#### Abstract

Using reversed martingale techniques, we prove the strong law of large numbres for independent Pettis-integrable multifunctions with convex weakly compact values in a Banach space. The Mosco convergence of reversed Pettis-integrable martingale of the form , where is a decreasing sequence of the sub -algebra of is provided.

#### 1. Introduction

The strong law of large numbers (SLLN) is used in a variety of fields including statistics, probability theory, and areas of economics and insurance. In recent years, SLLN has been extensively studied by several researchers. Let us mention Artstein and Hart [1], Castaing and Ezzaki [2], Etemadi [3], Ezzaki [4], Hess [5], and Hiai [6].

In the theory of integration in infinite-dimensional spaces, Pettis-integrability is a more general concept than that of Bochner-integrability. The purpose of this paper is to prove the SLLN for measurable and Pettis-integrable multifunctions by using the techniques of reversed martingale. The proof is based on the recent properties of Pettis-integrable multifunctions. See for example Akhiat et al. [7], Chowdhury [8], El Amri and Hess [9], Geitz et al. [10], Thobie and Satco [11], and Musial [12].

The paper is organized as follows.

In Section 2, we recall some definitions and results that will be used after. In Section 3, we prove the SLLN for Pettis-integrable multifunctions with convex weakly compact values.

#### 2. Notations and Definitions

Throughout this paper, we assume that is a complete probability space and is a decreasing sequence of sub--algebra of , such that . is a separable Banach space with the dual space .

Let be a dense sequence in with respect to the Mackey topology , (resp., ), and the closed unit ball of (resp., ). Let (resp., ) be the family of all nonempty convex and closed (resp., convex weakly compact) subsets of .

Given in , the distance function and the support function associated with are defined by For any in , we get A -valued sequence is called Mosco convergent to a closed convex set if , where If Mosco converges to in , we write

A measurable function is Pettis-integrable, if is scalarly integrable

(i. e. is integrable), and, for each , there exists in , such that is called the Pettis-integral of over . We will denote by the space of all measurable and Pettis-integrable -valued function defined on . We consider the space provided with the following topologies.

(i) The topology of the usual Pettis norm is as follows:

(ii) The topology is induced by the duality Recall that a sequence in converges to in this topology, if, for each and for all , one has

This topology is known as the weak topology and is denoted by -.

A multifunction is said to be measurable, if for every open set of the subset is an element of . The Effros -field on is generated by the subsets , so is measurable if, for any , one has .

Two measurable multifunctions and are said to be equal scalarly almost surely, if the following equality holds: For any convex and weakly compact measurable multifunction , the measurability of is equivalent to that of its support functional . (See [13].)

The tribe trace of on is defined by the set and is denoted by .

A measurable function is said to be a selection of , if, for any , . We denote by the set of all measurable selection of . It is known that a convex and closed valued multifunction in separable Banach space is measurable if and has a Castaing representation (i.e., there exists a sequence of measurable selections of such that for all , .

The distribution of the measurable multifunction on the measurable space is defined by Two measurable multifunctions and are said to be independent, if the following equality holds: The measurable multifunction is scalarly integrable, if, for any , the real function is integrable.

We say that the measurable multifunction is Pettis-integrable, if it is scalarly integrable and, for each , there exists such that is called the Pettis-integral of over . Let be the set of all -measurable and Pettis-integrable selections of .

The multivalued Pettis-integral of a -valued multifunction is defined by and is convex and compact and, for each , we have We will denote by the set of all Pettis-integrable -valued multifunctions.

Given a sub -algebra and a -valued Pettis-integrable multifunction, the Pettis conditional expectation of with respect to is a -measurable -valued Pettis-integrable multifunction denoted by which satisfies The two following propositions (see [7]) give a sufficient condition of the existence of the conditional expectation for a Pettis integrable -valued function and for a -valued Pettis-integrable multifunction.

Proposition 1. *Assume that is separable. Let be a sub -algebra of and let be a Pettis-integrable -valued function such that Then, there exists a unique -measurable Pettis-integrable -valued function denote by , which enjoys the following property; for every , one has *

Proposition 2. *Assume that is separable. Let be a sub--algebra of and let be a -valued Pettis-integrable multifunction such that Then, there exists a unique -measurable -valued Pettis-integrable multifunction, which enjoys the following property; for every , one has *

Akhiat et al. [14] extended the previous theorem in a -valued Pettis-integrable multifunction.

We close this section by the following useful corollaries.

Corollary 3 (see [15]). *Let be a decreasing sequence of a sub--algebra of and let ; set . Then, *

Corollary 4. *Let be a sub--algebra of and let be a -valued Pettis-integrable multifunction such that Then, for all , the following properties hold:*(1)* almost surely.*(2)* almost surely.*

*Proof. *(1) Let and be a dense sequence in with respect to the Mackey topology , then so and hence We conclude that For any , we have , then, by Proposition 2, we have . Since is a Pettis-integrable multifunction, And hence by uniqueness of the conditional expectation of , we obtain

#### 3. Strong Law of Large Numbers for Pettis-Integrable Multifunctions

Our first result is the following theorem.

Theorem 5. *Let be a decreasing sequence of sub--algebras of and set . Let be a separable Banach space and such that . Then *

*Proof. *Since , so, by Proposition 1, we have that exists and is in and provides a -measurable partition of such that (see [14]). Using Corollary 3, we obtain As , for every and for every , then On the other hand, Therefore,

Let us prove the following results which will be used after.

Proposition 6. *Assume that is separable. Let X and Y be two -valued Pettis-integrable multifunctions and be a sub--algebra of such that and . Then *

*Proof. *Let be a dense sequence in with respect to the Mackey topology . Since and are -valued Pettis-integrable multifunctions, then, for any in , there exist two sets and in , such that for all and

By Theorem 5.1.6 in [8], is -valued Pettis-integrable multifunction, then, for any in , there exists a set in such that On the other hand,Then

Therefore, On the other hand, By (31), then and therefore

Theorem 7. *Let and be in ; let be a random variable with values in a measurable space such that and have the same distribution. If and Then *

*Proof. *Since is Pettis-integrable with values in , . So by [11], there exists selection Pettis-integrable of , -measurable such that Let be a fixed element in , , and . Then, by Doob’s factorisation lemma, we can find a -measurable mapping from into (i.e., and a -measurable function from into (i.e., , which satisfy On the other hand, And By using the application And by the classical transfer theorem, we haveThen By combining relation (39), (43) and the fact that , we obtain Then In particular, if , for any , we have then and therefore And hence, by uniqueness of the conditional expectation of relative to ,

Before giving the principal result, we also need the following classical theorem (see p. 52 in [16]).

Theorem 8. *Let and be two sub--algebras of , is a -algebra generated by and , and let an integrable real measurable function. If and are independent of , then *

Now, we give the main result of this work.

Theorem 9. *Assume that is separable. Let be a sequence of independent measurable multifunctions in **Let , , and assume that *(i)*, and have the same distribution.*(ii)*, .** Then, we have the following assertions: *(1)*, *(2)*(3)*

*Proof. *The first equality follows from the Theorem 7.

Now let us prove the second equality. Let be a dense sequence in for the Mackey topology. Set , and .

Since and are independent of , and are independent of , then, by applying the Theorem 8, we have So, , then, by using Corollary 4, we obtain and since and are convex and weakly compact, then (i) Now, we show the last assertion. *Step 1*. We claim that

We have , so, by Proposition 6 and the first and the second assertions of the theorem, we obtainThen *Step 2*. We show that

(ii) We begin by proving that

Since is Pettis-integrable, then there exists Pettis-integrable selection of such that On the other hand , then exists and is in . Let ; by Corollary 4, exists and a. s.

Using Theorem 5, we have a.s.

Since by [17] Then and by (56),

Then (iii) Now, we show that

Let be a dense sequence in for the Mackey topology; we have then, by Corollary 3, Hence, there exists a negligible set ; for all and for every , we have So, for all and for all , Let and , then there exists in such that , which implies

Then Consequently, , then This yields

Corollary 10. *Under the same hypothesis of Theorem 9, we have *

*Proof. *By the previous theorem, we need only to check that Since is convex and weakly compact. Now let be a dense sequence in for the Mackey topology; we have On the other hand, is a Pettis reversed martingale, so, for each positive integer , set, and . Hence, for all any , the multifunction is -measurable. Moreover, by the previous theorem, we have Then Since, for every fixed positive integer , we have

Then by (70) Since, for all fixed integer , the multifunction is -measurable and by (71) is -measurable and so is -measurable. Then by the independence of and the Kolmogorov’s Zero-One law (see [18]), we conclude that for all Since and are -valued multifunctions and (72) is true for all , then

Therefore,

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.