# Strong Law of Large Numbers of Pettis-Integrable Multifunctions

Authors: Hamid Oulghazi; Fatima Ezzaki
Journal: Journal of Mathematics
 (2019)

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## 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(EBnX)n≥1, where (Bn)n≥1 is a decreasing sequence of the sub σ-algebra of F is provided.


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## Body

## 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 Section2, 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(Ω,F,P) is a complete probability space and (Bn)n≥1 is a decreasing sequence of sub-σ-algebra of F, such that B∞=⋂n≥1Bn. E is a separable Banach space with the dual space E∗.Let(ek∗)k∈N be a dense sequence in E∗ with respect to the Mackey topology τ(E∗,E), BE¯(resp., BE∗¯), and the closed unit ball of E (resp., E∗). Let cc(E)(resp., cwk(E)) be the family of all nonempty convex and closed (resp., convex weakly compact) subsets of E.GivenB in cc(E), the distance function and the support function associated with B are defined by (1)dx,B=inf⁡x-y,y∈B,x∈E,δ∗x∗,B=sup⁡x∗,y,y∈B,x∗∈E∗.For any C in cwk(E), we get (2)C=sup⁡x,x∈C.A cc(E)-valued sequence (Yn)n≥1 is called Mosco convergent to a closed convex set Y∞ if Y∞=s-liYn=w-lsYn, where (3)s-liYn=y∈E,∃yn→ya.s.yn∈Yn,∀n≥1,w-lsYn=y∈E,∃yk→yweakly,yk∈Ynk,∀k≥1.If (Yn)n≥1 Mosco converges to Y∞ in cc(E), we write M-lim⁡Yn=Y∞.A measurable functiong:Ω→E is Pettis-integrable, if g is scalarly integrable(i. e.x∗,g is integrable), and, for each A∈F, there exists xA in E, such that (4)∫Ax∗,gdP=x∗,xA,∀x∗∈E∗.xA is called the Pettis-integral of g over A. We will denote by PE1(F) the space of all measurable and Pettis-integrable E-valued function defined on (Ω,F,P). We consider the space PE1(F) provided with the following topologies.(i) The topology of the usual Pettis norm is as follows:(5)gPe=supx∗∈BE∗⁡∫Ωx∗,gdP.(ii) The topology is induced by the duality(PE1(X),L∞⊗E∗). Recall that a sequence (gn)n≥1 in PE1(X) converges to g in this topology, if, for each h∈L∞(F) and for all x∗∈E∗, one has (6)limn→+∞⁡∫Ωhωx∗,gnωdP=∫Ωhωx∗,gωdP.This topology is known as the weak topology and is denoted byw-Pe.A multifunctionX:Ω→cc(E) is said to be measurable, if for every open set U of E the subset X-(U)={ω∈Ω/X(ω)∩U≠∅} is an element of F. The Effros σ-field B on cc(E) is generated by the subsets U-={F∈cc(E)/F∩U≠∅}, so X is measurable if, for any B∈B, one has X-1(B)∈F.Two measurable multifunctionsX and Y are said to be equal scalarly almost surely, if the following equality holds: (7)δ∗x∗,X.=δ∗x∗,Y.a.s.∀x∗∈E∗.For any convex and weakly compact measurable multifunction X, the measurability of X is equivalent to that of its support functional δ∗(x∗,X). (See [13].)The tribe trace ofB on cwk(E) is defined by the set {cwk(E)∩B/B∈B} and is denoted by ξ.A measurable functionf:Ω→E is said to be a selection of X, if, for any ω∈Ω, f(ω)∈X(ω). We denote by SX1(F) the set of all measurable selection of X. It is known that a convex and closed valued multifunction X in separable Banach space is measurable if dom(X)∈F and X has a Castaing representation (i.e., there exists a sequence (fn)n≥1 of measurable selections of X such that for all ω∈Ω, X(ω)=cl{fn(ω),n≥1}).The distributionPΓ of the measurable multifunction Γ:Ω→cc(E) on the measurable space (cc(E),B) is defined by (8)PΓB=PΓ-1B,∀B∈B.Two measurable multifunctions Γ and Δ are said to be independent, if the following equality holds: (9)PΓ,Δ=PΓ⊗PΔ.The measurable multifunction X:Ω→cwk(E) is scalarly integrable, if, for any x∗∈E∗, the real function δ∗(x∗,X(.)) is integrable.We say that the measurable multifunctionX is Pettis-integrable, if it is scalarly integrable and, for each A∈F, there exists KA∈cwk(E) such that (10)∫Aδ∗x∗,XdP=δ∗x∗,KA,∀x∗∈E∗.KA is called the Pettis-integral of X over A. Let SXPe(F) be the set of all F-measurable and Pettis-integrable selections of X.The multivalued Pettis-integral of acwk(E)-valued multifunction X is defined by (11)∫ΩXdP=∫ΩfdP,f∈SXPeF,and ∫AXdP is convex and σ(E,E∗) compact and, for each A∈F, we have (12)∫Aδ∗x∗,XdP=δ∗x∗,∫AXdP,∀x∗∈E∗.We will denote by Pcwk(E)1(F) the set of all Pettis-integrable cwk(E)-valued multifunctions.Given a subσ-algebra B and a cwk(E)-valued Pettis-integrable multifunction, the Pettis conditional expectation of X with respect to B is a B-measurable cwk(E)-valued Pettis-integrable multifunction denoted by EBX which satisfies (13)∫AEBXdP=∫AXdP,∀A∈B.The two following propositions (see [7]) give a sufficient condition of the existence of the conditional expectation for a Pettis integrable E-valued function and for a cwk(E)-valued Pettis-integrable multifunction.Proposition 1.
Assume thatE is separable. Let B be a sub σ-algebra of F and let X be a Pettis-integrable E-valued function such that EB|X|∈[0,+∞[. Then, there exists a unique B-measurable Pettis-integrable E-valued function denote by EBX, which enjoys the following property; for every h∈L∞(B), one has (14)∫ΩhEBXdP=∫ΩhXdP.Proposition 2.
Assume thatE∗ is separable. Let B be a sub-σ-algebra of F and let X be a cwk(E)-valued Pettis-integrable multifunction such that EB|X|∈[0,+∞[. Then, there exists a unique B-measurable cwk(E)-valued Pettis-integrable multifunction, which enjoys the following property; for every h∈L∞(B), one has (15)∫ΩhEBXdP=∫ΩhXdP.Akhiat et al. [14] extended the previous theorem in a cc(E)-valued Pettis-integrable multifunction.We close this section by the following useful corollaries.Corollary 3 (see [15]).
Let(Bn)n≥1 be a decreasing sequence of a sub-σ-algebra of F and let f∈LE1(F); set B∞=⋂n≥1Bn. Then, (16)limn→+∞⁡EBnf=EB∞fa.s.Corollary 4.
LetB be a sub-σ-algebra of F and let X be a cwk(E)-valued Pettis-integrable multifunction such that EB|X|∈[0,+∞[. Then, for all f∈SXPe(F), the following properties hold:(1)
EBf(.)∈EBX(.) almost surely.(2)
δ∗(x∗,EBX(.))=EBδ∗(x∗,X(.)) almost surely.Proof.
(1) Letf∈SXPe(F) and (ek∗)k≥1 be a dense sequence in E∗ with respect to the Mackey topology τ(E∗,E), then(17)∫Aek∗,fdP≤∫Aδ∗ek∗,XdP,∀A∈B,so(18)ek∗,∫AfdP≤δ∗ek∗,∫AXdP,ek∗,∫AEBfdP≤δ∗ek∗,∫AEBXdP,and
hence(19)ek∗,EBf.≤δ∗ek∗,EBX.a.s.We conclude that(20)EBf.∈EBX.a.s.(2)
For anyA∈B, we have δ∗(x∗,∫AXdP)=∫Aδ∗(x∗,X)dP, then, by Proposition 2, we have δ∗(x∗,∫AEBXdP)=∫Aδ∗(x∗,X)dP. Since EBX is a Pettis-integrable multifunction, (21)∫Aδ∗x∗,EBXdP=∫Aδ∗x∗,XdP=∫AEBδ∗x∗,XdP.And hence by uniqueness of the conditional expectation ofδ∗(x∗,X), we obtain (22)δ∗x∗,EBX.=EBδ∗x∗,X.a.s.


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

Our first result is the following theorem.Theorem 5.
Let(Bn)n≥1 be a decreasing sequence of sub-σ-algebras of F and set B∞=⋂n≥1Bn. Let E be a separable Banach space and f∈PE1(F) such that EB∞f∈[0,+∞[. Then (23)limn→+∞⁡EBnf.=EB∞f.a.s.Proof.
SinceEB∞|f|∈[0,+∞[, so, by Proposition 1, we have that EBnf exists and is in PE1(Bn) and provides a B∞-measurable partition (Bk)k≥1 of Ω such that fk=f1Bk∈LE1(F) (see [14]). Using Corollary 3, we obtain (24)limn→+∞⁡EBnfk=EB∞fka.s.As Bk⊂B∞⊂Bn, for every n≥1 and for every k≥1, then (25)EBnfk=EBn1Bkf=1BkEBnf.On the other hand, (26)EB∞f=∑k=1+∞EB∞fk=∑k=1+∞limn→+∞⁡EBnfk==∑k=1+∞1Bklimn→+∞⁡EBnf=limn→+∞⁡EBnf.Therefore, (27)limn→+∞⁡EBnf.=EB∞f.a.s.Let us prove the following results which will be used after.Proposition 6.
Assume thatE∗ is separable. Let X and Y be two cwk(E)-valued Pettis-integrable multifunctions and B be a sub-σ-algebra of F such that EB|X|∈[0,+∞[ and EB|Y|∈[0,+∞[. Then (28)EBX+Y=EBX+EBYa.s.Proof.
Let(ek∗)k≥1 be a dense sequence in E∗ with respect to the Mackey topology τ(E∗,E). Since X and Y are cwk(E)-valued Pettis-integrable multifunctions, then, for any A in B, there exist two sets KA(X) and KA(Y) in cwk(E), such that for all k≥1δ∗(ek∗,KA(X))=∫Aδ∗(ek∗,X)dP and δ∗(ek∗,kA(Y))=∫Aδ∗(ek∗,Y)dP.
By Theorem 5.1.6 in [8], X+Y is cwk(E)-valued Pettis-integrable multifunction, then, for any A in B, there exists a set KA(X+Y) in cwk(E) such that (29)δ∗ek∗,KAX+Y=∫Aδ∗ek∗,X+YdP.On the other hand,(30)δ∗ek∗,KAX+KAY=δ∗ek∗,KAX+δ∗ek∗,KAY==∫Aδ∗ek∗,XdP+∫Aδ∗ek∗,YdP=∫Aδ∗ek∗,X+δ∗ek∗,YdP==∫Aδ∗ek∗,X+YdP=δ∗ek∗,KAX+Y.Then KA(X+Y)=KA(X)+KA(Y)a.s.
Therefore,(31)∫AX+YdP=∫AXdP+∫AYdPa.s.On the other hand, (32)∫Aδ∗ek∗,EBX+YdP=δ∗ek∗,∫AEBX+YdP=δ∗ek∗,∫AX+YdPBy (31), (33)δ∗ek∗,∫AX+YdP=δ∗ek∗,∫AXdP+∫AYdP=δ∗ek∗,∫AEBXdP+∫AEBYdP==∫Aδ∗ek∗,EBX+EBYdP,then (34)∫Aδ∗ek∗,EBX+YdP=∫Aδ∗ek∗,EBX+EBYdP,and therefore (35)EBX+Y=EBX+EBYa.s.Theorem 7.
LetX and Y be in Pcwk(E)1(F); let Z be a random variable with values in a measurable space (Σ,Γ) such that (Z,X) and (Z,Y) have the same distribution. If Eσ(Z)|X|∈[0,+∞[ and Eσ(Z)|Y|∈[0,+∞[. Then (36)EσZX.=EσZY.a.s.Proof.
SinceX is Pettis-integrable with values in cwk(E), SXPe(σ(X))≠∅. So by [11], there exists (fn)n≥1 selection Pettis-integrable of X, σ(X)-measurable such that (37)Xw=clfnω,n≥1.Let f:Ω→E be a fixed element in SXPe(σ(X)), h∈L∞(σ(Z)), and x∗∈E∗. Then, by Doob’s factorisation lemma, we can find a ξ-measurable mapping u from cwk(E) into E (i.e., (cwk(E),ξ)→u(E,B(E))) and a Γ-measurable function v from Σ into R (i.e., (Σ,Γ)→v(R,B(R))), which satisfy (38)f:Ω,F,P→XcwkE,ξ→uE,BE,fω=u∘Xω.h:Ω,F,P→ZΣ,Γ→vR,BR,hω=v∘Zω.On the other hand, (39)δ∗h⊗x∗,SXPeσX=∫Ωδ∗h⊗x∗,XdP=sup⁡∫Ωhωx∗,fdP,f∈SXPeσX.And (40)∫Ωhωx∗,fdP=∫Ωv∘Zωx∗,u∘XωdP.By using the application (41)Ω,F,P→gΣ×cwkE,Γ⊗ξ→φR,BRω→Z,X→voZωx∗,uoXω=hωx∗,fω.And by the classical transfer theorem, we have(42)∫Ωφ∘gωdP=∫Ωhωx∗,fωdP=∫Ωv∘Zωx∗,u∘XωdP==∫Σ⊗cwkEφz,BdPZ,Xz,B=∫Σ⊗cwkEvzx∗,uBdPZ,Xz,B==∫Σ⊗cwkEvzx∗,uBdPZ,Yz,B=∫Σ⊗cwkEφz,BdPZ,Yz,B==∫Ωv∘Zωx∗,u∘YωdP=∫Ωhωx∗,u∘YωdP.Then (43)∫Ωhωx∗,fωdP=∫Ωhωx∗,u∘YωdP.By combining relation (39), (43) and the fact that ∀B∈cwk(E),u(B)∈B,PYa.s., we obtain (44)∫Ωδ∗h⊗x∗,XdP=sup⁡∫Ωhωx∗,fωdP,f∈SYPeσY=∫Ωδ∗h⊗x∗,YdP.Then (45)δ∗h⊗x∗,SXPeσX=∫Ωδ∗h⊗x∗,XdP=∫Ωδ∗h⊗x∗,Y=δ∗h⊗x∗,SYPeσY.In particular, if h=1A, for any A∈σ(Z), we have (46)∫Ω1Aδ∗x∗,XdP=∫Ω1Aδ∗x∗,YdP,then (47)δ∗x∗,∫AXdP=δ∗x∗,∫AYdP, and therefore (48)δ∗x∗,∫AEσZXdP=δ∗x∗,∫AEσZYdP,And hence, by uniqueness of the conditional expectation of X relative to σ(Z), (49)EσZX.=EσZY.a.s.Before giving the principal result, we also need the following classical theorem (see p. 52 in [16]).Theorem 8.
LetB1 and B2 be two sub-σ-algebras of F, Φ=σ(B1,B2) is a σ-algebra generated by B1 and B2, and let f1 an integrable real measurable function. If f1 and B1 are independent of B2, then (50)EΦf1.=EB1f1.a.s.Now, we give the main result of this work.Theorem 9.
Assume thatE∗ is separable. Let (Xn)n≥1 be a sequence of independent measurable multifunctions in Pcwk(E)1(F).
LetSn=X1+X2+X3+⋯+Xn, Bn=σ(Sn,Xn+1,Xn+2,…), B∞=⋂n≥1Bn and assume that (i)
∀n,∀j∈{1,2,…,n}, (Sn,X1) and (Sn,Xj) have the same distribution.(ii)
∀n,∀j∈{1,2,…,n}, EBn|Xj|∈[0,+∞[.
Then, we have the following assertions:(1)
∀n,∀j∈{1,2,…,n}, Eσ(Sn)Xj(.)=Eσ(Sn)X1(.)a.s.(2)
EBnX1(.)=Eσ(Sn)X1(.)a.s.(3)
M-limn→∞Sn/n=M-limn→∞EBnX1(.)=EB∞X1(.)a.s.Proof.
The first equality follows from the Theorem7.
Now let us prove the second equality. Let(ek∗)k≥1 be a dense sequence in E∗ for the Mackey topology. Set B1=σ(Sn), B2=σ(Xn+1,Xn+2,…) and f1=δ∗(ek∗,X1).
SinceX1 and Sn are independent of B2, f1 and Sn are independent of B2, then, by applying the Theorem 8, we have (51)EBnf1.=EB1f1.a.s.So, EBnδ∗(ek∗,X1)=EB1δ∗(ek∗,X1), then, by using Corollary 4, we obtain (52)δ∗ek∗,EBnX1=δ∗ek∗,EB1X1,and since EBnX1(.) and Eσ(Sn)X1(.) are convex and weakly compact, then (53)EBnX1.=EσSnX1.a.s.
(i) Now, we show the last assertion.
Step 1. We claim that Sn/n=EBnX1(.)a.s.
We haveSn=∑i=1nXi=Eσ(Sn)(∑i=1nXi), so, by Proposition 6 and the first and the second assertions of the theorem, we obtain(54)EσSn∑i=1nXi=∑i=1nEσSnXi=∑i=1nEσSnX1=nEσSnX1=nEBnX1.Then (55)Snn=EBnX1.a.s.
Step 2. We show that M-limn→∞EBnX1=EB∞X1(.)a.s.
(ii) We begin by proving thatEB∞X1(.)⊂s-liEBnX1(.)a.s.
SinceX1 is Pettis-integrable, then there exists (gk)k≥1 Pettis-integrable selection of X1 such that (56)X1=clgk,k≥1.On the other hand EBn|X1|∈[0,+∞[, then EBnX1 exists and is in Pcwk(E)1(Bn). Let g∈SX1Pe(Bn); by Corollary 4, EBng exists and EBng(.)∈EBnX1(.) a. s.
Using Theorem5, we have limn→∞⁡EBng=EB∞g a.s.
Since by [17] (57)SEB∞XPe=EB∞f,f∈SXPeFThen EB∞g∈Ss-liEBnX1Pe and by (56), SEB∞X1(.)Pe⊂Ss-liEBnX1(.)Pea.s.
Then(58)EB∞X1.⊂s-liEBnX1.a.s.
(iii) Now, we show thatw-lsEBnX1(.)⊂EB∞X1(.)a.s.
Let(ek∗)k≥1 be a dense sequence in E∗ for the Mackey topology; we have (59)δ∗ek∗,EBnX1=EBnδ∗ek∗,X1,then, by Corollary 3, (60)limn→+∞⁡EBnδ∗ek∗,X1.=EB∞δ∗ek∗,X1.a.s.Hence, there exists a negligible set N=⋃k≥1Nk; for all k≥1 and for every ω∈Ω∖N, we have (61)limn→+∞⁡EBnδ∗ek∗,X1ω=EB∞δ∗ek∗,X1ω=δ∗ek∗,EB∞X1ωa.s.So, for all x∗∈E∗ and for all ω∈Ω∖N, (62)limn→+∞⁡δ∗x∗,EBnX1ω=δ∗x∗,EB∞X1ωa.s.Let ω∈Ω∖N and x∈w-lsEBnX1(ω), then there exists (xm)m≥1 in EBnmX1(ω) such that xm→wx, which implies limm→+∞x∗,xm=x∗,x.
Then(63)x∗,x=limm→+∞⁡x∗,xm≤limsupm→∞⁡δ∗x∗,EBnmX1==limsupm→∞⁡EBnmδ∗x∗,X1=δ∗x∗,EB∞X1ωa.s.Consequently, x∈EB∞X1(ω), then (64)w-lsEBnX1.⊂EB∞X1.a.s.This yields (65)M-limn→∞⁡Snn=M-limn→∞⁡EBnX1.=EB∞X1.a.s.Corollary 10.
Under the same hypothesis of Theorem9, we have (66)M-limn→∞⁡Snn=EB∞X1.=∫X1ωdPωa.s.Proof.
By the previous theorem, we need only to check that(67)EB∞X1.=∫ΩX1ωdPωa.s.Since ∫ΩX1(ω)dP(ω) is convex and weakly compact. Now let (ek∗)k≥1 be a dense sequence in E∗ for the Mackey topology; we have (68)limn→∞⁡δ∗ek∗,EBnX1=limn→∞⁡EBnδ∗ek∗,X1=EB∞δ∗ek∗,X1=δ∗ek∗,EB∞X1a.s.On the other hand, (EBnδ∗(ek∗,X1))n is a Pettis reversed martingale, so, for each positive integer m, set, Fm=σ(Xm,Xm+1,Xm+2,…) and F∞=⋂mFm. Hence, for all any n≥m, the multifunction ∑j=mnXj is Fm-measurable. Moreover, by the previous theorem, we have (69)1n∑j=1nδ∗ek∗,Xj=1n∑j=1m-1δ∗ek∗,Xj+1n∑j=mnδ∗ek∗,Xj==1nδ∗ek∗,∑j=1m-1Xj+1n∑j=mnδ∗ek∗,Xj=m-1nEBm-1δ∗ek∗,X1+1n∑j=mnδ∗ek∗,Xja.s.Then (70)1n∑j=1nδ∗ek∗,Xj=m-1nEBm-1δ∗ek∗,X1+1n∑j=mnδ∗ek∗,Xja.s.Since, for every fixed positive integer m, we have limn→∞m-1/nEBm-1δ∗(ek∗,X1)=0a.s.
Then by (70) (71)limn→∞⁡1n∑j=mnδ∗ek∗,Xj=limn→∞⁡1n∑j=1nδ∗ek∗,Xj=limn→∞⁡δ∗ek∗,EBnX1=EB∞δ∗ek∗,X1a.s.Since, for all fixed integer m, the multifunction 1/n∑j=mnXj(.) is Fm-measurable and by (71) EB∞X1 is Fm-measurable and so is F∞-measurable. Then by the independence of (σ(Xn))n∈N∗ and the Kolmogorov’s Zero-One law (see [18]), we conclude that for all k∈N∗(72)δ∗ek∗,EB∞X1.=EB∞δ∗ek∗,X1.=∫Ωδ∗ek∗,X1ωdPω=δ∗ek∗,∫ΩX1ωdPωa.s.Since EB∞X1 and ∫ΩX1(.)dP are cwk(E)-valued multifunctions and (72) is true for all k≥1, then EB∞X1(.)=∫ΩX1(ω)dP(ω)a.s.
Therefore,(73)M-limn→∞⁡Snn=EB∞X1.=∫ΩX1ωdPωa.s.



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