* and (. I" ) the bilinear form of duality; we keep the notations of section 1.


2.1. POLARITY

2.1.1. Conditional integrand of support function; conditional expectation of random s e t

Following the extension of concepts of measurability and integrability to random sets (see, for example, Castaing and Valadier [9] and Wagner [52]) several authors, such as Van Cutsum [50,51], Valadier [47-49], Neveu [34], Daur+s [10], Dynkin and Evstigneev [12], Kozek and Suchanecki [31], Hiai and Umegaki [24], have studied the notion of conditional expectation of a random set.

Let F be a d - random set of X, such that the set of all its (P-a.s.) integrable selections L~.(z~'), is non-empty. We can define the ~-conditional e.xpectation of F as tile essential supremum of the family of graphs (Gr E~u) , ,~ L',I.~,) (in the sense of Valadier [46]), that is the smallest (for P-a.s. equality) closed M-random set of X, denoted by E~F, such that for any tt ~ Ltl.(~'), E~u is a selection of E~F (see, for example, [48,49]). From {46, section 14], we deduce in particular the existence of some sequence (ltn) of ,zC-measurable and P-integrable selections of F such that the countable random set tO ,,Gr E'~u,, is dense in E~F, which is thus a ~-d-random set of X, that is, the distance function defined by:

dE*r: (to, x ) ~ I 2 • inf I x - . v [ c ~ E "~ l '(t~ )

is a ~-integrand [17, section 3.2.3]. With the preceding definition we can show [20, section 5.2] that ES~F is also

the unique closed ~-random set of X, such that

L~,r( ~ ) = cl E~ Llr( d ) ). (7)

where cl designs the closure for the norm of Llx($2. ~r P). So we find the same notion as Hiai and Umegaki [24].

Finally, when /" is convex and closed, E~3F can be characterized by its support fimction (see, for example, [3,48,10,24]). In the same way we will show that the support function of E~F is the conditional integrand (introduced in section 1.1) of the support function of F. Recall that the support function of a ~ (resp. ~r set F

87: (to, x* ) ~ /2 • X* --+ sup ( x l x * ) .vc F(~o)

is a ,sr (resp. ~r [39,17].

T H E O R E M

Let F be a ,zC-random set of X such that L ~ r ( d ) is non-empty. The support function of E ~ F is the unique conditional ~-integrand of the support function of F:


Since the support functions of two sets C and C ' coincide if and only if C and C' have the same closed convex hull, we have:

COROLLARY 1

Let F be a ~- random set of X such that L ~ ( d ) is non-empty. Ee(-doF) and -coESSF coincide almost surely.

Before proving the theorem, we give two other corollaries which extend results obtained by Hiriart Urruty [25, section 35] in the particular case ~ = {~, ~2}.

For any convex closed ~'-random set i v of X, we denote by iv* its polar set, that is, F* = {(,0, x * ) ~ $2 • X*: 8~-(w, x * ) 4 1}, which is a convex closed Ja~-random set of X*. We denote by Jr the gauge of iv defined on ~2 • X by: jr(~o, x) ~ inf{c >~ 0: x ~ c/'(co)}.

Recall that Jr is the support function of the polar set of F; so it is a ~-integrand whose conditional integrand is obviously given by the preceding theorem:

COROLLARY 2 Let F be a convex closed ~- random set of X containing 0. The gauge of /"

admits a conditional ~-integrand which satisfies:

E ~ ( j r ) =J}s'(r*~)* P-a.s.

COROLLARY 3

Let F be a convex closed cone (with 0 as origin) d - r andom set of X. We have the following P-a.s. equalities: (i) F~(F *) = ( E ~ F ) * P-a.s. (ii) Ee(F*)=(FeeF) * P-a.s.

(iii) E ~ ( j r ) = j F , ~ r P-a.s.

Proof Since F is a convex closed cone of origin 0, so are F*, E e F and FeF. Now,

using theorems 1.8, 2.1.1, corollary 2 and the following equalities, the conclusion is easily reached.

= - . ) = E (Sr) = 8F, r =

and

= E (sr.) = =

Proof of the theorem The support function of iv is null at the origin and minorized by an integrable

function; so from 1.3.1, it is sufficient to verify that the integral functionals


associated with 8~- and 8e,, r coincide on L . v , ( ~ ). From [39], we have the following equalities:

I8~ (8,.) and I8,;, ' = (Is~_,,r)

The polarity is taken in the duality (L~(, L~.. ). For any u E L ~ . ( ~ ) , we have:

I * (~:,,,,) (u)= sup (v u) t ,~ L~,q.( ~ )

= sup (E:ew, u) (1) u,~ L'r(~)

= sup (w, u> 1.2.2 w~ L~.(.~)

= (18,.)* (u).

2.1.2. Conditional integrand of a conjugate function

Let f : ~XX--*-~ be a ~-integrand; its conjugate function in the duality (X, X*), defined on O x X * by:

f* (w, x * ) = sup ( ( x l x * ) - f ( w , x)) , x ~ X

is a ..~-integrand [36] and a .~a~-inte~rand [17] if, moreover, f is a lower semicontinuous d-d-integrand. Given a ~-integrand g on ~ • X*, we can form by a similar operation its conjugate function g*, which is still an ~-integrand. So we will denote f * * the conjugate function of f *

We will see that the conditional expectation of the epigraph of f * is the epigraph of the conjugate function of the conditional integrand of f .

THEOREM

Let f: ~ 5< X ~ [ - oo, + oo] he a ~r If there exists some u~ ~a~( .~ r for which f (u l )+ is integrable then f *

admits a unique conditional ~-integrand E ~ ( f * ) . If, moreover, there exists some vo ~ s (~) for which f * ( v 0) + is integrable

then E~( f* ) satisfies the following equality:

ep i (E-e ( f* ) ) * = c-oE-e(epi f ) P-a.s.

The previous result allows us to compare the conjugate function of the conditional integrand with the conditional expectation of the conjugate function:

COROLLARY 1

Let f be a convex 1.s.c. proper ag-integrand on /2 > X. If there exists some u~ ~&~ and v 1 ~s r for which f(u~) and f*(vl) are integrable then


f admits a unique conditional .~-integrand ES~f which satisfies, for any zaC-measurable and integrable function v: (i) (Eeef)*(E~v) <~ E a ( f * ( v ) ) P-a.s.

In particular, we have: (ii) (E~f) * <~ Ee~(f *) P-a.s.

Generally, the conditional integrand of the conjugate function does not coincide with the polar function of the conditional integrand. In other terms, the last inequality can be strict. For example, let us consider some indicator mapping of a closed convex zae-random set of X which contains a N-measurable and integrable selection. From theorems 1.8 and 2.1.1, we have:

*

(E Be) =6~,,r and Ee~(6~-)=Se*,,r

and E~F and FeF cannot be equal without the coincidence of f ' and Eal " [21]. If we apply this result to the conjugate function of a convex 1.s.c. proper

~-integrand, we obtain an inequality of Jensen's type (see Kozek and Suchaneki [31, section 7.1]):

Vu~.LPxl(Z~r f(Eeeu)<~Ea(f(u)) P-a.s. (8)

As a consequence of the previous property, we establish that the restriction of the conditional expectation operator to an integral space L~ is still a linear contractant projector of norm 1 when q0 is a Young's ~-integrand; so we obtain an extension of Neveu's result where cp was supposed without parameter [34, chapter 9 23].

Recall that q~: 12 • X---, [0, + ~ ] is a Young's ~-integrand if it is a convex, 1.s.c., even, coercive and continuous at the origin ~-integrand. L~o is the linear subspace of L ~ generated by A ~ + and equipped with the gauge of the convex { 1~ ~< 1}, denoted by I1" II ~-

COROLLARY 2

Let us consider a Young's ~-integrand cO on s • X, such that L~ is contained in L~r(s~r E e is a linear contractant projector of norm 1 on L~.

Proof Let w ~ L~; for any X > 0 such that q~(Xw) is integrable, we obtain from (2)

applied to q~ and u = Xw:

~ ( . , E ~ ( ? , w ) ( . ) ) = ~ ( . , X E % ( - ) ) ~ E ~ ( ~ ( ., Xw(-)))

and consequently the following inequality:

Ir(XEe(w)) <~ I~(Xw).

In par t icular , f r o m X = 1/11 w II ~ we obta in : II g~w II ~ ~ II w II ~.

140 A. Truffert / Conditional expectation of mtegrands

Proof of the theorem (i) Existence of E ~ ' ( f * ) : f * is the composition 1.6.3) of 8r / with g:

( ~ o , . x - * ) ~ w x X * - - - , ( x * , - 1 ) ~ X * x R [9. chapter 12]. Because e p i f is a za/| ~R-random set of X x [R which adnfits ( lq. f ( t q ) + ) as integrable selection, 8" , is, from 2.1.1. a conditional integrand of 8~*p~/ and from 1.6.3 we deduce E cpi f that 8t*%pi/.g is a conditional ~-integrand of f * .

(ii) From Fenchel-Young's inequality

V~ e e, Vx* ~ X* f* (w, x * ) > / (u , (~ ) Ix*) - f (w , u,(w)),

so A; . contains L ~ . ( ~ ) which is P-discreteIy dense in ~.0 ( ~ ) and the uniqueness results from the theorena in section 1.3.

1 E%pi f is still an epigraph because, if (u,,, c~,,),, is some sequence of Lepi/(,5r such that u , G r E~(u , , c5,) is dense in E%pi f then we have:

E~'epi/(~o) c lx{ ( ~" :~,, = E ( , , , ) ( ~ ) ,.o . E ( < , ) ( ) ) . , , e r ~ }

= cl,. ( ( E ~ ( , , , ) ( , ~ ) . e"~(,~,.)(,~) + , . ) . ,, ~ r~. , .~ Q ) .

so E'%pi f is the epigraph of some l.s.c..~-d-integrand h and we have

E"'a(f*) = 8 " , . = E cpi f g h*

and

epi( E:~"(f*)) = epi h* * = co epi h = ~ { E-'%pi f )

since (-Ivo) - E:~"(f*(v0) + ) is a continuous affine function less than h.

Proof of corollao ~ 1 From tile theorem applied to f * , f admits a unique conditional ~-integrand

satisfying:

epi (E '~f) = E~X(epi f * ) P-a.s. (9)

Let v ~ 5q~. ( ~ ) and /2~ = { ~o ~ s E"'*( f * ( t, ))( ~, ) + < + c~ }. To show (i), it is sufficient to verify that the following inequality,

is satisfied for any ~-measurable set B: If B meets ~2~, (10) is obviously satisfied; otherwise, let us consider for each integer n: ~,,= {~oE B: E ~ ( f * ( v ) ) ( w ) ~<n},

On = Olga,, -t- t.;ll_gal i

and

o# = vl a + Vlla,.

Since (v,,, f * (u , ) ) is an integrable selection of epi f * , E~(epi f * ) admits (E%,,, E~'~(f*(u,))) for selection. Because ( E v B, E ~ ( f * ( v n ) ) ) is the P-a.s.


limit of this sequence, it is a selection of epi (E~[)* from (9) and the inequality (10) is satisfied.

Since 5~ ( ~ ) is P-discretely dense in SPa. ( ~ ) , (ii) immediately results from the lemma in section 1.3 and the definition of E ~ ( f *).

2.2. SUBDIFFERENTIAL CALCULUS

For any convex l.s.c. ~J-integrand f on $2 x X and any u ~o_90~!(~4) we call the subd(fferential o f f at u, the following closed convex ~- random set of X*"

of(.) = x * : x * ) =

A function c, of L!~v.(~J) is a selection of ~f(u) if t,,(~0) is almost surely a subgradient of f (w, .) at the point u(co):

V x e X f(~o, x ) - f ( ~ , u ( o ~ ) ) > / ( x - u ( ~ 0 ) l t , ( ~ o ) ) .

Moreover, if f is a ~-d-integrand then 0f (u) is a ~JC'-random set. 2.2.1. In [3, section 3] and [9, chapter 81 Bismut and Valadier studied the

subdifferential of the conditional integrand for an essentially bounded ~-measurable function. They used the subdifferentials of the integral functionals If and I E ,/ in the dualities < L~( x,r ), Lk.. ( ~.~ )) and { L~f( ~ ), LIx . (~ ) ) . They supposed that 1/ is finite and continuous at a point of L ~ ( N ) and proved the following equality:

Olt!,f(u ) = Ee(Ol[(u)) + As(~)l,:.V(u)), (11)

where for any closed convex set C, we denote by As C the recession cone of C, that is the greatest cone A such that A + C c C , also equal to r where x is an arbitrary point of C; from this fact, if F is a closed convex non-empty J - r andom set of X*, then

A s F = ((co, x * ) ~ 2 • x ' G A s F(~o)}

is a ,xC-random set of X*. Using Rockafellar's results on polarity [37], relation (11) disintegrates and

becomes:

= + (12)

We will see that the previous formula is still satisfied even if u is not essentially bounded. On the other hand, we will point out that the hypothesis of continuity can be replaced by any condition which ensures the exactness and the lower semicontinuity of the infimum convolution of the conjugate functions of I/ and 8LI:~ ~ in the duality (L~,(,,r L ~ ( , ~ ) * ) .

For any function F defined on L~.(,~r we denote by F * (resp. F * ) its conjugate function in the duality (L~, L~*) (resp. (L~ , L~..)). Recall that the dual space of L ~ ( , ~ ) is the direct sum of L~,.(sr and the space 5 p of the linear


continuous singular forms on L~(~r the conjugate function of an integral functional is given by [38]:

(/i) (0+')=/:'(0)+

We denote by W/~ (resp. ~f* ) the set of all the functions such that ( I / ) * VS~ca ) (resp. I/.V6L, Z(~) ) is exact and | (resp. * o)-lower semicontinuous.,.7., We can

express the support functions of L ~ ( ~ ) by using (L~( .~)) ~ and (L~(~ ' ) ) • (orthogonals of this set, in the corresponding duality) which are the kernels of the conditional expectation projectors E ~ and E a ~174 the latter is also equal, from 1.2.2 lemma 1, to the transpose function i s of the embedding map from L ~ ( ~ ) to LT(~').

Finally, for any ~r f on ~2 x X, we denote by ~ the following subset of 50:

THEOREM Let f: ~2 • X --* R be a proper convex 1.s,c. JaC-integrand, such that there exist

some uo~ ~s and ol ~.s162 for which f(uoo) and f ( v , ) are integrable, and u ~ Z # ~ such that f (u) is integrable.

We have the following equalities:

n V7 = : (

COROLLARY 1 Under the assumptions of the theorem, if, moreover:

A + L~ (,~) is a closed linear subspace of L~ (,.~'), (13)

where A denotes the positive cone generated by A-~" (in particular, if If is finite and continuous in L~(,~r at a point of L~(~ ' ) ) , then

Proof Attouch-Brezis's assumption (13) ensures the exactness and the lower semicon-

tinuity of the infimum-convolution of ( I i ) ~ and 8~7(~) for any point of L~x. (~r [2], in other terms, Vp = L~. (~r Then, we easily deduce the corollary from the following lemma:

A. Truffert / Conditional expectation of integrands

LEMMA 1 Under the assumptions of the theorem, we have:

Ee(Llalo,) (,sd)) C L~.~,/(,)(~).

143

Proof Any integrable selection v of 8f (u) satisfies:

VwP-a.s.,VxcX f(w, x ) - f ( w , u(co))>/(x-u(~o)lv(w)). Let us consider h the affine integrand (. - u I v); since A~- contains u +,LP~'(~)

it is P-discretely dense in s176 and we obtain from the corollary in section 1.3 and 1.2.2:

V~oP-a.s., Vx ~ X: ES~f(~o, x) - E~f(~o, u(co)) >i (x - u(~) l Ee~v(~o)), so E~v is a selection of OEe~f(u).

Applying the preceding corollary to the support function of a closed convex zaC-random set of X, we obtain a formula similar to the one given by Valadier [9, chapter 8 and 3]:

COROLLARY 2 Let F be a closed convex d - r andom set of X* such that L~r(ea r is non-empty.

If x A ~ + + L ~ ( ~ ) is a closed linear subspace of L ~ ( d ) , then

L ~ , . r ( ~ ) = E~(Llr(s~)) + Lk~E~r)(~).

Proof Let f be the support function of F; from 2.1.1, the support function of E~I" is

the unique conditional ~-integrand of f . Because the subdifferentials of f and E ~ f at the origin are equal to /7 and E~F , the preceding result is an easy consequence of corollary 1.

Valadier gave a version of theorem 2.1.2. The following more precise form illustrates the previous result:

COROLLARY 3 Let f : $2 X X ~ - ~ be a proper convex 1.s.c..~r which satisfies the

following properties: (1) there exists some vl ~s162162 such that f * ( v l ) is integrable; (2) I! is finite and continuous in L~(~r at some point voo of L ~ ( ~ ) ;

(3) dora E ~ f = dom f P-a.s.


For any V ~ x . x ~ l , . . . . . . . . . . o . . . . x - ~ ] , . , , - ~.,,,,~,,,,~,,.,, ,.,.p.,.t,,..,,, equal to v, such that: (i) ( E~f)*(v) = E~e(f*)(O), (ii) I j . (O) = Min{ I / . ( w ) : w ~ L ' v . (~ ' ) and E e w = v}.

Before giving the proof, recall that the recession cone of the epigraph of a proper convex l.s.c. ~-integrand h is the epigraph of the asymptotic functional 1.6.4 of h [33, section 6.8.3] which can be characterized by the following equality [32, section 6.8.5]:

I,~ = 62~,, ,,.. (14)

Proof Let h be the support function of the epigraph of f * . Since It is finite and

continuous at v~o, Ih is finite and continuous at (v~, - 1). So from corollary 2 and 2.1.2, we deduce the following equality:

1 . ~ 1 1 Lcpi,eV,.(N') = E ( L C p i / . ( d ) ) + LA~.,r ). (15)

By applying (14) to f * and ( E e l ) * (see the preliminary remarks) we can deduce from (3) that the recession cones of the epigraphs of f * and epi(E~ coincide almost surely. From this (15) becomes:

1 = E e ( 1 1 ( d ) + , (d)) Lepi J * LAsl epi J *

and so

L:p,E..t,. ( ~ ) = E~(L~ , / . ( ~ ) ) .

( ]6)

(17)

Let v ~5~ If (E~f )*(u) is not integrable, then (i) is satisfied for each P ~ 2 ~ . . ( d ) with conditional expectation equal to v from 2.1.2, corollary 1. Otherwise, (v, (E~f)*(v)) is an integrable selection of epi( E~f )* and from (17), we deduce the existence of some D satisfying the first assertion of the corollary.

On the other hand, we obtain from 2.1.2, corollary 1:

I/ .( (~) = I, e v , . ( v ) <~ inf{ I j . (w) Lw~ L'x.( d ) and E'~ = u},

and ~ also satisfies the second assertion.

To prove the theorem we use a technical lemma characterizing the set of integrable selections of the recession cone of 8E~f(u).

Recall that the recession cone of a non-empty closed convex set C of X* can be characterized (see for example [9, chapter 1, 7]) by:

AsC = (dom 8~. )*.

In particular, if C = 8F(x) , then the support function of C is the lower semicontinuous regularization of the directional derivative of F at the point x [32, section


6.4.8]; the closure of its domain thus coincides with the tangent cone of the domain of F at the point x (denoted by T(x, dom F)). In this case As C is therefore equal to the normal cone of the domain of F at the point x, denoted by N(x, dom F).

When u is essentially bounded, Valadier [9, chapter 8, 38] proved the following equality:

tk~,aE ,,,, , , ( ~ = A s ( / & . f , . ) ( ~ ) ) .

In the case where u is only ..~-measurable, we have:

L E M M A 2 1 Let w ~ L ~ . ( ~ ) ; w belongs to LAs(aE:~/[u))(,.~) if and only if ( u l w ) is

minorized by an integrable function and satisfies:

P l ' o o f

Let w ~ Llx. (..~) such that (u I w ) is minorized by an integrable selection, then

fo.(ulw) d P = lim s + uja,~ w) dP

for any sequence (fa,,) of a P-covering of a2; choosing (fa,,) such that ula, ' E L~(..~) for every integer n, we obtain

fn(u I w) d P ~ 8,(~f+ n L?~{~)(W ) . (18)

1 t If w ~ LA.~OE~,/O,,(~ ), then (u ' [w) ~< ( u l w ) P-a.s. for any u ~ ay, n because for P-a.s. co, ,,(co) belongs to the normal cone of the domain of EUf(co, .) at the point u(co); so ( u l w ) is minorized by the integrable function ( u~ I w) and in (18) we have an equality.

Conversely, if we have an equality in (18), then

~ ( u l w ) d P = sup f ( u ' ] w ) dP,

because I/ and IE, / coincide on L ~ ( ~ ) . The conditions for "interchange of sup and integral" being satisfied [37,6], we deduce the following P-a.s. equality:

VwP-a.s. (u(co) I w(co)) = sup (x lw(w)) . x ~ d o r a E:'tf(~a, . )

Hence, w is a selection of the recession cone of 3E~ef(u).


Proof of theorem (1) Let t ,~L~E,/ (ul ( ,~)C~Wf*; if f ~ * ( ( u [ v ) - ( f ( u ) ) d P = + ~ , the result is

obviously reached because v = E % is a selection of 3 f (u ) . Otherwise, let w + s realize the exactness of the in f imum-convo lu t ion at v;

then i| and v = E~(v+ w)-E~w. To conclude, it is sufficient to 1 verify that v + w is a selection of Of(u) and - E ~ w ~ LA~OE 90 ,~ ( ' ~ ) n i |

(a) We begin by proving the following equality:

f~ 8% . ( ( u [ v ) - f ( u ) ) d P = I / . ( v + w ) + :v~+nL.~(e,(s)~N (19)

We have:

(zj)| | - = V'8c~ .~ ( v )

= (;i+

= - inf ~ [ ( f ( u ' ) - ( u ' [ v ) ) d P u ' ~ L~:( ~ ') J,f2

= - inf fs2(eSef(u')-(u ' lvl)dP u'~L~:

= s d P [37]

= ~ ( ( u [ v ) - f ( u ) ) dP,

because v is a selection of 3E~ef(u) and f(u) is integrable. (b) By (19), (u I v) and f * ( v + w) are integrable; so (u I w) has an integrable

non-negat ive par t f rom Fenchel ' s inequali ty; thus we deduce f rom 1.2.2:

fs(ul-w)dP=s +oo. (20)

So ( u l w ) is integrable and we obta in f rom (19) and (20):

s * s alP. (21) ( f ( v + w ) - ( u l E % ) ) d P ~

From this fact, v + w is a selection of Of(u) and

8~r , n t ,~(j , ( s ) = ~ ( u [ , - E~Sw) dP= 65A7. ~,_~,~,(i| (22)

and we conclude by combin ing (22) and l e m m a 2. (2) Conversely, let us consider

1 1 i | . vEE (La/,,,,(si'))+LA~,~eO.(,,,,(N)n (5"/e)


1 There exists some w ~ LA~{OE,*/(,,~(~) A i|176 and some z with null conditional expecta t ion satisfying:

t, - w + z is a selection of 3f(u). (23)

F rom lemma 1, v - w is a selection of 3Eef(u) and so is v; thus it is sufficient to verify that v ~ ~7/*" since w ~ LA.,OE,/I,,~)(N'), f rom l emma 2, ( u l w ) is minorized by an integrable funct ion and

~ ( u ]w) d P = 3'e*A,, n L?(~21(W) - (24)

On the other hand w ~ i| so there exists a singular l inear cont inuous form s such that its restriction to L~(. .~) coincides with w and satisfying f rom (24)'

~ ( u ]w) d P = 8.~< n L ~ , j , ( S ) . (25)

We can suppose without restriction that the,..:.,infimum-convolution of ( I f ) | and

8L~;c~ is finite at v. Let t h +s~ ~ ( L ~ . ( ~ ) ) ~ ) such that v+v~ + . h is in the domain of (lj.) | From (23), we have:

f*( t ,+ z - w ) + (u[w) <~ f*(v+v~) + ( u l z - v , ) , (26) ~3.S.

and, from (25) and (26), we deduce

-o~ < ( I / ) | s ) d P

If,(c, + vl) + ~(u! z - v,) dP.

Since (u l z - t , l ) has an integrable negative par t and z has a condi t ional expecta t ion equal to O, we have also:

~< sup ( u', - eevl )

| < G;: nz_~ ~,~(sl).

Hence, x - w + s realizes the exactness of the in f imum-convo lu t ion at v. F rom (26) and (23) we obta in the following equalities:

I i . ( v + z - w ) + ~ ( u l w ) d P = fa ( u l v + z ) - f ( u ) d P

=s ((uIv)-f(u)) dP,


because z has a conditional expectation equal to 0. Thus the infimum-convolution coincides with f~((ulv ) - f ( u ) ) d P (see (1, a) above) and so it is lower semicontinuous at v.

To verify the following inclusion:

it is sufficient to remark that any point realizing the exactness of the infimum- convolution of I / , and 6~{~) also realizes the exactness of ( I / ) | | V6L~t~ ). The proof is then similar to (1, a).

(4) Lemma 1 ensures an inclusion. For the other one, we must verify that any v of ESe(L~/I,)(~)) belongs to ~Tfl. For such v, there exists some w with null conditional expectation such that v + w has an integrable negative part and so we have from 1.2.2:

f E'f(.)dP+ff*(v+w)dP=f(.Io)dP. Since v is a selection of 8E~f(u), we deduce the following equality:

ff*(.+w) dP= f dP. For any z with null conditional expectation, we obtain by 2.1.2 corollary 1

* f / ff*(v+w) de= (E~f ) (v+E~z)dP<... * ( v + z ) d P .

The infimum-convolution thus coincides with f~*((ulv ) - f ( u ) ) dP, and, so, it is exact and lower semicontinuous at v.

2.2.2. Commutation between expectation and subdifferential operators," directional derivative

Let h be a convex function of X such that h(Y) ~ R. We denote by h'(Y; x) the directional derivative of h at Y for the direction x:

h'(Y; x ) = inf h ( Y + ~ x ) - h ( ~ ) ~,>0

For ~ fixed, h'(Y; .) is convex and positively homogeneous. The notion of directional derivative is closely connected to the one of subdifferential, as shown by the following equality [32, section 6.4.8]:

(h ' (Y; .))* = 6ah(.~). (27)

Let f : 9 X X--* R be a convex ~-integrand and u ~ , o ( ~ ) such that f (u) takes P-almost surely real values.

We call the directional derivative of f at u the map associating to each (~o, x) ~ t2 x X the directional derivative of f (w, �9 ) at u(~o) for the direction x.


That is a convex, positively homogeneous ~-integrand and a ~r if, moreover, f is a 1.s.c. d-d-integrand.

We denote by d ( f ; u) the lower semicontinous regularization of the directional derivative of f at u. The closure of its domain is equal to the tangent cone of the domain of f at u:

T(u, dom f ) = {(w, x ) ~ f2 • X: x ~ r(u(w), dom f ( ~ , . ))},

admitting for polar set (2.1.1), the normal cone of the domain o f f at u:

N(u, dora f ) = ((co, x* ) ~ $2 • X*: x* E U(u( t0 ) , dom f ( w , .))}.

When 0f(u) admits a selection, we deduce from the relation (27):

el(T; u) = 8~(,,) P-a.s. (28)

As a consequence of theorem 2.2.1, we give some characteristic conditions allowing commutation between conditional expectation and subdifferential operators.

THEOREM Let f: $2 • X-~ ~ be a proper convex 1.s.c. ,,~-integrand such that there exist

some u~ ~2~o~(~,) and v 1 ~&~ for which f (u~) and f*(vl) are integrable. We suppose that ~/* is equal to L l x , ( d ) .

Let u be a N-measurable function such that f(u) is integrable and Of(u) admits an integrable selection.

The following conditions are equivalent:

(i) OE~f(u) = E~(Of(u)) P-a.s.

(i ') J(E~f; u ) = E ~ ( d ( f ; u)) P-a.s.

(ii) As(OEef(u)) = As(E~(Of(u))) P-a.s.

(ii ') dora d(E~f; u) = dom E ~ ( c ] ( / ; u)) P-a.s.

COROLLARY Under the assumptions of the theorem, if we suppose that one of the equiv-

alent following conditions is satisfied:

(a) T(u, dom E-~f) = F~(T(u, dom f ) ) P-a.s.;

(b) N(u , dom E~f) =E~(U(u , dom f ) ) P-a.s.,

then E ~ f ( u ) and ~EC~f(u) coincide P-almost surely.

Remark The conditions, (a) and (b), are sufficient, but not necessary as shown by the

following example.


h(co, .)

i/~ / i

1 X (-2,o)

(1,2)

0,0)

Fig. 1. Commutation example without sufficient conditions.

Let I2=]0,1[, ,xg=~10,q , ~ '={4, ,I2}, X = R 2 and let f be defined by: f(w, x, y ) = h ( ~ , x)+6r~)(x, y) where, for fixed co, h(~, -) and F(o~) are represented by fig. 1. f admits ~{(x.y):3y~<2x+4and-2~<x~<0} as conditional integrand. Conditional expectation and subdifferential operators commute even though

T(0, dom Eef) c_F~(T(O, dom f ) ) = {(x, y) : y>~x and y>~0}.

The preceding example also proves that the cones T(0, Fedom f ) and F~(T(0, dom f) ) do not always coincide, since:

T(0, F e d o m f ) = {(x, y)" y>~2xand y>_-0}.

Proof of corollary The equivalence of (a) and (b) results from 2.1.1 corollary 3. Let us show that if

(a) is satisfied, then so is condition (ii') of the theorem. From 1.6.1, we have:

T(u, dom f ) = dora aT(f; u) D dora E ~ ( d ( f ; u)) a . s .

and thus

F~(T(u, dom f ) ) z dom E ~ ( g ( f ; u)). a . s .

From (a), we deduce the following inclusion:

dom(cT(Eef; u)) ~.dom Ee(d(f; u))

and that (ii') is satisfied because the other inclusion is always true.

Proof of theorem (i) ~ , ( i ' ) : Two closed convex ~-random set~ coincide P-almost surely if their

support functions are identical. The support function of E~(3f(u)) is, from 2.1.1,


a conditional N-d-integrand of the support function of 0f(u) , so the equivalence results from (28) applied to f and E'ef.

(ii) ~ (ii ') is immediately reached from the preceding, since the sets interven- ing in (ii') are the polar sets of the recession cones of (ii).

To conclude we must prove that (ii) implies (i). Since Lae~U~,,)(~) is closed in L~( . (~) , we deduce from 2.2.1 lemma 1"

c

On the other hand, by (ii) we have from theorem 2.2.1:

1 1 L~e.q~,,)(~ ) c LE.,~at(,,,(~ ) + LA~E,a/O,,(~' )

1 C LE:qa/(,,~)(~),

so ~)E~/(u) and Eg(Sf(u)) coincide P-almost surely, because they are the same integrable selections.

2.2.3. Conditional closure

In the optimal recourse problems, it is very important, notably when construct- ing dual solutions by some dynamic process, to know when the equality

L~e ,/,u, ( N ' ) = E e ( L~I(,,, (at ' )) (29)

is satisfied (see for example [40,41,45]). Equation (29) is equivalent to commutation of conditional expectation and

1 subdifferential operators and the closure of E (La/o,)(sd)) in Llx.(..~). We have seen in the previous section that a sufficient condition to have commutation is

Ee(N(u, dom f ) ) = N(u, dora E~f) P-a.s.

This equality does not ensure the closure of E~ as shown by the following example (see fig. 2).

y = X

~x

X

(o,-i) �9 E~(N(O,F)) = ~(0, F~F)

but (0,-i) ~ EB(L~(o.V) (~))

Fig. 2. Example for lack of closure.


Y

i/~ / 1

1 x 0 1 x

Z(u) = af~(u,.)

E~Z = OE~f2 (0)

( 0 , 1 ) 6 Er3Y but (0 , .1) ~ E ~ (L~ z (~))

Fig. 3. Further example without closure.

Let ~2=]0,1[, ~ r ,'~'l~ "~ '={~, $2}, X=II~ 2, u = O and f , = S t - where F = {(~, (x, y)) ~ n x >/~x}.

It is tempting to suppose, on top of commutation, that the tangent cone of the domain (or still the closure of the domain) of f at u is a closed ~-random set. The following shows that this condition is not sufficient yet (see fig. 3).

In the context of the preceding example, let us consider f2 = 6" where X={(co , (x , y ) ) ~ $ 2 • 2- 0~<x~<l and y = ~ o - l x } . Since X is compact, the domain of fz is equal to Q • ~2. Let us represent X(~) (for a fixed ~o) and E~X as in fig. 3.

So we must add more assumptions. From 2.2.1 corollary 1: we can deduce the following result:

T H E O R E M

Let f: ~2 • X ~ R be a proper 1.s.c. convex d-integrand such that there exist some u~r ~-~c,~ and o 1 ~s for which f(u,~) and f * ( v l ) are integrable. We suppose that V~ is equal to L1r , (~ ) . ./

Let u be a ,~-measurable function such that f ( u ) is integrable and Of(u) admits an integrable selection.

If the following condition

T( u, dom E~I ) = T( u, dora f ) P-a.s.

is satisfied, then we have the equality

Remarks (1) The example given in 2.2.2 shows that the suggested condition is sufficient

but not necessary.


(2) The assumptions are less restrictive than the one given by the author in [45] and by Rockafellar and Wets [41].

Proof From corollary 2.2.1, each integrable selection of OEef(u) can be written

| E~176 + w), where v ~ L~/(,,)(..~') and w ~ LNc,,.d .... e , / l ( ~ ) . SO by hypothesis we immediately deduce that v + w is a selection of 0f(u). Any integrable selection of OEef(u) therefore belongs to Ee(L~/~, , l (d)) . The theorem results from 2.2.1 lemma 1.

2.2.4. O-subdifferential

Let us consider 0~.L-P~162 ') and u ~ c~~176 For any 1.s.c convex Jae-integrand f on ~ • X such that f (u ) takes almost surely real values, we call then O-subdifferential of f at u, the closed convex ~-random set

Oof(u ) = {(co, x*) ~ f2 • X*: /(co, u(co)) + / * ( c o , x* ) ~< (u (co ) Ix* )

+0(co)}. A function v of L~ r is thus a selection of Oof(u) if and only if v(co) is for

P-a.s. co, a O( co )-subgradient of f(co, -) at u(co) [26; section 11], i.e.,

V x ~ X , f(co, x ) - f ( c o , u ( c o ) ) > ~ ( x - u ( c o ) [ v ( c o ) ) - 0 ( c o ) .

In this section, we propose to establish the 0-subdifferential of the conditional integrand of f at u.

For any function 8 of .s176176 ( ~ ) , we denote by E ~ - ( 0 ) the inverse image of { 0} by the conditional expectation operator:

= E O=O P-a.s.}. The following result is an extension of Hiriart Urruty's theorem [25, section

14], established under conditions more restrictive, in the particular case ~ = {~, ~}:

THEOREM Let f : ~2 • X-~ • be a proper 1.s.c. convex d-integrand such that there exist

some uoo c s for which f * ( v ~ ) is integrable. Let 0 ~L#~ and u ~ , o ( ~ ) be such that f ( u ) is integrable. We suppose the following conditions are satisfied:

(i) I/ is finite and continuous in L ~ ( d ) at a point of L~(~ ' ) ,

(ii) dora E~ = dom f P-a.s. Then OoEef(u) is characterised by:

O~E:~-O


Cont r a ry to the asser t ions of Papageorg iou in [35, p. 9], the d e m o n s t r a t i o n given by Hir ia r t Ur ru ty for ~ = {t~, /2}, in the contex t of finite d imens iona l l inear spaces, can be adap t ed to the case of some sub-o-f ie ld and some Banach space with separable dual.

Proof We use the same a rguments as in 2.2.1 l emma 1, to verify that the fol lowing

inclusion

I c I

is sat isf ied for any O ~ E~-O. On the o ther hand, if L, is an in tegrable select ion of OoE~f(u) it sat isf ies:

( E ~ f ) ( t , ) + E # f ( u ) - ( u l v ) < ~ O P-a .s .

F r o m 2.2.2 corol la ry 3, app l i ed to f * and v, we can assert the exis tence of 3 ~ . s such that :

E ~ t 3 = v and ( E ~ f ) ( t , ) = E ~ ( f * ( O ) ) P-a .s .

If f~*((u] ~ 3 ) - f ( u ) ) d P = + oo, the p r o b l e m is obv ious ly o b t a i n e d since, in this case. ~ E Lla,,n,)(ar

Otherwise, (u]~3) has an in tegrable negat ive par t and so admi ts , by 1.2.2, ( u l v ) as cond i t iona l expecta t ion .

Let O = f * ( 0 ) + f ( u ) - (u [ ~). F r o m the preceding , we easi ly deduce that

O~2~~ E~O<~O and g~L~a, j -~, ,~(~)

consequent ly , v EOal3 ~ EOa( 1 = L ~,~t.~,,~(~)). The growth p r o p e r t y in 8 (for P-a.s. inclusion) of the 0-subdi f fe rent ia l , pe rmi t s

us to l imit our s tudy to funct ions O admi t t i ng 0 as cond i t i ona l expec ta t ion , and so we can conclude.

R e f e r e n c e s

[1] Z. Arstein, Limit laws for multifunctions applied to an optimization problem, in: Proc. Catania Conf. on Multifunctions and lntegrands ed. G. Salineni, Lecture Notes in Mathematics 1091 (Springer, 1983) p. 66.

[2] H. Anough and H. Brezis, Duality for the sum of convex functions in general Banach spaces, in: Proc. Conf on Aspects of Mathematics and Applications, ed. J.A. Barroso (Elsevier Science, Amsterdam, 1986).

[3] J.M. Bismut, Intdgrales convexes et probabilitds, J. Math. Anal. Appl. 42 (1973) 639. [4] A. Bourass, Comparaison des fonctionnelles intdgrales sur les sdlections d'une multiapplication

mesurable (III). Conditions de croissance et semi-continuitd forte et en mesure d'une fonction- nelle intdgrale, Travaux du Sdminaire Analyse Convexe de Montpellier 11 (1982).

[5] A. Bourass and A. Truffert, Ensembles fermds ddcomposables dans les espaces intrgraux de type Orlicz, Travaux du Sdminaire Analyse Convexe de Montpellier 15 (1982).


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