Approximating integrably bounded fuzzy random variables in terms of

Approximating integrably bounded fuzzy random variables in terms of the \generalized" Hausdor® metric MIGUEL LOPEZ-DIAZ

and MARIA ANGELES GIL

Departamento de Estad¶³stica, I.O. y D.M., Universidad de Oviedo 33071 Oviedo, SPAIN

ABSTRACT Consider the class of the mappings from a Euclidean space to the unit interval [0;1] (that is, the class of the fuzzy sets of this space) which are upper semicontinuous, the closure of their supports are compact, and the inverse images of the singleton f1g are nonempty sets. When the metric d1 , which generalizes the Hausdor® metric, is de¯ned on the preceding class, the resulting metric space is complete but nonseparable. As a consequence of this nonseparability, given a probability space and an integrably bounded fuzzy random variable associated with it, one cannot ensure the existence of a sequence of simple fuzzy random variables d1 -converging almost surely to the former one. However, we are going to show in this paper that for any given integrably bounded fuzzy random variable we can guarantee the existence of a sequence of integrably bounded fuzzy random variables whose ®-level functions are simple random compact sets, and which d1 -converges at every point to the former variable. On the basis of this result we will obtain in this paper a constructive de¯nition of integrably bounded fuzzy random variables, and an operational way to compute their fuzzy expected values.

1. INTRODUCTION The concept of fuzzy random variable was introduced by Puri and Ralescu [11] as a generalization of random variables and random sets, and as an important tool for representing imprecise data associated with the outcomes of a random experiment. Some studies have been developed in connection with fuzzy random variables, most of them focused on the fuzzy expected value of a fuzzy random variable, on sequences of fuzzy random variables, on special types of fuzzy random variables, and on di®erent properties and applications of them (see, 1

for instance, Puri and Ralescu [10], [11], [12], Klement et al. [7], Ban [3], Stojakovi¶c [13], [14], Gil and L¶ opez-D¶³az [6]). The aim of this paper is to state an approximation result in terms of the metric d1 : if (­; A; P ) is a probability space and X is an integrably bounded fuzzy random variable associated with it, we can construct a sequence fXm gm of integrably bounded fuzzy random variables with simple ®-level functions such that limm!1 d1 (Xm (!); X(!)) = 0; for all ! 2 ­:

After some preliminaries on random sets and fuzzy random variables which will be recalled in Section 2, we will introduce an approximation result concerning random sets in Section 3; in Section 4 we present the above commented approximation result for fuzzy random variables. Finally, in Section 5 we derive some constructive results for fuzzy random variables and their fuzzy expected values. 2. PRELIMINARIES ON RANDOM SETS AND FUZZY RANDOM VARIABLES ¡ ¢ Let K(Rp ) Kc (Rp ) be the collection of non-empty compact (and convex) subsets of the Euclidean space Rp : This space can be endowed with a linear structure induced by the scalar multiplication and the Minkowski addition, that is, ¸A = f¸a : a 2 Ag;

A + B = fa + b : a 2 A; b 2 Bg;

for all A; B 2 K(Rp ); and ¸ 2 R: If d is the Hausdor® metric on K(Rp ), which for A; B 2 K(Rp ) is given by d(A; B) = max f sup inf ja ¡ bj; sup inf ja ¡ bj g; a2A b2B

b2B a2A p

where j:j denotes the Euclidean norm, then (K(R ); d) is a complete metric space, and (Kc (Rp ); d) is a closed subspace (see Debreu, [5]). Given a measurable space (­; A) and the metric space (K(Rp ); d), a random set (often referred to as a random compact set - see, for instance, Artstein and Vitale, [1] -) associated with (­; A) is a Borel measurable mapping X : ­ ! K(Rp ): If X : ­ ! K(Rp ) is a set-valued mapping, X is a 2

random set if and only if X ¡1 (C) = f! 2 ­ : X(!) \ C 6 = ;g 2 A for all p C 2 K(R ) (see Debreu, [5]). If X is a random set, the mapping denoted by jjXjj and de¯ned by jjX(!)jj = d(f0g; X(!)) = supx2X(!) jxj for all ! 2 ­ is a random variable. Given a probability space (­; A; P ), Aumann [2] has de¯ned the expected value EX of the random set X (with respect to P ) as follows: EX = f Ef : f 2 L1 (­; A; P ); f 2 X a:s: [P ] g; where Ef is the (classical) expected value of the random variable f (with respect to P ). If X is a random set and jjXjj 2 L1 (­; A; P ); then EX 2 K(Rp ): In particular, if X is a simple random convex set (that is, Im X is a ¯nite subset of Kc (Rp ), so that there exists a partition fAi gni=1 ½ A of ­ P such that X(!) = ni=1 ÂAi (!) Ki for all ! 2 ­, with Ki 2 Kc (Rp ), and  meaning the indicator function), then EX =

n X

P (Ai ) Ki

i=1

(see Byrne [4]). A fuzzy set of Rp is a mapping V : Rp ! [0; 1]: From now on, we will denote by V® the ®-level of V (that is, V® = fx 2 Rp : V (x) ¸ ®g) for all ® 2 (0; 1]; and by V0 the closure of the support of V (that is, V0 = cl fx 2 Rp : V (x) > 0g). ¡ ¢ Let F (Rp ) Fc (Rp ) be the class of the fuzzy sets V satisfying the follow=; ing conditions: i) V is upper semicontinuous, ii) V0 is compact, iii) V1 6 ¡ ¢ and iv) V® is convex for all ® 2 [0; 1] : F(Rp ) can be linearized with the fuzzy addition and the fuzzy product by an scalar (based on Zadeh's extension principle - see [15] -). On the other hand, F (Rp ) can be endowed with the metric d1 de¯ned as a possible generalization of the Hausdor® metric d as follows: d1 (V; W ) = sup d(V® ; W® ); for all V; W 2 F(Rp ); ®2(0;1]

3

and which is sometimes referred to as the generalized Hausdor® metric. (F (Rp ); d1 ) is a complete metric space (see Puri and Ralescu, [11]), but it is not separable (see Klement et al., [7]). Given a measurable space (­; A) and the metric space (F(Rp ); d1 ) a fuzzy random variable associated with (­; A) is a Borel measurable function X : ­ ! F(Rp ): If X is a fuzzy random variable, the mapping X® : ­ ! K(Rp ); de¯ned ¡ ¢ by X® (!) = X(!) ® is a random set for all ® 2 [0; 1] (see Klement et al., [7], for ® = 0). A fuzzy random variable X is said to be an integrably bounded fuzzy random variable associated with the probability space (­; A; P ) if and only if jjX0 jj 2 L1 (­; A; P ): In this case, the fuzzy expected value of X, is the unique e e ® = EX® for all ® 2 (0; 1]. EX e with the property (EX) fuzzy set of Rp , EX, can be proven to belong to F(Rp ) (see Puri and Ralescu, [11]). 3. APPROXIMATING RANDOM SETS Since (K(Rp ); d) is a separable metric space, given a random set X one can guarantee the existence of a sequence of simple random sets fXm gm such that limm!1 d(Xm (!); X(!)) = 0; a:s: [P ]: We now propose a special sequence satisfying this approximation result at every point ! 2 ­, which will be used in Section 4. Proposition 3.1. Let (­; A; P ) be a probability space and let X be a random set associated with (­; A). We consider the p-cubes Cjm1 ;:::;jp

=

p Y

i=1

[¡m + ji

1 2m¡1

; ¡m + (ji + 1)

1 2m¡1

];

ji = 0; 1; 2; : : : ; 2m2m¡1 ¡ 1; i = 1; : : : ; p: Then, the mapping de¯ned as ½S Cjm1 ;:::;jp if X(!) ½ [¡m; m]p ; j1 ;:::;jp :X(!)\C m =; 6 j1 ;:::;jp Xm (!) = [¡m; m]p otherwise, is a simple random set, and limm!1 d(Xm (!); X(!)) = 0; for all ! 2 ­: 4

Proof: Let

[

¢m (!) =

Cjm1 ;:::;jp :

j1 ;:::;jp : X(!)\C m =; 6 j1 ;:::;jp

If C 2 K(Rp ); then we have that = ;g = f! 2 ­ : ¢m (!) \ C 6

[

j1 ;:::;jp : C\C m =; 6 j1 ;:::;jp

= ;g: f! 2 ­ : X(!) \ Cjm1 ;:::;jp 6

Since X is a random set, we have that f! 2 ­ : X(!)\Cjm1 ;:::;jp 6 = ;g 2 A; and hence ¢m is a random set. Analogously, f! 2 ­ : X(!) ½ [¡m; m]p g 2 A; which implies that Xm is a random set and, obviously, it is a simple random set. On the other hand, for each ! 2 ­ we have that X(!) 2 K(Rp ); so that there exists l 2 N with X(!) ½ [¡l; l]p ; whence for all m ¸ l we can conclude that X(!) ½ Xm (!) and therefore d(Xm (!); X(!)) = supa2Xm (!) inf b2X(!) ja ¡ bj:

If a 2 Xm (!); there exists j1 ; : : : ; jp ; such that a 2 Cjm1 ;:::;jp and Cjm1 ;:::;jp = ;: If b 2 Cjm1 ;:::;jp \ X(!); then ja ¡ bj ∙ p=2m¡1 and hence d(X(!); \X(!) 6 Xm (!)) ∙ p=2m¡1 ; which guarantees that limm!1 d(Xm (!); X(!)) = 0; for all ! 2 ­: 4. APPROXIMATING INTEGRABLY BOUNDED FUZZY RANDOM VARIABLES In this section, for any given probability space, (­; A; P ) and an integrably bounded fuzzy random variable X associated with it, we are going to obtain a sequence of integrably bounded fuzzy random variables fXm gm such that limm!1 d1 (Xm (!); X(!)) = 0; for all ! 2 ­; and so that for each m 2 N, the variables Xm has simple ®-level functions ¡ ¢ Xm® (with Xm® : ­ ! K(Rp ) such that Xm® (!) = Xm (!) ® for all ! 2 ­) whatever ® 2 (0; 1] may be. 5

Firstly, we recall a supporting result (Negoita and Ralescu, [9]), which will be used in this section. Lemma 4.1. Let M be a set and let fM® ; ® 2 [0; 1]g be a family of subsets of M such that i) M0 = M; ii) ® ∙ ¯ implies M¯ ½ M® ; iii) ®1 ∙ ®2 ∙ : : : ; with limn!1 ®n = ® implies M® = \1 n=1 M®n ; then, the function Á : M ! [0; 1]; de¯ned by Á(x) = sup f® 2 [0; 1] : x 2 M® g; satis¯es fx 2 M : Á(x) ¸ ®g = M® ; for all ® 2 [0; 1] (that is, Á de¯nes a fuzzy set of M ). Theorem 4.2. Let (­; A; P ) be a probability space and let X be an integrably bounded fuzzy random variable. Then, there exists a sequence of integrably bounded fuzzy random variables fXm gm with simple ®-level functions such that lim d1 (Xm (!); X(!)) = 0; for all ! 2 ­: m!1

Proof: For each ® 2 (0; 1]; let fYm® gm denote the sequence of simple random sets such that limm!1 d(Ym® (!); X® (!)) = 0 for all ! 2 ­, suggested in Proposition 3.1. We now de¯ne the mappings Xm so that, given ® 2 (0; 1], ½ ® if X0 (!) ½ [¡m; m]p ; Ym (!) Xm® (!) = p [¡m; m] otherwise. Since X0 is a random set, then f! 2 ­ : X0 (!) ½ [¡m; m]p g 2 A; whence the mappings Xm® are simple random sets. We are now going to show that Xm are integrably bounded fuzzy random variables satisfying that limm!1 d1 (Xm (!); X(!)) = 0; for all ! 2 ­: To this purpose, we are ¯rst going to prove that Xm (!) 2 F(Rp ); for all m 2 N and ! 2 ­. If ! satis¯es that X0 (!) 6 ½ [¡m; m]p ; then we can obviously conclude that Xm 2 F (Rp ) for all m 2 N and ! 2 ­. 6

If ! satis¯es that X0 (!) ½ [¡m; m]p , and ® < ¯; we have that X¯ (!) ½ X® (!); and hence for all ji = 0; 1; : : : ; 2m2m¡1 ¡ 1; i = 1; : : : ; p; = ;g ½ f! 2 ­ : X® (!) \ Cjm1 ;:::;jp 6 = ;g; f! 2 ­ : X¯ (!) \ Cjm1 ;:::;jp 6 which implies in accordance with the construction in Proposition 3.1 that Ym¯ (!) ½ Ym® (!), whence Xm¯ (!) ½ Xm® (!):

To prove the condition iii) in Lemma 4.1, we now consider the sequence f®n gn " ® along with an x 2 \1 n=1 Xm®n (!); and we are going to show ®n ® that x 2 Xm® (!) (or equivalently, if x 2 \1 n=1 Ym (!); then x 2 Ym (!)): ®n m If x 2 \1 n=1 Ym (!); then for all n 2 N we have that x 2 Cj1n ;:::;jpn with = ; for some j1n ; : : : ; jpn (depending on n), whence Cjm1n ;:::;jpn \ X®n (!) 6 m there exists b®n 2 Cj1n ;:::;jpn \ X®n (!) such that b®n ; x 2 Cjm1n ;:::;jpn :

Since jjX0 (!)jj < 1; fb®n gn is a bounded sequence, which ensures the existence of a convergent subsequence we will denote in the same way, and b0 being the limit of this subsequence. Since for each ®n there exists j1n ; : : : ; jpn such that b®n ; x 2 Cjm1n ;:::;jpn ; then there exists a p-cube Cjm1 ;:::;jp such that b0 ; x 2 Cjm1 ;:::;jp . On the other hand, b®n 2 X®n (!) for all n 2 N, which implies b0 2 X® (!) and hence [ x2 Cjm1 ;:::;jp j1 ;:::;jp : X® (!)\C m =; 6 j1 ;:::;jp

whence x 2 Ym® (!), and consequently x 2 Xm® (!); so that Xm (!) is a fuzzy set for all m 2 N and ! 2 ­.

Since for each ! 2 ­; we have that jjXm0 (!)jj ∙ p1=2 m; and Xm® (!) 2 K(Rp ) for all ® 2 (0; 1]; we can conclude that Xm (!) 2 F (Rp ) for all m 2 N and ! 2 ­. Before examining the measurability of Xm , we are going to analyze the approximation result in the present theorem. Thus, for all ! 2 ­; and for all ® 2 (0; 1], in virtue of Proposition 3.1 if m ¸ jjX0 (!)jj we have that d(Xm® (!); X® (!)) = d(Ym® (!); X® (!)) ∙ 7

p 2m¡1

;

which implies d1 (Xm (!); X(!)) ∙

p 2m¡1

;

and hence lim d1 (Xm (!); X(!)) = 0 for all ! 2 ­:

m!1

To prove the measurability and integrable boundedness of Xm , we consider a closed set of the topology induced by d1 , B(V; ") = fW 2 F(Rp ); d1 (V; W ) ∙ "g, where V 2 F(Rp ) and " > 0: Then, for each m 2 N we have that (Xm )¡1 (B(V; ")) = f! 2 ­ : Xm (!) 2 B(V; ")g = f! 2 ­ : d1 (Xm (!); V ) ∙ "g = f! 2 ­ : d(Xm® (!); V® ) ∙ "; for all ® 2 (0; 1]g \ = f! 2 ­ : d(Xm® (!); V® ) ∙ "g ®2(0;1]

=

\

f! 2 ­ : d(Xm® (!); V® ) ∙ "g

®2(0;1]\Q

=

\

(Xm® )¡1 (B d (V; "));

®2(0;1]\Q

where B d (V; ") = fC 2 K(Rp ) : d(V® ; K) ∙ "g: Since Xm® are random sets, then (Xm® )¡1 (B d (V; ")) 2 A for all ® 2 (0; 1]; whence (Xm )¡1 (B(V; ")) 2 A: Since fB(V; "); V 2 F (Rp ); " > 0g is a generator of the Borel ¾-¯eld associated with F(Rp ), Xm is a fuzzy random variable and jjXm0 (!)jj ∙ p1=2 m; so that Xm is integrably bounded. It should be emphasized that similar results can be established when an Fc (Rp )-valued integrably bounded fuzzy random variable X is considered, so that the existence of a sequence of Fc (Rp )-valued integrably bounded fuzzy random variables approximating the ¯rst one can be guaranteed. To prove this existence, we have to replace the random sets Xm in Proposition 3.1 by Zm = co Xm (where co means the convex hull) which are also random sets and d(co Xm ; X) ∙ d(Xm ; X) (see Debreu, [5]), which implies the approximation result.

8

5. SOME IMPLICATIONS The results in Section 4 allow us to establish a characterization of integrably bounded fuzzy random variables in terms of d1 -limits of sequences of integrably bounded fuzzy random variables with simple ®-level functions. To formalize this characterization, we ¯rst state a supporting result Proposition 5.1. Let (­; A; P ) be a probability space and let X be an integrably bounded fuzzy random variable. Let fXm gm be the sequence of integrably bounded fuzzy random variables suggested in Theorem 4.2. Then, there exists a function h : ­ ! R+ ; h 2 L1 (P ) such that jjXm0 (!)jj ∙ h(!) for all ! 2 ­; and m 2 N; m > 1: Proof: Since X is integrably bounded, there exists g 2 L1 (P ) with jjX0 (!)jj ∙ g(!) for all ! 2 ­: If X0 (!) 6 ½ [¡m; m]p ; then m ∙ g(!); whence 1

1

jjXm0 (!)jj = d(f0g; Xm0 (!)) = d([¡m; m]p ; f0g) ∙ p 2 m ∙ p 2 g(!): If X0 (!) ½ [¡m; m]p ; then for all ® 2 (0; 1] we have that jjXm® (!)jj = d(f0g; (Xm® (!)) = d(f0g; Ym® (!)) ∙ d(f0g; X® (!)) + d(X® (!); Ym® (!)) ∙ g(!) + 1

p 2m¡1

∙ g(!) + p2 ;

so that the function h(!) = p 2 g(!) + p2 for all ! 2 ­ satis¯es the required conditions. The following result gives a constructive de¯nition of integrably bounded fuzzy random variables in terms of d1 -convergences. Theorem 5.2. Let (­; A; P ) be a probability space. Then X is an integrably bounded fuzzy random variable associated with it if and only if there exists a sequence of integrably bounded fuzzy random variables fXm gm with simple ®-level functions, and a mapping h 2 L1 (P ) such that i) limm!1 d1 (Xm (!); X(!)) = 0 for all ! 2 ­; ii) jjXm0 (!)jj ∙ h(!) for all ! 2 ­; and m 2 N: Proof: The existence of a sequence fXm gm satisfying the conditions i) and ii) can be ensured whatever the integrable bounded fuzzy random variable X may be, in virtue of Theorem 4.2 and Proposition 5.1. The su±9

ciency of the conditions i) and ii) above to guarantee the measurability and integrable boundedness of the limit variable can be immediately derived. The following result establishes a procedure to compute the fuzzy expected value of an integrably bounded fuzzy random variable as the d1 -limit of the fuzzy expected values of a sequence of integrably bounded fuzzy random variables with simple ®-level functions. Thus, Theorem 5.3. Let (­; A; P ) be a probability space and let X be an integrably bounded fuzzy random variable. Then, there exists a sequence of integrably bounded fuzzy random variables fXm gm with simple ®-levels mappings such that e m ; EX) e lim d1 (EX = 0: m!1

Proof: To prove this result, we only need to consider the preceding ones and the extension of the (classical) Lebesgue dominated convergence theorem in Puri and Ralescu [11].

The preceding result is especially useful when we deal with Fc (Rp )valued random variables. Thus, on the basis of Theorem 3.1 the expected value of a general integrably bounded Fc (Rp )-valued random variable can be obtained as the d1 -limit of a sequence of expected values of Fc (Rp )-valued random variables whose ®-level functions are simple random convex sets, the computation of these last expected values being immediate (see Section 2). 6. ILLUSTRATIVE EXAMPLE As an illustration of the results in this paper, we now present the following example: ¡ ¢ Example: Let (­; A; P ) = (0; 1); M(0;1) ; m be the Lebesgue space in (0; 1); and let X : ­ ¡! Fc (R2 ) be the integrably bounded fuzzy random variable whose ®-level function is given by ½ [0; !]2 if ® > ! (X(!))® = [0; 2]2 otherwise In accordance with the constructive procedure, we have that the fuzzy random variables Xm : ­ ¡! Fc (R2 ) whose (simple) ®-level functions are 10

given by

∙

¸2 1 ¡1 (Xm (!))® = m¡1 ; 2 + m¡1 I[®;1) (!) 2 2 2 m¡1 3 ¸2 2 X¡1 ∙ ¡1 k +I(0;®) (!) 4 ; 1 ¡ m¡1 I£ 2m¡1 ¡k¡1 2m¡1 ¡k ¢ (!)5 m¡1 ; m¡1 2 2 2m¡1 2 k=0

(IA being the indicator function of A) satis¯es the convergence result. p 2 Indeed, if ® ∙ !; then we have that ((X (!)) (X(!)) ) = , ; d H m ® ® m¡1 2 ´ h m¡1

m¡1

¡k¡1 2 whereas if ® > ! and ! 2 2 2m¡1 ; 2m¡1¡k we obtain that dH ((Xm (!))® ; p £ ¡1 ¤2 k 2 (X(!))® ) = dH ( 2m¡1 conclude ; 1 ¡ 2m¡1 ; [0; !]2 ) ∙ 2m¡1 ; whence we can p 2 that d1 (Xm (!); X(!)) = sup®>0 dH ((Xm (!))® ; (X(!))® ) ∙ 2m¡1 : Consequently,

lim d1 (Xm (!); X(!)) = 0:

m!1

On the other hand, in accordance with the results stated in this paper we can also guarantee that e e m ; EX) = 0; lim d1 (EX

m!1

e ® ) = 0 uniformly in ®: e m )® ; (EX) or equivalentely, limm!1 dH ((EX ¢ £ j Since for ® 2 2m¡1 ; 2j+1 ; j 2 f0; 1; : : : ; 2m¡1 ¡ 1g; we have that m¡1 ∙

e m )® = ¡1 ; 2 + 1 (EX 2m¡1 2m¡1 +

2m¡1 X¡1

k=2m¡1 ¡j

∙

¡1

2m¡1

;1 ¡

k 2m¡1

=

∙

¸2

¡1

2m¡1

1 2m¡1

;2+

+

µ

1 2m¡1

¸2

(1 ¡ ®) ¶∙

j +1 ¡® 2m¡1 ¸2

¡1

j +1 ; 1 + m¡1 m¡1 2 2

¸2

(1 ¡ ®)

∙ µ ¶µ ¶¸2 µ ¶∙ ¸2 1 ¡1 j+1 j j +1 ¡1 j +1 + + ; ¡® ; 1 + m¡1 ; 2 22m¡3 2m¡1 2m¡1 2m¡1 2m¡1 2 11

e m ))® ; [0; 2(1 ¡ ®) + and limm!1 dH ((EX

®2 2 2 ] ),

we obtain that

∙ ¸ 2 2 ® e ® = 0; 2(1 ¡ ®) + (EX) : 2

7. CONCLUDING REMARKS The results in this paper can be applied in di®erent settings. The ¯rst application is that presented and illustrated in Section 5. The results are also useful to develop further theoretical results concerning the integration of fuzzy random variables, since Theorem 4.2 obviously allows us to establish a constructive de¯nition of fuzzy random variables. Thus, some studies concerning integrably bounded fuzzy random variables and their fuzzy expected values could be developed by ¯rst focusing on simple random sets and later using the approximation results we have presented to achieve general conclusions. Finally, it should be emphasized that given an integrably bounded fuzzy random variable, there exists a sequence of simple fuzzy random variables such that the sequence of their ®-level functions d-converges to the ®-level function of the ¯rst variable everywhere (L¶ opez-D¶³az and Gil [8]). This approximation result would permit us also to obtain the expected value of the ®-level of an integrably bounded fuzzy random variable as the d-limit of the expected values of the ®-level functions of a sequence of simple fuzzy random variables. Acknowledgements The research in this paper has been supported in part by DGES Grant No. PB95-1049 and a Grant from the Vicerrectorado de Investigaci¶ on of the University of Oviedo. Their ¯nancial support is gratefully acknowledged. Miguel L¶ opez-D¶³az wants also to thank Professor George J. Klir and his group in the University of Binghamton because of their comments in connection with the research in this paper during his visit to Binghamton, which was ¯nancially supported in part by the Vicerrectorado de Investigaci¶ on of the University of Oviedo and by the Grant No. F30602-94-1-0011 (sponsored by the Rome Laboratory, USAF) (Professor Klir). 12

References [1] Artstein, Z. and Vitale, R.A. (1975) A strong law of large numbers for random compact sets. Ann. Prob. 3, 879-882. [2] Aumann, R.J. (1965) Integrals of set-valued functions. J. Math. Anal. Appl. 12, 1-12. [3] Ban, J. (1992) Ergodic theorems for random compact sets and fuzzy variables in Banach spaces. Fuzzy Sets and Systems 44, 71-82. [4] Byrne, C. (1978) Remarks on the set-valued integrals of Debreu and Aumann. J. Math. Anal. Appl. 62, 243-246. [5] Debreu, G. (1967) Integration of correspondences. Proc. Fifth Berkeley Symp. Math. Statist. Prob., 351-372. Univ of California Press, Berkeley. ¶ pez-D¶³az, M. (1996) Fundamentals and Bayesian analyses of [6] Gil, M.A. and Lo decision problems with fuzzy-valued utilities. Int. J. Approx. Reason. 15, 203-224. [7] Klement, E.P., Puri M.L. and Ralescu, D.A. (1986) Limit theorems for fuzzy random variables. Proc. R. Soc, Lond. A 407, 171-182. ¶ pez-D¶³az, M. and Gil, M.A. (1997) Constructive de¯nition of fuzzy random [8] Lo variables. Statist. Probab. Lett. (to appear). [9] Negoita, C.V. and Ralescu, D. (1975) Applications of Fuzzy Sets to System Analysis. J. Wiley, New York. [10] Puri, M.L. and Ralescu, D. (1985) The concept of normality for fuzzy random variables. Ann. Probab. 13, 1373-1379. [11] Puri, M.L. and Ralescu, D. (1986) Fuzzy random variables. J. Math. Anal. Appl. 114, 409-422. [12] Puri, M.L. and Ralescu, D. (1991) Convergence theorems for fuzzy martingales, J. Math. Anal. Appl. 160, 107-122. ¶, M. (1992) Fuzzy conditional expectation. Fuzzy Sets and Systems 52, [13] Stojakovic 53-60. ¶, M. (1994) Fuzzy random variables, expectation, and martingales. J. [14] Stojakovic Math. Anal. Appl. 184, 594-606. [15] Zadeh, L.A. (1975) The concept of a linguistic variable and its application to approximate reasoning. Parts 1,2, and 3. Information Sci. 8, 199-249; 8, 301-357; 9, 43-80.

13