# Lebesgue Decomposition Theorem and Weak Radon-Nikodým Theorem for Generalized Fuzzy Number Measures

Authors: Cai-Li Zhou; Fu-Gui Shi
Journal: Journal of Function Spaces
 (2015)

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## Abstract

The Lebesgue type decomposition theorem and weak Radon-Nikodým theorem for fuzzy valued measures in separable Banach spaces are established.


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## Body

## 1. Introduction

The topic of set valued and fuzzy valued measures has received much attention in the past few decades because of its usefulness in several applied fields like mathematical economics and optimal control. Significant contributions to set valued measures were made by Artstein [1], Cascales et al. [2, 3], Hiai [4], Papageorgiou [5], Stojaković [6], Zhang et al. [7], Zhou and Shi [8], and others. Fuzzy valued measure is a natural generalization of set valued measures. Hence the study of fuzzy valued measures is usually connected with set valued measures. Contributions in this field were made, among others, by Puri and Ralescu [9], Bán [10], Stojaković [11, 12], M. Stojaković and Z. Stojaković [13, 14], Xue et al. [15], and Park [16].The main results of this paper fall into two main parts. It is well known that Lebesgue type decomposition theorem and Radon-Nikodým theorem are very important results in measure theory. As an extension of Lebesgue type decomposition theorem for vector measures, Zhang et al. [7] obtained Lebesgue type decomposition theorem for set valued measures. In the first part we generalize these results to generalized fuzzy number measures. On the other hand, Wu et al. [17] obtained Radon-Nikodým theorem for generalized fuzzy number measures using Bochner integral. But it is well known that Pettis integrability is a more general concept than that of Bochner integrability in the theory of integration in infinite dimensional spaces. In the second part, we obtain weak Radon-Nikodým theorem for generalized fuzzy number measures using Pettis type integral.The paper is structured as follows. In Section2, we state some basic concepts and preliminary results. In Section 3, the Lebesgue type decomposition theorem and weak Radon-Nikodým theorem for generalized fuzzy number measures will be established.


## 2. Preliminaries

Throughout this paper, let(Ω,A,μ) be a complete finite measure space, where Ω is a nonempty set, A is a σ-algebra of subsets of Ω, and μ is a measure. Let (X,·) be a real separable Banach space with its dual space X∗. LetP0(X)={A⊂X:Ais a nonempty subset ofX},Pbfc(X)={A∈P0(X):Ais boundedclosedconvex},Pwkc(X)={A∈P0(X):Ais weakly compact and convex}.ForA,B∈Pf(X), the Hausdorff metric dH of A and B is defined by (1)dHA,B=max⁡supx∈A⁡infy∈B⁡x-y,supy∈B⁡infx∈A⁡x-y.Note that (Pwkc(X),dH) is a complete metric space. The number |A| is defined by A=dHA,0=supx∈A⁡x for each A∈Pf(X).We will denote byσ(·,A) the support function of a set A⊂X defined by (2)σx∗,A=supx∈A⁡x∗,x,x∗∈X∗.The support function satisfies the following properties: σ(x∗,A+B)=σ(x∗,A)+σ(x∗,B) and σ(x∗,λA)=λσ(x∗,A) for all A,B∈P0(X) and λ≥0.Definition 1 (see [15]).
Letu~:X→[0,1]. One denotes u~α={x∈X:u~(x)≥α} for each α∈(0,1]. u~ is called a generalized fuzzy number if, for each α∈(0,1], u~α∈Pwkc(X). Let Fwkc(X) denote the set of all generalized fuzzy numbers on X.Foru~,v~∈Fwkc(X), we define u~+v~ as follows: (3)u~+v~x=supx=y+z⁡min⁡u~y,v~z.Obviously, we have (u~+v~)α=u~α+v~α for each α∈(0,1], and therefore u~+v~∈Fwkc(X). In the set Fwkc(X) we define dH∞ by (4)dH∞u~,v~=supα∈0,1⁡dHu~α,v~α.(Fwkc(X),dH∞) is a metric space.Theorem 2 (see [18]).
Ifu~∈Fwkc(X), then one has the following:(1)
u~α∈Pwkc(X) for all α∈(0,1].(2)
u~α⊇u~β for 0<α≤β≤1.(3)
If{αn}n∈N is a nondecreasing sequence in [0,1] converging to α∈(0,1], then u~α=⋂n=1∞u~αn.
Conversely, if{Aα:α∈(0,1]}⊆P0(X) satisfies (1)–(3) above, then there exists a u~∈Fwkc(X) such that u~α=Aα for each α∈(0,1].Theorem 3 (see [19]).
LetMα∈Pwkc(X), {Mαn}n∈N⊂Pwkc(X), and αn↗α, Mαn⊃Mαn+1⊃Mα; then σ(x∗,Mαn) converges to σ(x∗,Mα) for each x∗∈X∗ if and only if Mα=⋂n=1∞Mαn.Definition 4 (see [4]).
Let(Ω,A) be a measurable space. The mapping M:A→P0(X) is said to be a set valued measure if it satisfies the following two conditions:(1)
M(∅)={0};(2)
ifA1,A2,… are in A, with Ai∩Aj=∅ for i≠j, then (5)M⋃i=1∞Ai=∑i=1∞MAi,where∑i=1∞‍M(Ai)=x∈X:x=∑n=1∞‍xi(unc.conv.),xi∈M(Ai),i≥1.Particularly,M:A→Pwkc(X) is a set valued measure if and only if σ(x∗,M(·)) is a real valued measure for all x∗∈X∗.As for single valued measures, we have the notion of total variation|M| of M. For A∈A we define M(A)=sup∑i=1n‍MAi, where the supremum is taken over all finite measurable partitions {A1,…,An} of A. We call that M is of bounded variation if MΩ<∞. We call that M is of σ-bounded variation if there exists a countably measurable partition {An}n∈N such that each restriction of M to An is of bounded variation. We call that M is continuous absolutely about μ if, for any A∈A, μ(A)=0; then M(A)={0}, denoted as M≪μ. We call that M is singular about μ if there exists N∈A, μ(N)=0, such that, for any A∈A, M(A∩Nc)={0}, denoted as M⊥μ.Definition 5 (see [15]).
Let(Ω,A) be a measurable space. The mapping M~:A→Fwkc(X) is called a generalized fuzzy number measure if it satisfies the following two conditions:(1)
M~(∅)=0~, where 0~ is indicator function of {0}.(2)
IfA1,A2,… are in A, with Ai∩Aj=∅ for i≠j, then (6)M~⋃i=1∞Ai=∑i=1∞M~Ai,where(∑i=1∞M~(Ai))(x)=sup{⋀i=1∞M~(Ai)(xi):x=∑i=1∞xi(unc.conv.)}.We call thatM~ is continuous absolutely about μ if, for any A∈A, μ(A)=0; then M~(A)=0~, denoted as M~≪μ.Theorem 6 (see [15]).
The mappingM~:A→Fwkc(X) is a generalized fuzzy number measure if and only if there exists a family of set valued measures Mα:A→Pwkc(X),α∈(0,1] satisfying the following three conditions:(1)
For arbitraryα,β∈(0,1] and A∈A, if α≤β, then Mα(A)⊇Mβ(A).(2)
For arbitrary{αn}n∈N⊆(0,1] and α∈(0,1] such that α1≤α2≤⋯ and limn→∞⁡αn=α, then MαA=⋂n=1∞‍MαnA,∀A∈A.(3)
For arbitraryA∈A, one has (7)M~Ax=sup⁡α:x∈MαA,α∈0,1,ifα:x∈MαA,α∈0,1≠∅,0,ifα:x∈MαA,α∈0,1=∅.
Note that for a generalized fuzzy number measureM~:A→Fwkc(X) the set valued measure Mα:A→Pwkc(X) is determined by (8)MαA=x∈X:M~Ax≥α;that is, Mα(A)=[M~(A)]α.


## 3. Main Results

In this section, we first give the Lebesgue type decomposition theorem for generalized fuzzy number measures. Our result is a generalization of Zhang et al.’s result [7]. And then, we obtain weak Radon-Nikodým theorem for generalized fuzzy number measures.Theorem 7 (see [7] Lebesgue type decomposition theorem for set valued measures).
LetM:A→Pwkc(X) be a set valued measure. Then there exists a unique pair of weakly compact and convex set valued measures Mc and Ms such that(1)
Mc≪μ, Ms⊥μ;(2)
M=Mc+Ms;(3)
for arbitraryx∗∈X∗, (9)σx∗,M·=σx∗,Mc·+σx∗,Ms·is the Lebesgue decomposition of σx∗,M(·).In what follows, we first give the definition of singular for generalized fuzzy number measure about crisp measure and give two lemmas before we establish the main result.Definition 8.
Let(Ω,A,μ) be a finite real valued measure space and let M~:A→Fwkc(X) be a generalized fuzzy number measure. One calls that M~ is singular about μ if there exists N∈A, μ(N)=0, such that, for any A∈A, M~(A∩Nc)=0~, denoted as M~⊥μ.Lemma 9.
Let(Ω,A,μ) be a finite real valued measure space and let M~:A→Fwkc(X) be a generalized fuzzy number measure. Then(1)
M~≪μ if and only if M~α≪μ for all α∈(0,1];(2)
M~⊥μ if and only if M~α⊥μ for all α∈(0,1].Proof.
(1) Suppose thatM~≪μ. For any A∈A, if μ(A)=0 then M~(A)=0~, which implies that M~α(A)={0} for all α∈(0,1]; that is, M~α≪μ for all α∈(0,1].
On the contrary, suppose thatM~α≪μ for all α∈(0,1]. For any A∈A, if μ(A)=0 then M~α(A)={0}. If M~(A)≠0~, then there exists x∈X such that M~(A)(x)>0, which leads to the conclusion that there exists α∈(0,1] such that x∈M~α(A). This contradicts the fact that M~α(A)={0} for all α∈(0,1]. Hence, M~(A)=0~. This implies that M~≪μ.
(2) Suppose thatM~⊥μ. If there exists N∈A, μ(N)=0, then, for any A∈A, M~(A∩Nc)=0~ which implies that M~α(A∩Nc)={0} for all α∈(0,1]. It follows that M~α⊥μ for all α∈(0,1].
Conversely, suppose thatM~α⊥μ for all α∈(0,1]. According to Definition 8, if there exists N∈A, μ(N)=0, then, for any A∈A and α∈(0,1], we have M~α(A∩Nc)={0}, which implies that M~(A∩Nc)=0~. Hence, M~⊥μ.Lemma 10.
Let{αn}n∈N⊆(0,1] such that αn↗α and let {Mαn(·)}n∈N be a family of weakly compact and convex set valued measure such that Mα(A)⊂Mαn+1(A)⊂Mαn(A) for any A∈A. Then one has the following:(1)
limn→∞⁡σx∗,Mαn(·) is a real valued measure for each x∗∈X∗.(2)
IfMαn≪μ for each αn∈(0,1], then limn→∞⁡σx∗,Mαn·≪μ for each x∗∈X∗.(3)
IfMαn⊥μ for each αn∈(0,1], then limn→∞⁡σx∗,Mαn·⊥μ for each x∗∈X∗.Proof.
(1) SinceMαn:Ω→Pwkc(X) is a set valued measure for any n∈N, σ(x∗,Mαn(·)) is a real valued measure for each x∗∈X∗ and any n∈N. Further, (10)MαA⊂⋯⊂MαnA⊂⋯⊂Mα1Aimplies (11)σx∗,MαA≤⋯≤σx∗,MαnA≤⋯≤σx∗,Mα1Afor each x∗∈X∗ and A∈A, which shows that {σ(x∗,Mαn(A))}n∈N is a monotone decreasing and bounded sequence for each x∗∈X∗ and A∈A. This ensures that limn→∞⁡σx∗,MαnA exists. In the following, we claim that limn→∞⁡σx∗,Mαn· is a real valued measure for each x∗∈X∗. Obviously, limn→∞⁡σ(x∗,Mαn(∅))=0. Let {Aj}j∈N be a sequence of pairwise disjoint elements of A. By countable additivity of set valued measure and Theorem 6.1.1 [7], we have (12)limn→∞⁡σx∗,Mαn⋃j=1∞Aj=limn→∞⁡σx∗,∑j=1∞MαnAj=limn→∞⁡∑j=1∞σx∗,MαnAjfor each x∗∈X∗. Sine for each Aj∈A and αn∈(0,1], we have(13)σx∗,MαnAj≤x∗MαnAj≤x∗Mα1Aj≤x∗Mα1Aj,∑j=1∞x∗Mα1Aj=x∗∑j=1∞Mα1Aj=x∗Mα1⋃j=1∞Aj<∞,which implies that (14)limn→∞⁡σx∗,Mαn⋃j=1∞Aj=limn→∞⁡∑j=1∞σx∗,MαnAj=∑j=1∞limn→∞⁡σx∗,MαnAj.This shows the countable additivity of limn→∞⁡σ(x∗,Mαn(·)) for each x∗∈X∗.
(2) Suppose thatMαn≪μ for each αn∈(0,1]. If, for A∈A, μ(A)=0, then Mαn(A)={0} which implies that σ(x∗,Mαn(A))=0 for each x∗∈X∗. It follows that limn→∞⁡σ(x∗,Mαn(A))=0 for each x∗∈X∗. Thus limn→∞⁡σ(x∗,Mαn(·))≪μ for each x∗∈X∗.
(3) Suppose thatMαn⊥μ for each αn∈(0,1]. If there exists N∈A, μ(N)=0, then, for arbitrary A∈A, Mαn(A∩Nc)={0} which implies that σ(x∗,MαnA∩Nc)=0 for all α∈(0,1]. It follows that limn→∞⁡σ(x∗,Mαn(A))=0 for each x∗∈X∗. Thus limn→∞⁡σ(x∗,Mαn(·))⊥μ for each x∗∈X∗.In the following, we give the Lebesgue type decomposition theorem for generalized fuzzy number measures.Theorem 11.
LetM~:A→Fwkc(X) be a generalized fuzzy number measure. Then there exists a unique pair of generalized fuzzy number measures M~c and M~s such that (15)M~=M~c+M~s,M~c≪μ,M~s⊥μ.Proof.
Firstly, we show the existence of decomposition. IfM~:A→Fwkc(X) is a generalized fuzzy number measure, then, by Theorem 6, M~α:A→Pwkc(X) defined by (16)M~αA=x∈X:M~Ax≥αis a set valued measure for each α∈(0,1]. It follows from Theorem 7 that there exists a unique pair of weakly compact and convex set valued measures M~α(c) and M~α(s) such that (17)M~α=M~αc+M~αs,M~αc≪μ,M~αs⊥μfor each α∈(0,1]. Further,(18)σx∗,M~α·=σx∗,M~αc·+σx∗,M~αs·is the Lebesgue decomposition of σ(x∗,M~α(·)) for any x∗∈X∗ and α∈(0,1].
To complete the proof, by Theorem6, we show that {M~α(c)}α∈(0,1] and {M~α(s)}α∈(0,1] define two generalized fuzzy number measures M~c and M~s such that (19)M~=M~c+M~s,M~c≪μ,M~s⊥μ.From (18), we have(20)σx∗,M~αA=σx∗,M~αcA+σx∗,M~αsAfor each A∈A, x∗∈X∗, and α∈(0,1]. Since M~:A→Fwkc(X) is a generalized fuzzy number measure, by Theorem 6, for arbitrary α,β∈(0,1] with α≤β and A∈A, we have M~β(A)⊆M~α(A). It follows that σ(x∗,M~β(A))≤σ(x∗,M~α(A)) for each A∈A and x∗∈X∗ if α≤β. According to the uniqueness of Lebesgue decomposition for real valued measures, we have (21)σx∗,M~βcA≤σx∗,M~αcA,σx∗,M~βsA≤σx∗,M~αsAfor each x∗∈X∗ and A∈A, which imply (22)M~βcA⊆M~αcA,M~βsA⊆M~αsAfor each A∈A. Now let {αn}n∈N be a nondecreasing sequence in [0,1] converging to α∈(0,1]. We use Theorem 3 to show that (23)M~αcA=⋂n=1∞M~αncA,M~αsA=⋂n=1∞M~αnsAfor any A∈A. By Theorem 7, we can conclude that there exists a unique pair of weakly compact and convex set valued measures M~αn(c) and M~αn(s) such that (24)M~αn=M~αnc+M~αns,M~αnc≪μ,M~αns⊥μfor any αn∈(0,1]. It follows that(25)σx∗,M~αnA=σx∗,M~αncA+M~αnsA=σx∗,M~αncA+σx∗,M~αnsAfor any x∗∈X∗ and A∈A. Lemma 10 ensures that (26)limn→∞⁡σx∗,M~αnc·,limn→∞⁡σx∗,M~αns·exist. Then, from (25), we have(27)limn→∞⁡σx∗,M~αnA=limn→∞⁡σx∗,M~αncA+σx∗,M~αnsA=limn→∞⁡σx∗,M~αncA+limn→∞⁡σx∗,M~αnsA.Again by Theorem 6, we have(28)M~αA=⋂n=1∞M~αnAfor any A∈A. It follows from Theorem 3 that(29)limn→∞σx∗,M~αnA=σx∗,M~αAfor each x∗∈X∗. Then (27) and (29) imply relation(30)σx∗,M~αA=limn→∞⁡σx∗,M~αncA+limn→∞⁡σx∗,M~αnsAfor each x∗∈X∗ and A∈A. Lemma 10 shows that (31)limn→∞⁡σx∗,M~αnc·,limn→∞⁡σx∗,M~αns·are two real valued measures such that(32)limn→∞⁡σx∗,M~αnc·≪μ,limn→∞⁡σx∗,M~αns·⊥μ.Then, by uniqueness of Lebesgue decomposition for real valued measure, (20) and (30) imply (33)σx∗,M~αcA=limn→∞⁡σx∗,M~αncA,σx∗,M~αsA=limn→∞⁡σx∗,M~αnsAfor any A∈A. It follows from Theorem 3 that (34)M~αcA=⋂n=1∞M~αncA,M~αsA=⋂n=1∞M~αnsAfor each A∈A. Then {M~α(c)(·)}α∈(0,1] and {M~α(s)(·)}α∈(0,1] satisfy conditions (1) and (2) of Theorem 6, respectively. For any A∈A, let (35)M~iAx=sup⁡α∈0,1:x∈M~αiA,ifα∈0,1,x∈M~αiA≠∅,0,ifα∈0,1,x∈M~αiA=∅, where i=c,s. Then M~c and M~s define two generalized fuzzy number measures such that (36)M~cAα=M~αcA,M~sAα=M~αsAfor any A∈A and α∈(0,1]. Since M~α(c)≪μ, M~α(s)⊥μ for all α∈(0,1], by Lemma 10, M~c≪μ and M~s⊥μ.
In the following, we claim thatM~=M~c+M~s. Obviously, for any A∈A, M~c(A)+M~s(A)∈Fwkc(X) and for any α∈(0,1],(37)M~cA+M~sAα=M~cAα+M~sAα=M~αcA+M~αsA=M~αA.Then, if M~(A)(x)>0, (38)M~Ax=sup⁡α∈0,1:x∈M~αA=sup⁡α∈0,1:x∈M~cA+M~sAα=M~cA+M~sAx.If M~(A)(x)=0, then {α∈(0,1]:x∈M~α(A)}=∅, which implies that, for any α∈(0,1], x∉M~α(A). Hence, for any α∈(0,1], x∉(M~c(A)+M~s(A))α; that is, {α∈(0,1]:x∈(M~c(A)+M~s(A))α}=∅, which implies that (M~c(A)+M~s(A))(x)=0. It follows that M~(A)(x)=(M~c(A)+M~s(A))(x).
Lastly, we show the uniqueness of decomposition. Suppose that there exist generalized fuzzy number valued measuresM~cj,M~sj, j=1,2, such that(39)M~=M~cj+M~sj,M~cj≪μ,M~sj⊥μ,j=1,2.Then, for any α∈(0,1], (40)M~α=M~c1α+M~s1α,M~c1α≪μ,M~s1α⊥μ,M~α=M~c2α+M~s2α,M~c2α≪μ,M~s2α⊥μ.According to the uniqueness of Lebesgue decomposition of set valued measures, we have(41)M~c1α=M~c2α,M~s1α=M~s2α,which implies that M~c1=M~c2, M~s1=M~s2. This completes the proof.In the following, we generalize weak Radon-Nikodým theorem for set valued measures [20] to generalized fuzzy number measures.LetF:Ω→Pf(X) be a set valued function. F(ω) is said to be scalarly measurable if, for each x∗∈X∗, the real valued function σ(x∗,F(ω)) is measurable. F:Ω→Pf(X) is said to be weakly integrable bounded if the real valued function |x∗F|:Ω→R defined by |x∗F|(ω)=supx∈F(ω)|〈x∗,x〉| is integrable for each x∗∈X∗. f:Ω→X is called a measurable selector of F:Ω→Pf(X) if f(ω)∈F(ω) for each ω∈Ω. A measurable selector f(ω) of F(ω) is called a Pettis integrable selector of F(ω) if f(ω) is Pettis integrable. We denote by SwF the set of all Pettis integrable selectors of F(ω). F:Ω→Pf(X) is said to be Aumann-Pettis integrable if SwF≠∅. In this case, the Aumann-Pettis integral of F:Ω→Pf(X) is defined by (42)w∫AFωdμω=w∫Afωdμω:f∈SwF,∀A∈A.We denote by Lwkcw(Ω,X) the set of all Aumann-Pettis integrable, weakly compact, and convex set valued functions.Theorem 12 (see [20]).
Suppose thatX has the RNP and X∗ is separable. Let M:A→Pwkc(X) be a μ-continuous set valued measure of bounded variation. M has a weak Radon-Nikodým derivative F∈Lwkcw(Ω,X) if and only if M is of σ-bounded variation.The mappingF~:Ω→Fwkc(X) is called a measurable fuzzy mapping, if F~α(ω)={x∈X:F~(ω)(x)≥α} is a measurable set valued mapping for each α∈(0,1]. F~(ω) is called weakly integrable bounded if F~α(ω) is weakly integrable bounded set valued mapping for each α∈(0,1].Definition 13 (see [19]).
The mappingF~:Ω→Fwkc(X) is said to be weakly integrable, if, for each A∈A, there exist u~A∈Fwkc(X) such that, for each α∈(0,1], (43)u~Aα=w∫AF~αωdμω.In this case u~A=(w)∫A‍F~(ω)dμ(ω) is called the weak integral of F~(ω) over A.
LetLwkcw(Ω,X) denote the set of all weakly integrable bounded measurable fuzzy mappings.Definition 14.
LetM~:A→Fwkc(X) be a generalized fuzzy number measure. If there exists a weakly integrable bounded measurable fuzzy mapping F~:Ω→Fwkc(X) such that (44)M~A=w∫AF~ωdμω,∀A∈A,then one calls that F~:Ω→Fwkc(X) is a weak Radon-Nikodým derivative for M~ with respect to μ.
We call thatM~ is of σ-bounded variation if M~α is of σ-bounded variation for each α∈(0,1]. In the following we show the existence of weak Radon-Nikodým derivatives for generalized fuzzy number measures.Theorem 15.
Suppose thatX has the RNP and X∗ is separable. Let M~:A→Fwkc(X) be a μ-continuous generalized fuzzy number measure of σ-bounded variation. Then M~ has a weak Radon-Nikodým derivative F~∈Lwkcw(Ω,X); namely, there exists a weakly integrable bounded measurable fuzzy mapping F~:Ω→Fwkc(X) such that (45)M~A=w∫AF~ωdμωfor each A∈A.Proof.
SinceM~:A→Fwkc(X) is a generalized fuzzy number measure, by Theorem 6, M~α:A→Pwkc(X) defined by (46)M~αA=x∈X:M~Ax≥αis a set valued measure for each α∈(0,1]. Since M~:A→Fwkc(X) is μ-continuous and is of σ-bounded variation, M~α:A→Pwkc(X) is μ-continuous and is of σ-bounded variation for each α∈(0,1]. According to Theorem 12, for each α∈(0,1] there exists Fα∈Lwkcw(Ω,X) which is a weak Radon-Nikodým derivative for M~α with respect to μ; namely, Fα satisfies(47)M~αA=w∫AFαωdμωfor each α∈(0,1] and A∈A.
In the following, we will show that the family{Fα(ω)}α∈(0,1] defines a weakly integrable bounded measurable fuzzy mapping F~(ω). Firstly, we prove that the family {Fα(ω)}α∈(0,1] defines a generalized fuzzy number F~(ω) for each ω∈Ω. We will prove that all the conditions of Theorem 2 are satisfied. Obviously, Fα(ω)∈Pwkc(X) for all α∈(0,1] and ω∈Ω. For 0<β≤α≤1, we have M~α(A)⊆M~β(A) for any A∈A which implies that (48)w∫AFαωdμω⊆w∫AFβωdμω.Then, by properties of support function, we have(49)σx∗,w∫AFαωdμω≤σx∗,w∫AFβωdμωfor each x∗∈X∗. It follows from Lemma 4.3 [19] that (50)∫Aσx∗,Fαωdμω≤∫Aσx∗,Fβωdμωfor each x∗∈X∗. Since this is true for all A∈A we deduce that (51)σx∗,Fαω≤σx∗,Fβω.Also, since Fα(ω) and Fβ(ω) are weakly compact and convex, we have Fα(ω)⊆Fβ(ω). Let {αn}n∈N⊆[0,1] be a nondecreasing sequence converging to α∈(0,1]. By Theorem 6, we have M~α(A)=⋂n=1∞‍M~αn(A) which implies that (52)w∫AFαωdμω=⋂n=1∞w∫AFαnωdμωfor each A∈A. It follows from Theorem 3 that (53)limn→∞⁡σx∗,w∫AFαnωdμω=σx∗,w∫AFαωdμω.By Lemma 4.3 [19], we have(54)limn→∞⁡∫Aσx∗,Fαnωdμω=∫Aσx∗,Fαωdμω.Since Fαn(ω) is weak integrable bounded and (55)σFαnωx∗≤x∗Fα1ωfor all n∈N, ω∈Ω, and x∗∈X∗, by Lebesgue’s dominated convergence theorem and (54), we obtain that (56)limn→∞⁡∫Aσx∗,Fαnωdμω=∫Alimn→∞⁡σx∗,Fαnωdμω=∫Aσx∗,Fαωdμωwhich implies that(57)limn→∞⁡σx∗,Fαnω=σx∗,Fαω.Again, by Theorem 3, we have σx∗,Fα(ω)=⋂n=1∞‍σx∗,Fαn(ω). Now, applying Theorem 2, we get that the family {Fα(ω)}α∈(0,1] generates the unique generalized fuzzy number F~(ω) such that F~α(ω)=Fα(ω) for each ω∈Ω. Since Fα(ω) is weakly integrable bounded set valued measurable function for each α∈(0,1], F~(ω) is a weakly integrable bounded measurable fuzzy valued function.
To prove thatF~(ω) is a weak Radon-Nikodým derivative for M~ with respect to μ, we will show that (58)M~A=w∫AF~ωdμωfor each A∈A. By Definition 13, (59)w∫AF~ωdμω∈FwkcXsuch that (60)w∫AF~ωdμωα=w∫AF~αωdμωfor each α∈(0,1] and A∈A. Together with (47), we have (61)w∫AF~ωdμωα=w∫AF~αωdμω=w∫AFαωdμω=M~αA=M~Aαfor each α∈(0,1] and A∈A, which implies that (62)M~A=w∫AF~ωdμωfor each A∈A. This completes the proof.



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*Source: 576134-2015-11-25.xml*