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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

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 [

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. [

The paper is organized as follows.

In Section

Throughout this paper, we assume that

Let

Given

A measurable function

(i. e.

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

(ii) The topology is induced by the duality

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

A multifunction

Two measurable multifunctions

The tribe trace of

A measurable function

The distribution

We say that the measurable multifunction

The multivalued Pettis-integral of a

Given a sub

Assume that

Assume that

Akhiat et al. [

We close this section by the following useful corollaries.

Let

Let

(1) Let

then

so

hence

We conclude that

For any

And hence by uniqueness of the conditional expectation of

Our first result is the following theorem.

Let

Since

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

Assume that

Let

By Theorem 5.1.6 in [

Therefore,

Let

Since

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

Let

Now, we give the main result of this work.

Assume that

Let

Then, we have the following assertions:

The first equality follows from the Theorem

Now let us prove the second equality. Let

Since

(i) Now, we show the last assertion.

We have

(ii) We begin by proving that

Since

Using Theorem

Since by [

Then

(iii) Now, we show that

Let

Then

Under the same hypothesis of Theorem

By the previous theorem, we need only to check that

Then by (

Therefore,

No data were used to support this study.

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