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Some monotone properties for the expectation offuzzy random variablesDaisy, M. H. Huang aa Department of Mathematics College of Science and Engineering , Tainan Woman'sCollege of Arts and Technology , Tong Liao , 028043 , People's Republic of China E-mail:Published online: 18 Jun 2013.

To cite this article: Daisy, M. H. Huang (2004) Some monotone properties for the expectation of fuzzy random variables,Journal of Information and Optimization Sciences, 25:3, 453-460, DOI: 10.1080/02522667.2004.10699620

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Some monotone properties for the expectation of fuzzy randomvariables

Dabuxilatu Wang∗

Department of Mathematics

College of Science and Engineering

Inner Mongolia University for Nationalities

Tong Liao, 028043

People’s Republic of China

E-mail : dbxlt0@yahoo.com

Abstract

An order on a class of fuzzy sets is induced by a non-empty convex cone. Based onthe ordering some monotone properties for the expectation of fuzzy random variables in thesense of Puri-Ralescu are obtained.

Keywords : Fuzzy random variables, expectation, order relations.

1. Introduction

Fuzzy random variables by [10] have been introduced as an extensionof real- or vector-valued random variables as well as random sets basedon measurable set-valued functions (cf. [1, 3]). One of the most commonand useful characteristics of a random variable is its expected value.The concept of expectation plays an important role as a central notionin Probability theory, Statistics and decision theory. The expectationof fuzzy random variables has been presented by [10] as an extensionof the expectation of random variable based on the Aumann integralof measurable set-valued functions (cf. [1]), whose properties such asthe linearity and the dominated convergence have been examined in

∗Current corresponding address : Mr. Dabuxilatu Wang c/o Professor Masami Yasuda,Department of Mathematics and Informatics, Faculty of Science, Chiba University, Yayoi-cho, Inage-ku, Chiba 263-8522, Japan.——————————–Journal of Information & Optimization SciencesVol. 25 (2004), No. 3, pp. 453–460c© Taru Publications 0252-2667/04 $2.00 + .25

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[10, 8, 13]. However, as far as the author is aware, the monotone propertiesof the expectation of multi-dimensional fuzzy random variables withrespect to some order relation induced by a cone have not been treated inthe literature. M. Kurano et al. [6] has proposed an order relation for a classof multi-dimensional fuzzy sets by a convex cone, which is thought as anextension of the fuzzy max order (cf. [12]) on a class of one-dimensionalfuzzy sets to the multi-dimensional case. For the further discussion, thereader are referred to [14].

In this paper, we give some monotone properties of the expectationof multi-dimensional fuzzy random variables with respect to the orderrelation given in [6], which leads to the monotone convergence theorem.

The rest of the paper is organized as follows : In Section 2 we presentpreliminary definitions and results which will be used later; In Section 3some reasonable monotone properties of expectation of fuzzy randomvariables w.r.t. this order of fuzzy sets are obtained.

2. Preliminaries

Let (Rn, ‖ · ‖) be the n -dimensional Euclidean space with theassociated norm. Let C(Rn) be the set of all non-empty compact convexsubset of Rn and K be a non-empty closed convex cone of Rn . Let(C(Rn), dH) be the Hausdorff metric space, that is, If A, B ∈ C(Rn) ,

dH(A, B) = max

supa∈A

infb∈B

‖a− b‖ , supb∈B

infa∈A

‖a− b‖

where ‖a − b‖ =√

∑ni=1(ai − bi)2 , (a1, . . . , an), (b1, . . . , bn) ∈ Rn . We

denote by B the Borel field on Rn induced by the usual metric in Rn .For A ∈ C(Rn) , we define the magnitude of A by ‖A‖ = dH(A, 0) =supx∈A ‖x‖ .

Let F(Rn) denote the class of the fuzzy subsets of Rn, u : Rn → [0, 1] ,such that: (i) u is upper semicontinuous (i.e., the α -level sets of u ,uα = x ∈ Rn : u(x) ≥ α , are closed for all 0 < α ≤ 1) ; (ii) u isnormal (i.e., u1 6= ∅) ; (iii) u is convex (i.e., uα is convex subset of Rn forall 0 < α ≤ 1) ; and (iv) the closure of the support of u, u0 = clx ∈ Rn :u(x) > 0 , is compact.

By F0(Rn) , we denote the class of all fuzzy subsets on Rn satisfyingconditions (i), (ii) and (iv) mentioned above.

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SOME MONOTONE PROPERTIES 455

We will consider that the F(Rn) is endowed with the uniform metricd∞ (cf [11]), i.e., for u, v ∈ F(Rn) ,

d∞(u, v) = supα∈(0,1)

dH(uα , vα) .

Note that (F(Rn), d∞) is a complete metric space but not separable(cf. [3], pp. 57 or [10]).

Definition 2.1. (1) A pseudo-order relation 4K on Rn , is defined as: Fora, b ∈ Rn , a 4K b if and only if b− a ∈ K . (2) A pseudo-order relation4K on C(Rn) is defined as: For A, B ∈ C(Rn), A 4K B if and only ifA + K ⊃ B and B− K ⊃ A . Note that this definition is equivalent to thecorrespondent definition in [6].

Definition 2.2. For u, v ∈ F(Rn) , u 4K v if and only if uα 4K vα for allα ∈ [0, 1] . For n = 1, K = [0, ∞) , u 4K v (denoted by u 41 v later)if and only if u−α ≤ v−α , u+

α ≤ v+α , where uα = [u−α , u+

α ] , vα = [v−α , v+α ]

namely, 41 here is equivalent to the fuzzy max order (cf. [12, 6]).

The dual cone of K is denoted by K+ , and

K+ := a ∈ Rn | a · x ≥ 0 , ∀ x ∈ K

where · denotes the inner product on Rn .

Lemma 2.1 ([6]). For u, v ∈ F(Rn) , u 4K v if and only if a · uα 41 a · vα

for all a ∈ K+ and α ∈ [0, 1] .

Let (Ω, A, P) is a probability space.

Definition 2.3 ([10]). A mapping X : Ω → F0(Rn) is called fuzzy randomvariable, if for any α ∈ [0, 1], (ω, x), x ∈ Xα(ω) ∈ A × B . whereXα(ω) := x ∈ Rn, X(ω)(x) ≥ α, (α ∈ (0, 1]) and X0(ω) := clx ∈Rn |, X(ω)(x) > 0 .

Fuzzy random variable X is said to be integrably bounded if for eachx ∈ Xα(ω) , ω ∈ Ω , there exists hα ∈ L1(Ω) , such that ‖x‖ ≤ hα(ω) ,where L1(Ω) denotes the set of all real integrable functions on Ω .

Definition 2.4 ([10]). The expectation E(X) of a fuzzy random variableX is a fuzzy set v ∈ F0(Rn) such that for any α ∈ [0, 1] , vα = x ∈Rn : v(x) ≥ α =

∫Xα , where

∫Xα := ∫Ω f dP, f ∈ S(Xα) denotes the

Aumann integral of set-valued functions Xα , and S(Xα) denotes the set

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of all integrable selections of Xα (α ∈ [0, 1]) w.r.t. the probability measureP (cf. [1, 10]).

Note that if P is a non-atomic probability measure, then E(X) ∈F(Rn) , i.e., E(X) is convex. We henceforth suppose that the probabilitymeasure is non-atomic.

It is easy to check that for n = 1 , (E(X))α = [E(X−α ), E(X+

α )] for allα ∈ [0, 1] .

Lemma 2.2 (Theorem 4.2 of [6]). Let um ⊂ F(Rn) and u ∈ F(Rn) suchthat um 4K um + 1(m ≥ 1) and d∞(um, u) → 0(m → 0) . Then it holds thatu1 4K u .

3. Monotone properties of the expectation

In this section, we investigate some monotone properties of theexpectation of fuzzy random variables w.r.t. the pseudo-order 4K

induced by convex cone K as well as the uniform metric d∞ .

Definition 3.1. For fuzzy random variables X, Y, X 4K Y meansXα(ω) 4K Yα(ω) for all α ∈ [0, 1] and ω ∈ Ω .

Lemma 3.1. If X, Y are integrably bounded fuzzy random variables valued inF(R) and X 41 Y , then E(X) 41 E(Y) .

Proof. Trivial. ¤

Theorem 3.1. If X, Y are integrably bounded fuzzy random variables valued inF(Rn) and X 4K Y , then E(X) 4K E(Y) .

Proof. We first prove that for any a a ∈ Rn , it holds that

a · E(Xα) = E(a · Xα),

ie.,

a ·∫

XαdP =∫

a · XαdP (1)

where a · D = a · x | x ∈ D for and D ∈ C(Rn) . In fact, from theproperty of Bochner integral (cf. [4, 5]), we have

a ·∫

XαdP = a ·∫

f dP | f ∈ S(Xα)

=∫

a · f dP | f ∈ S(Xα)

. (2)

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SOME MONOTONE PROPERTIES 457

Obviously, a · f ∈ S(a · Xα) for all f ∈ S(Xα) such that a · S(Xα) ⊂S(a · Xα) , which implies the inclusion ⊆ in (1). Since a · Xα(ω) is convexand compact for each ω ∈ Ω , we can write a · Xα = [δ−α , δ+

α ] , whereδ−α (ω) = maxx∈Xα(ω) a · x , δ+

α (ω) = minx∈Xα(ω) a · x . Applying theBorel selection theorem of [2], there exists f , f ∈ S(Xα) such that δ− =a · f , δ+ = a · f . Observing that

∫a · XαdP = [

∫δ−α dP,

∫δ+α dP] , together

with the inclusion ⊆ in (1), we obtain (1). Now, we prove E(X) 4K E(Y) .By Lemma 2.1, X 4K Y means that a · Xα(ω) 41 a · Yα(ω) for alla ∈ K+,α ∈ [0, 1], ω ∈ Ω . Applying Lemma 3.1, we obtain

∫a · XαdP 41

∫a ·YαdP ,

Thus, by (1), it follows that

a ·∫

XαdP 41 a ·∫

YαdP

for all a ∈ K+ . Here, using Lemma 2.1 again, we have∫

XαdP 4K∫

YαdP,i.e., E(X) 4K E(Y) . This completes the proof. ¤

Let Rn+ be the subset of entrywise non-negative elements in Rn .

Lemma 3.2. Let K ⊂ Rn+ . For any x, y ∈ Rn with x 4K y, let

g(x, y) := (x + K) ∩ (y− K)

Then,

‖g(x, y)‖ ≤ 2‖x‖+ ‖y‖ .

Proof. Let z ∈ g(x, y) . Then, there exists k, k′ ∈ K such that z = x + k =y− k′ . Since k + k′ ≥ k ≥ 0 , with componentwise inequalities, we have‖y− x‖ = ‖k + k′‖ ≥ ‖k‖ . Thus, we have

‖z‖ ≤ ‖x‖+ ‖k‖ ≤ ‖x‖+ ‖y− x‖ ≤ 2‖x‖+ ‖y‖.

This completes the proof. ¤

Lemma 3.3. Let K ⊂ Rn+ . For any A, B, C ∈ C(Rn) with A 4K B 4K C , let

G(A, C) :=⋃

x4k y,x∈A,y∈Cg(x, y) .

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Then, it holds that G(A, C) ⊃ B and

‖G(A, C)‖ ≤ 2‖A‖+ ‖C‖.

Proof. It follows from Lemma 3.2 directly. ¤

Definition 3.2. Let um, u ∈ F(Rn)(m ∈ N) . um converges to u inmetric d∞ if and only if

limm→∞ d∞(um, u) = 0

and denoted by um →d∞ u; um weakly converges to u in metric d∞ ifand only if

limm→∞ dH(um,α , uα) = 0

except for at most countable α ∈ [0, 1] and denoted by um →w·d∞ u .(cf. [7]).

Theorem 3.2 (Monotone Convergence Theorem). Let K ⊂ Rn+ . Let

Xm, m = 1, 2, . . ., X be integrably bounded fuzzy random variables valuedin F(Rn) such that Xm(ω) →d∞ X(ω) , P -a.s. ω ∈ Ω . Suppose thatXm(ω) 4K Xm+1(ω) , P -a.s. ω ∈ Ω(m ≥ 1) . Then, E(Xm) →d∞ E(X) .

Proof. By Lemma 2.2, we have that

X1(ω) 4K Xm(ω) 4K X(ω)

P -a.s. ω ∈ Ω for all m ≥ 1 , which implies that

X1,α(ω) 4K Xm,α(ω) 4K Xα(ω)

P -a.s. ω ∈ Ω for all m ≥ 1 . By Lemma 3.3,

‖Xm,α(ω)‖ ≤ 2‖X1,α(ω)‖+ ‖Xα(ω)‖ ≤ 2‖X1,0(ω)‖+ ‖X0(ω)‖.

Since X1 and X are integrably bounded, there exists h, h′ ∈ L1(Ω)such that ‖X1,0(ω)‖ ≤ h(ω) and ‖X0(ω)‖ ≤ h′(ω), (ω ∈ Ω) , whence‖Xm,α(ω)‖ ≤ 2h(ω) + h′(ω) for m ≥ 1 and α ∈ [0, 1] . Thus,applyingthe dominated convergence type theorem (Theorem 4.3 of [10]), We obtain

E(Xm) →d∞ E(X) .

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SOME MONOTONE PROPERTIES 459

This completes the proof. ¤

Theorem 3.3 (Weak Monotone Convergence Theorem). Let K ⊂ Rn+ . Let

Xm, m = 1, 2, . . ., X be integrable bounded fuzzy random variables valuedin F(Rn) such that Xm(ω) →w·d∞ X(ω) , P -a.s. ω ∈ Ω . Suppose thatXm(ω) 4K Xm+1(ω) , P -a.s. ω ∈ Ω(m ≥ 1) . Then, E(Xm) →w·d∞ E(X) .

Proof. For any α ∈ [0, 1] and m ≥ 1 , it holds that (cf. [10])

dH(E(Xm,α), E(Xα)) ≤∫

dH(Xm,α , Xα)dP . (3)

From the hypotheses of the Theorem 3.3, we have dH(Xm,α(ω), Xα(ω)) →0(m → ∞) P− a.s. ω ∈ Ω except for at most countable α ∈ [0, 1] . Onthe other hand, By the same way of the proof of Theorem 3.2, we obtain

dH(Xm,α(ω), Xα(ω)) ≤ dH(Xm,α(ω), 0) + dH(Xα(ω), 0)≤ 2(h(ω) + h′(ω)),

where h, h′ ∈ L1(Ω) are given in the proof of Theorem 3.2. Thus,by the classical Lebesgue dominated convergence theorem, we have∫

dH(Xm,α , Xα)dP → 0(m → ∞) except for at most countable α ∈ [0, 1] .This leads by (3) that E(Xm) →w·d∞ E(X) . This completes the proof. ¤

Acknowledgements. The author greatly appreciates Professor M. Yasuda,Professor J. Nakagami, Professor M. Kurano of Chiba University, Japanfor their valuable comments and corrections which led to considerableimprovements of the manuscript.

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Received December, 2002 ; Revised February, 2003

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