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Fuzzy valued measure
FUZZY
sets and systems ELSEVIER
Fuzzy Sets and Systems 65 (1994) 95-104
Fuzzy valued measure Mila Stojakovi6* Department of Mathematics, Faculty of Engineering, University of Novi Sad, Trg Dositeja Obradovica 6, Yu-21000 Novi Sad, Yugoslavia Received January 1993; revised April 1993
Abstract
The main purpose of this paper is to introduce and develop the notion of a fuzzy valued measure in separable Banach space. This definition of fuzzy valued measure is a natural generalization of the set-valued measure. Radon-Nikod~m theorems for fuzzy valued measure are established.
Key words: Fuzzy valued measure; Radon-Nikod~m derivative
I. Introduction
The concept of fuzzy set was introduced by Zadeh in 1965. Subsequent developments focused on applications of this concept to pattern recognition and system analysis, among other areas. The topic of set valued and fuzzy valued measures has received much attention in the last few years because of its usefulness in several applied fields like mathematical economics and optimal control. Significant contributions in this area were made by Hiai, [4], [5], [6] Papageorgiou, [13], Puri, Ralescu [16], Ban [1], Zhang Wen Kin, Li Teng [19], [20]. In these works appeared several different definitions of the notion of set valued and fuzzy valued measure. The purpose of this paper is to give and compare different definitions of fuzzy valued measure whose values are fuzzy sets defined on a real separable Banach space and this problem is investigated in Section 3. One of advantages of this concept of fuzzy valued measure is that it enables us to use the rich mathematical apparatus of the theory of set-valued measures. Let (f2, ~/) be a measurable space with ~¢ a a-field of measurable subsets of the set f2. Let Z be a real separable Banach space and ~c(Z) the family of nonempty, convex, compact subsets of X and let ~-c(Z) be the set of all fuzzy sets u:z--* [0, 1] such that u~ = {x~z:u(x)>1 ct}~c(X) for all ~ ( 0 , 1]. The mapping J / C : d ~ ~c(X) such that for every sequence {Ai}i~N of pairwise disjoint elements of ~¢ the next equality is satisfied,
* This research was supported by Science Fund of Serbia, grant number 0401A, through Matemati6ki institut,
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M. Stojakovi? / Fuzzy Sets and Systems 65 (1994) 95-104
96
where
,.~(ai)
(x) =
i=1
sup
.~¢(hi)(xi):x = I.i=1
xi, i=1
Plxl]l < oo i=1
we shall call fuzzy valued measure. If Jr' is a fuzzy valued measure then for every ~e(0, 1], ./t', : ~¢ ~ ~c(Z) defined by ~t',(A) = [,/g(A)],, A e d , is a set-valued measure. Conversely, if {-It', }~(o, xj is a family of compact, convex set valued measures such that Jr', ~ Jc'p if 0 < fl < ~ ~< 1, then Jr': ~¢ ~ ~-()0 defined by de(A)(x) = sup {a:x ~ , ( A ) } is a fuzzy valued measure. The notion of a measurable mapping with values in space of fuzzy sets (fuzzy random variable) was introduced by Kwakernaak [10] and Puri, Ralescu [14]. The papers of Klement, Puri, Ralescu [8], [15], [16], Kaleva [7], Kruse [9], Ban [2], Lowen [1 i], Stojakovi6 [17], [18] are related to this topics. The section 4 is devoted to fuzzy random variable and some Radon-Nikod3)m type theorems are proved. One can define the integral of measurable fuzzy valued function F: f2 ~ ~-~ ()0 by
where (f2, d , / ~ ) is a-finite measure space. Then the function defined by ~t'(A) = f a Fd/~ is a p-continuous fuzzy valued measure of bounded variation. If ~ t ' : d ~ - c ( X ) is /~-continuous fuzzy valued measure then there exists measurable function F: £2 --* ~-c(Z) such that J / ( A ) = fA Fd#,
Ae~/.
The results of this paper generalize some results of Hiai [6], Papageorgiou [13], Purl, Ralescu [16], Ban [1].
2. Preliminary notions on set valued measures
Let (f2, ~¢) be a measurable space with ~¢ a a-field of measurable subset of the set f2. Let (X, I I ' H ) be a real separable Banach space, ~(X) the family of all nonempty subsets of X and ~c(Z) the set of all convex compact subsets of z. If AeC~(X) the number Ial is defined by IAI = sup Ilxll. x~A
For two closed bounded sets A, B e ~ ( Z ) denote the Hausdorff metric of A and B, H ( A , B) = max t sup inf II x - y rl, sup inf IIx - y II}. I xeA yeB
y~B xe.A
A set valued, set function M : ~¢ ~ ~(X) is said to be a set valued measure (multimeasure) if it satisfies the following two requirements:
M. Stojakovi6 / Fuzzy Sets and Systems 65 (1994) 95-104
97
(i) M(.) is countably additive, in the sense that given any sequence {A. }.~N of pairwise disjoint elements of ~¢ we have that
M(U= 1 A,,)= o=1 ~ where
~ M(A.)={x6Z:x= n=l
~ x.. ~ ,]x.[]
n=l
(ii) M(0)= {0}. As for single valued measures, we have the notion of total variation [M](-) of M('). For A ¢ ~ / w e define IMI(A) = sup ~ IM(A,)I, i=1
where the supremum is taken over all finite measurable partitions {A1 . . . . . A, } of A. If IMI(f2) < ~ , then we say that M(-) is of bounded variation. It is easy to see that in this case the sums in the definition of Y.,,~ 1 M(A,,) are absolutely convergent. Finally, we say that M(- ) is #-continuous, where p is a single valued vector measure if and only if for any Ae~¢ for which ~(A) = 0 we have M(A) = {0}.
3. Fuzzy valued measure Let ~-,(Z) be a set of fuzzy sets u : X ~ [0, 1] such that 1. u~ =: {x~z:u(x ) >1~}~¢(Z), a~(O, 1] 2. u 1 5 0 . The mapping dg: d ~ ~-c(Z) is a fuzzy valued measure if it satisfies 1. Jr'(0) = I~0~ where I/o~ is indicator function of {0} 2. if {A.} cz~/, a~c~ aj = O, i # j
n
1
n=l
where
Jg(A.)(x) = sup n=l
JI(A.)(x.), x = Ln=l
x., n=l
llx. N < ~
•
n=l
Theorem 1. If Jg : ~t ~ ~c(Z) is a fuzzy valued measure, then d,g~: d -~ Pc(Z), where
Mt~(A) = {x6z:~(A)(x) >1 ~}, is a set valued measure for all ~ ( 0 , 1]. Proof. Since J l ( A ) ~ 0 and dll(A)c.Ag.(A) for all cte(0, 1] and all Ae~¢, for the sequence {A.}.~u of mutually disjoint subsets of X, we have
~t,(A.)= n=l
~,(A.)=~t, n=l
A. n
M. Stojakovik / Fuzzy Sets and Systems 65 (1994) 95-104
98
which means that ~,~= acC-/,(A,) is nonempty, bounded set for all ~¢(0, 1]. From
x:
¢#[(Ai)(xO, x =
sup
x.
i
i=1
Ilxill <
IIx~lt < ~
>1
i=1
~,.A[(Ai)(xi)>/0~}
i=1
i=1
i=!
we get the additivity of . ~ .
Theorem 2. Let J / , , c~e(0, 1], be a family of compact, convex set valued measures and let ~¢¢p(A) ~ Jg~(A ) for all 0 < ~ < fl <~ 1 and all A ~ ¢ . Then ~[(A)(x) = sup,~tO, ll{X6~t',(A)} is a fuzzy valued measure. Proof. From compactness of ¢¢/,(A) we have
MI=(A) =
N ,8
for all ~6(0, 1] and all A~M, and from definition of ~ we have that
(.l¢(A)),={x"
sup {x6,aCl,(A)} } = n.lCp(A) ~ ( 0 , 1]
# <~
which means that J/~(A) = (~(A))~. If {A. }.EN is a sequence of pairwise disjoint subsets of Z, then .At'
A
x) = sup
i
x~..gg,
~tE(0, 1]
Ai i
=
sup
ta
i=1
i=1
~¢¢(Ai)(xl):x =
t.i= 1
..¢[,(Ai i= 1
L a ~ ( 0 , 1]
= sup
xe
~te(O, 1]
x. i= 1
[Ixill <
~
=
~t'(h~)(x), i= 1
that is, ~ ' is additive.
4. Radon Nikod~m derivative Hereafter, let (12, d,/~) be a a-finite measure space. Let L(12, d , ~, Z) = L denotes the Banach space of measurable functions f : £2 ~ Z such that the norm [[flJL = So [[f(og)I[ d/z is finite and let h be a Hausdorff metric in L. If F: ~2 ~ ~c(Z) is a measurable multivalued function, then a measurable function f : f2 ~ X is called a measurable selection of F iff(~o)~F(o)) for all o2~f2 and
SF = {f~L :f(co)eF(co)p - a.e.}.
M. Stojakovik / Fuzzy Sets and Systems 65 (1994) 95-104
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The set SF is a closed subset of L. A multivalued function F: f2 ~ ~c (X) is called integrably bounded if there exists/~ - integrable function s: f2 ~ R such that supx~r,o)ll x I1 ~ s(~o) for all coef2. F is integrably bounded if and only if SF is nonempty and bounded. Let L,e(t2,d , #, ;t) = S~ denote the space of all integrably bounded multivalued functions where two functions F , , F2eL,~' are considered to be identical if Fl(CO) = F2(co)/~-a.e. For Fx, F 2 e ~ we can define
A(F1, F2) = f n(F~(co), FE(tO))d/~ 3
where H is a Hausdorff metric in ()~, II II). £a is a complete metric space with respect to the metric A ([4]). If F is a measurable multivalued function F: f2 ~ ~(X), then the integral of F is defined by
Feron ([3]) and Puri and Ralescu ([14]) introduced the notion of a fuzzy random variable as a function F :Q ~ #-(R") subject to certain conditions. The next definition is slightly different from that in [14]. A fuzzy random variable is a function X : f~ --* ~-d)0, where #-c()~) denotes all functions (fuzzy sets of Z) X(. ): Z ~ [0, 1] such that 1. {xe)~: X(~o)(x) >~ ~} = X~(~o) is nonempty, convex and compact for all xe(0, 1]. 2. X~ : O ~ #dZ), is measurable function for all ee(0, 1]. A fuzzy random variable X is called integrably bounded if X~ is integrably bounded for all ee(0, 1], i.e. for all ~e(0, 1], X~e~e. Let A(~, s~, ~, Z) = A be the set of all integrably bounded fuzzy random variables. For the convenience of the reader we state here one result that we shall need in the sequel. Lemma 1 ([12]). Let M be a set and let {M~:cte[O, 1]} be a family of subsets of M such that (1) Mo = M (2) o t ~ < f l ~ M a~_M~ (3) Otl ~< 0~2 ~....lim.+ooct, = o t ~ = M~ = 0 . = 1 M ~,. Then, the function dp:M~ [0, 1] defined by ~(x) = sup{ere[O, 1]:xeM~} has the property {xe M : ck(x) >~or} = M~for every ore[O, 1]. For all
X, YeA we can define the function @ : A x A --* R
@(X, Y) = sup A (X~, Y~). ~>0
Two fuzzy variables X, YeA are considered to be identical if ~ ( X , Y) = 0. (A, 9 ) is a complete metric space. [17-1 In [18] the integral Sa Xd# of XeA is introduced by
(fxd,)=fox d.,
ae(O, 1].
and
(f Xdp)(x)=
sup
{xefaX~dl.~ }.
• e(O, 1]
If A e~¢, SAX dp is the integral of the restriction
X IA.
that
100
M . S t o j a k o v i k / F u z z y S e t s a n d S y s t e m s 65 (1994) 9 5 - 1 0 4
A fuzzy valued measure d¢:~¢ ~ ~-c(Z) is of bounded variation if ~/~ :~¢ ~ ~c(Z) is a set valued measure of bounded variation for all 0re(0, 1] and ,/¢ is p-continuous if A e ~ with #(A) = 0 implies that JC(A) = I~o~ where I~0~ is the indicator function of {0}. Theorem 3. ,./¢(A) =
If XeA(f2, d, p, Z), then f a Xda
is a p-continuous fuzzy valued measure of bounded variation. Proof. We shall show that Jr' satisfies the condition of countable additivity. If {A~}~N is the sequence of pairwise disjoint subsets of Z and A = Ui°°__~A~, then, using results from [18]
•gI~(A)=[JC(A)]==(fAXdp)=fAX~d# • Since X,,e..W, we have
i=1
. Ai
i=1
From [18] we know that JCa(A)cJI~(A) for all 0 < ct < fl ~< 1 and all Ae~q¢. Now, we can use Theorem 2. In order to prove that ¢g is of bounded variation we proceed as follows: Since XeA, there exists integrably bounded function h~ : D ~ R such that IX~I(~) : IX=(~)I :
sup IIx[I ~< h,(co) for all coet2. x~Xo(~O)
Further, according to Theorem 2.2 [4], we obtain I~t',lff2)= sup ~. I~¢,(Ai)I = sup ~. i=1
=sup~ i=1
sup Ilxll
i= 1 xe,ll,(AO
fAfdP
sup f¢Sx
= sup
~
supf feSx
sup ]lxlld# = sup ~ i= 1
~
A i xeX,(o~)
i= 1
Ilflldp A~
IX~(co)ld# Ai
h~(og)dp=fah~(og, dll
f , Ai
So, we have proved that if XeA then J l : d ~ ~c(Z) defined by ¢g(A) = ~a X dp is a fuzzy valued measure. Let ..¢t' : d ~ ~c(;t) be a fuzzy valued measure and let XeA. We call X a Radon-Nikod~,m derivative of ./¢ with respect to p if ..¢t'(A)= ~ Xdp, JA
Aed.
In the next theorem we present the existence of Radon-Nikod~m derivative of fuzzy valued measure of bounded variation.
M. StojakoviO / Fuzzy Sets and Systems 65 (1994) 95-104
101
Theorem 4. If z has the Radon-Nikodj~m property, Z* is separable and J [ : s l ~ ~¢(Z) is a #-continuous fuzzy
valued measure of bounded variation, then ,At has Radon-Nikod~m derivative contained in A. ProoL The function ,At : ~¢ ~ ~-¢(X) is a/~-continuous fuzzy valued measure of bounded variation which means that ~g, : d ~ ~ ( g ) defined by
is a/~-continuous set valued measure of bounded variation. According to Theorem 4.5 and Remark after Theorem 4.6 [6], Jr', has a unique Radon-Nikod#m derivative X,~£~'. We need to prove that the family {X, },~¢o.1~ define one and only one fuzzy set X defined by X(o2)(x)= sup {xeX,(o2)}, ~t~(0.1]
that is, we shall show that all the conditions of Lemma 1 are satisfied. The construction of Radon--Nikod~,m derivative X, of set valued measure ~t', is given in [-6] (Theorem 4.3, Theorem 4.4). The sequence {m~}~N of selections of Jr', is such that JI,(A) = cl{m~(A)}~N for all A ~ and X,((~) = C--o{f~(o2)}~N, wheref~ are Radon-Nikod~,m derivative of m~. It is obvious that relation
< fl=~Cta(A)=~I,(A)
for all A6~¢
implies that
=:~-'~{ f ~(o)) } j e N = ~ { f J(oJ) } j~N =~ X#(~)= X~t(OO) for all ~o~0. If we consider the sequence ~ < ~2 <
"'"
,lim,~t.
= ~ > O,~ie[O, 1], we have to prove that
x:(o~) = (~ x:,(~o). i=l
Let Ta be the set of selection of ~t'a, fl~{ct, ~ , ..., ~ .... }. Since JI,(A)=,/t',,(A) for all A ~ ¢ it is obvious that T , ~ T,, for all i~N, that is T , ~ n~=l T,,. If we suppose that there exists m from N ~ T,, such that me T,, then
m6T,~ ~m(A)~Jg,,(A) for all i~N and all A6~¢, and there exists Aez¢ such that m(A)¢ ,I,[,(A). It is a contradiction with ~¢/,(A) = n i ~ l ~t',,(A). So, we have proved that T, = N,~l T,,. If we denote by Sa the set of measurable selections of Xa, fie{or, ~1 .... }, then from the relation X,(o))cX,,(to) for all ieN and all o2~t2, we get that S, c N I ~ I S , . In order to prove that S, = N/~lS,, we assume that there exists fE L such that f e N i~ 1s,, and f ¢ S,. Let the sequence {J~ },~Nc N ~ ~S,, be such that l i m , ~ J ~ = f , J~ is a Radon-Nikod~m derivative for some m,~ni~iT~,neN. This means that l i m . . ~ IIL - f IIL = 0 ~ lim._~~ ~a II~(tn) - f(co)II d/~ = 0..~ lim._~~ j~(~o) = f(o~) g-a.e. If B = ~, /~(B) = 0 is such that l i m . ~ ( c o ) =f(o~)for all o~¢P\B, then let the sequence {f.}.~N be defined by f,(to) = j~(og), o 2 ~ \ B , f.(¢o) = f(aO,
co~B.
102
M. Stojakovi5 / Fuzzy Sets and Systems 65 (1994) 95-104
It is obvious that IIL--~IIL=0 for all neN, lim.~oof.=f (i.e. l i m . ~ [ I f . - - f l l L : 0 ) and lim,~ ~of. (~o) = f(~o) for all ~oe t2. From closeness of 0 ~= iS~, it follows that {f, }.~ ~ 0 i~S~,, • Now, we can take a countable field ~¢o = A such that {fn} ~ L(f2, .~t~, #, Z), where ~ 1 is the a-field generated by ~¢o ([6]). From compactness of ~t',(A)= ~I°°_I~,,(A), for all AeMo, since {~Af.d#},~u=JC~(A), the sequence { ~af, d/z}.~u has a convergent subsequence. Thus, by the Cantor diagonal process, there exists a subsequence {g.} of {f.} such that {~a g,d;t}.~u is convergent for every A=~¢o. Since IIg,(og)[I ~< IZ(og)l/~-a.e., where Z = Aa is defined by Z(~o)= cl{f(~o)}, f s 0 L I S~,, to el2, the sequence {~Ag, dkt}.~ is convergent and lim f g.dp = m(A) n~oo A
for all A ~ .
Z is a finitely additive set function and since IIm(A)II ~ I ~ I(A) for all A ~ ~¢1 it follows that m is countable additive and m is a selection of ~t',. From the extension of the Lebesgue dominated convergence theorem T h e n m:~q~ 1 ~
m(A) ~--lira f gnd]~ = f lim gnd~ n~ A a n~
and, since {g. }.,N is subsequence of {f. }.~N, we get
re(A) =
ff d#
which means that feS~, that is S~ = (]~=1S~,. To see that X~(o~) = (] i~ aX~(a~) for all oJ~O, we denote by Y(o~) the set (]i~ iX~,(°~), ¢oeY2. Since the family of sets {X~,(o~)}i~N has the finite intersection property, Y(~o)4= 0 and Y(o~) is a compact set. Also limi~ ~H(Y(o~), X~,(m)) = 0 for all ~ o ~ , which implies that limio ooX~,& Y and from completeness of (.L~, A ) we get that YeLlow. Further, we have the next implications.
> ~:::.X~(oJ)~X~,(o~)~X~(o~)~ N X~,, i=1
that is, X~(~o)~ Y(o~) for all oJ~g2. From Y ~ A a and Lemma 1.1. [-4] it follows that there exists the sequence {h.}.~N~L such that Y(o~) = cl{h.(~o)} for all o ~ O and
Y(og)~X.,(~)=:. {h.}.~N=S.,=~ {h.}.~u~ (~ S., = S~ i=1
~cl{h,(~o)}~cl{g(m):geS~}
for all wsg2,
which means that Y(~o)~ X~(oJ). So, we have proved
Y(o~) = N X~,(~o) = X~(~o) i=1
for all o~eQ.
Theorem 5. Let Z has the Radon-Nikod~m property, Z* be separable and ~ sub-a-algebra of d . If X e A then
there exists E(XI~)eA(f2, ~ ) such that A
Xd# = f
A
E(Xr~-)d#
for all A ~ .
M. Stojakovid / Fuzzy Sets and Systems 65 (1994) 95-104
103
Proof. Since X e A , from Theorem 3 there exists ./4 : ~¢ ~ ~c(X) such that
Jt'(A) = C Xd#, JA
A~¢.
The restriction of ~¢/on ~ is defined by ./t'(A) = f Xd#, JA
A~.
That function is fuzzy valued measure on ~ and from Theorem 4 there exists YeA(t2, ~ ) which is RadonNikod~m derivative of .A( such that J I ( A ) = ~ Ydp, dA
A~.
That Y we denote by E ( X I ~ ) and call conditional expectation of X. It is obvious (Theorem 2 [18,]) that
E(X~I~) = (E(XI~-))~. Further, uniqueness of the existence of conditional expectation E(XI °~) follows from the uniqueness of conditional expectation for random set X,, ~e(0, 1]. Theorem 6. Let ~t[ : d ~ ~c(Z) be a fuzzy valued measure. Then ~l[ has a unique Radon-Nikod~m derivative
contained in A if and only if the following conditions are satisfied : (1) ~t' is #-continuous, (2) Jr' is of bounded variation, (3) 9iven A e d , 0 < #(A) < o% there exists a B ~ A and a compact subset C of z such that #(B) > 0 and ,Ag'o(B')/p(B')cC for all B ' c B with #(B') > 0 (where ~¢[o(A ) = supp J4 (A ) = {x e z : Jg (A ) (x) > 0}). Proof. Since ,g~(B')cJCo(B') for all ~E(0, 1,] we get
.//~(B') - C #(B')
-
JCo(B') C C, #(B')
and all the conditions of Theorem 5.2 [6] are satisfied for ./¢~ : ~¢ ~ ~'c(Z). Further we can proceed as in the proof of Theorem 5. Theorem 7. If . X / " : d ~ ~c(X), n = 1, 2, are fuzzy valued measures of bounded variation which are #-
continuous and such that for all J I ~ ( A ) ~ ~¢[2(A) for all ~te(0, 11 and all A e ~ then (d~g~/d#)(og)c (djt'2/d#)(~o) p-a.e, where d~¢~ and d# are Radon-Nikodj~m derivative of ~g~ and p, ~e(0, 1,]. Proof. The proof is based on Proposition 5.1. [13].
References [1] J. B~n, Radon-Nikod~,m theorem and conditional expectation of fuzzy-valued measures and variables, Fuzzy Sets and Systems 34 (1990) 383-392. [2] J. B~n, A remark on the metric structure of the space of integrably bounded fuzzy variables, Acta Math. Univ. Comenianea LVIII-LIX (1990) 119-126.
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M. StojakoviO / Fuzzy Sets and Systems 65 (1994) 95-104
[3] R. Feron, Ensambles aleatoires flous, C.R. Acad. Sci. Paris Ser. A 282 (1976) 903-906. [4] F. Hiai and H. Umegaki, Integrals, conditional expectations and martingals of multivalued functions, J. Multivar. Anal. 7 (1977) 149-182. [5] F. Hiai, Convergence of conditional expectations and strong laws of large numbers for multivalued random variables, Trans. Amer. Math. Soc. 291(5) (1985) 613-627. [6] F. Hiai, Radon-Nikod~,m theorems for set valued measures, J. Multivar. Anal. 8 (1978) 96-118. [7] O. Kaleva, The calculus of fuzzy valued functions, Appl. Math. Lett. 3(2) (1990) 55-59. [8] P. Klement, M. Puri and D. Ralescu, Limit theorems for fuzzy random variables, Proc. R. Soc. Land. A 407 (1986) 171-182. [9] R. Kruse, The strong law of large numbers for fuzzy random variables, Inf. Sci. 28 (1982) 233-241. [10] H. Kwakernaak, Fuzzy random variables, Inf. Sci. 15 (1978) 1-29. [11] R. Lowen, Convex fuzzy sets, Fuzzy Sets and Systems 3 (1980) 291-310. [12] C.V. Negoita and D. Ralescu, Applications of Fuzzy Sets to Systems Analysis (Wiley, New York, 1975). [13] N. Papageorgiu, On the theory of Banach space valued multifunctions, J. Multivar. Anal. 17 (1985) 185-227. [14] M. Puri and D. Ralescu, Fuzzy random variables, J. Math. Anal. Appl. 114 (1986) 409422. [15] M. Purl and D. Ralescu, Differentials of fuzzy functions, J. Math. Anal. Appl. 91 (1983) 552-558. [16] M.L. Puri and D.A. Ralescu, Convergence theorem for fuzzy martingale, J. Math. Anal. Appl. 160 (1991) 107-122. [17] M. Stojakovir, Fuzzy conditional expectation, Fuzzy Sets and Systems 52(1) (1992) 151-158. [18] M. Stojakovir, Fuzzy random variable, martingales, J. Math. Anal. Appl. (to appear). [19] Zhang Wen-Xin, Li Teng, Ma Ji-Feng and Li Ai-Jie, Set valued measure and fuzzy set-valued measure, Fuzzy Sets and Systems 36 (1990) 181-188. [20] Zhang Wen-Xin and Li Teng, The representation theorem of set valued measure, Acta Math. Sinica 31(2) (1988) 201-208.
sets and systems ELSEVIER
Fuzzy Sets and Systems 65 (1994) 95-104
Fuzzy valued measure Mila Stojakovi6* Department of Mathematics, Faculty of Engineering, University of Novi Sad, Trg Dositeja Obradovica 6, Yu-21000 Novi Sad, Yugoslavia Received January 1993; revised April 1993
Abstract
The main purpose of this paper is to introduce and develop the notion of a fuzzy valued measure in separable Banach space. This definition of fuzzy valued measure is a natural generalization of the set-valued measure. Radon-Nikod~m theorems for fuzzy valued measure are established.
Key words: Fuzzy valued measure; Radon-Nikod~m derivative
I. Introduction
The concept of fuzzy set was introduced by Zadeh in 1965. Subsequent developments focused on applications of this concept to pattern recognition and system analysis, among other areas. The topic of set valued and fuzzy valued measures has received much attention in the last few years because of its usefulness in several applied fields like mathematical economics and optimal control. Significant contributions in this area were made by Hiai, [4], [5], [6] Papageorgiou, [13], Puri, Ralescu [16], Ban [1], Zhang Wen Kin, Li Teng [19], [20]. In these works appeared several different definitions of the notion of set valued and fuzzy valued measure. The purpose of this paper is to give and compare different definitions of fuzzy valued measure whose values are fuzzy sets defined on a real separable Banach space and this problem is investigated in Section 3. One of advantages of this concept of fuzzy valued measure is that it enables us to use the rich mathematical apparatus of the theory of set-valued measures. Let (f2, ~/) be a measurable space with ~¢ a a-field of measurable subsets of the set f2. Let Z be a real separable Banach space and ~c(Z) the family of nonempty, convex, compact subsets of X and let ~-c(Z) be the set of all fuzzy sets u:z--* [0, 1] such that u~ = {x~z:u(x)>1 ct}~c(X) for all ~ ( 0 , 1]. The mapping J / C : d ~ ~c(X) such that for every sequence {Ai}i~N of pairwise disjoint elements of ~¢ the next equality is satisfied,
* This research was supported by Science Fund of Serbia, grant number 0401A, through Matemati6ki institut,
SSDI 0 1 6 5 - 0 1 1 4 ( 9 3 1 E 0 1 2 8 - F
M. Stojakovi? / Fuzzy Sets and Systems 65 (1994) 95-104
96
where
,.~(ai)
(x) =
i=1
sup
.~¢(hi)(xi):x = I.i=1
xi, i=1
Plxl]l < oo i=1
we shall call fuzzy valued measure. If Jr' is a fuzzy valued measure then for every ~e(0, 1], ./t', : ~¢ ~ ~c(Z) defined by ~t',(A) = [,/g(A)],, A e d , is a set-valued measure. Conversely, if {-It', }~(o, xj is a family of compact, convex set valued measures such that Jr', ~ Jc'p if 0 < fl < ~ ~< 1, then Jr': ~¢ ~ ~-()0 defined by de(A)(x) = sup {a:x ~ , ( A ) } is a fuzzy valued measure. The notion of a measurable mapping with values in space of fuzzy sets (fuzzy random variable) was introduced by Kwakernaak [10] and Puri, Ralescu [14]. The papers of Klement, Puri, Ralescu [8], [15], [16], Kaleva [7], Kruse [9], Ban [2], Lowen [1 i], Stojakovi6 [17], [18] are related to this topics. The section 4 is devoted to fuzzy random variable and some Radon-Nikod3)m type theorems are proved. One can define the integral of measurable fuzzy valued function F: f2 ~ ~-~ ()0 by
where (f2, d , / ~ ) is a-finite measure space. Then the function defined by ~t'(A) = f a Fd/~ is a p-continuous fuzzy valued measure of bounded variation. If ~ t ' : d ~ - c ( X ) is /~-continuous fuzzy valued measure then there exists measurable function F: £2 --* ~-c(Z) such that J / ( A ) = fA Fd#,
Ae~/.
The results of this paper generalize some results of Hiai [6], Papageorgiou [13], Purl, Ralescu [16], Ban [1].
2. Preliminary notions on set valued measures
Let (f2, ~¢) be a measurable space with ~¢ a a-field of measurable subset of the set f2. Let (X, I I ' H ) be a real separable Banach space, ~(X) the family of all nonempty subsets of X and ~c(Z) the set of all convex compact subsets of z. If AeC~(X) the number Ial is defined by IAI = sup Ilxll. x~A
For two closed bounded sets A, B e ~ ( Z ) denote the Hausdorff metric of A and B, H ( A , B) = max t sup inf II x - y rl, sup inf IIx - y II}. I xeA yeB
y~B xe.A
A set valued, set function M : ~¢ ~ ~(X) is said to be a set valued measure (multimeasure) if it satisfies the following two requirements:
M. Stojakovi6 / Fuzzy Sets and Systems 65 (1994) 95-104
97
(i) M(.) is countably additive, in the sense that given any sequence {A. }.~N of pairwise disjoint elements of ~¢ we have that
M(U= 1 A,,)= o=1 ~ where
~ M(A.)={x6Z:x= n=l
~ x.. ~ ,]x.[]
n=l
(ii) M(0)= {0}. As for single valued measures, we have the notion of total variation [M](-) of M('). For A ¢ ~ / w e define IMI(A) = sup ~ IM(A,)I, i=1
where the supremum is taken over all finite measurable partitions {A1 . . . . . A, } of A. If IMI(f2) < ~ , then we say that M(-) is of bounded variation. It is easy to see that in this case the sums in the definition of Y.,,~ 1 M(A,,) are absolutely convergent. Finally, we say that M(- ) is #-continuous, where p is a single valued vector measure if and only if for any Ae~¢ for which ~(A) = 0 we have M(A) = {0}.
3. Fuzzy valued measure Let ~-,(Z) be a set of fuzzy sets u : X ~ [0, 1] such that 1. u~ =: {x~z:u(x ) >1~}~¢(Z), a~(O, 1] 2. u 1 5 0 . The mapping dg: d ~ ~-c(Z) is a fuzzy valued measure if it satisfies 1. Jr'(0) = I~0~ where I/o~ is indicator function of {0} 2. if {A.} cz~/, a~c~ aj = O, i # j
n
1
n=l
where
Jg(A.)(x) = sup n=l
JI(A.)(x.), x = Ln=l
x., n=l
llx. N < ~
•
n=l
Theorem 1. If Jg : ~t ~ ~c(Z) is a fuzzy valued measure, then d,g~: d -~ Pc(Z), where
Mt~(A) = {x6z:~(A)(x) >1 ~}, is a set valued measure for all ~ ( 0 , 1]. Proof. Since J l ( A ) ~ 0 and dll(A)c.Ag.(A) for all cte(0, 1] and all Ae~¢, for the sequence {A.}.~u of mutually disjoint subsets of X, we have
~t,(A.)= n=l
~,(A.)=~t, n=l
A. n
M. Stojakovik / Fuzzy Sets and Systems 65 (1994) 95-104
98
which means that ~,~= acC-/,(A,) is nonempty, bounded set for all ~¢(0, 1]. From
x:
¢#[(Ai)(xO, x =
sup
x.
i
i=1
Ilxill <
IIx~lt < ~
>1
i=1
~,.A[(Ai)(xi)>/0~}
i=1
i=1
i=!
we get the additivity of . ~ .
Theorem 2. Let J / , , c~e(0, 1], be a family of compact, convex set valued measures and let ~¢¢p(A) ~ Jg~(A ) for all 0 < ~ < fl <~ 1 and all A ~ ¢ . Then ~[(A)(x) = sup,~tO, ll{X6~t',(A)} is a fuzzy valued measure. Proof. From compactness of ¢¢/,(A) we have
MI=(A) =
N ,8
for all ~6(0, 1] and all A~M, and from definition of ~ we have that
(.l¢(A)),={x"
sup {x6,aCl,(A)} } = n.lCp(A) ~ ( 0 , 1]
# <~
which means that J/~(A) = (~(A))~. If {A. }.EN is a sequence of pairwise disjoint subsets of Z, then .At'
A
x) = sup
i
x~..gg,
~tE(0, 1]
Ai i
=
sup
ta
i=1
i=1
~¢¢(Ai)(xl):x =
t.i= 1
..¢[,(Ai i= 1
L a ~ ( 0 , 1]
= sup
xe
~te(O, 1]
x. i= 1
[Ixill <
~
=
~t'(h~)(x), i= 1
that is, ~ ' is additive.
4. Radon Nikod~m derivative Hereafter, let (12, d,/~) be a a-finite measure space. Let L(12, d , ~, Z) = L denotes the Banach space of measurable functions f : £2 ~ Z such that the norm [[flJL = So [[f(og)I[ d/z is finite and let h be a Hausdorff metric in L. If F: ~2 ~ ~c(Z) is a measurable multivalued function, then a measurable function f : f2 ~ X is called a measurable selection of F iff(~o)~F(o)) for all o2~f2 and
SF = {f~L :f(co)eF(co)p - a.e.}.
M. Stojakovik / Fuzzy Sets and Systems 65 (1994) 95-104
99
The set SF is a closed subset of L. A multivalued function F: f2 ~ ~c (X) is called integrably bounded if there exists/~ - integrable function s: f2 ~ R such that supx~r,o)ll x I1 ~ s(~o) for all coef2. F is integrably bounded if and only if SF is nonempty and bounded. Let L,e(t2,d , #, ;t) = S~ denote the space of all integrably bounded multivalued functions where two functions F , , F2eL,~' are considered to be identical if Fl(CO) = F2(co)/~-a.e. For Fx, F 2 e ~ we can define
A(F1, F2) = f n(F~(co), FE(tO))d/~ 3
where H is a Hausdorff metric in ()~, II II). £a is a complete metric space with respect to the metric A ([4]). If F is a measurable multivalued function F: f2 ~ ~(X), then the integral of F is defined by
Feron ([3]) and Puri and Ralescu ([14]) introduced the notion of a fuzzy random variable as a function F :Q ~ #-(R") subject to certain conditions. The next definition is slightly different from that in [14]. A fuzzy random variable is a function X : f~ --* ~-d)0, where #-c()~) denotes all functions (fuzzy sets of Z) X(. ): Z ~ [0, 1] such that 1. {xe)~: X(~o)(x) >~ ~} = X~(~o) is nonempty, convex and compact for all xe(0, 1]. 2. X~ : O ~ #dZ), is measurable function for all ee(0, 1]. A fuzzy random variable X is called integrably bounded if X~ is integrably bounded for all ee(0, 1], i.e. for all ~e(0, 1], X~e~e. Let A(~, s~, ~, Z) = A be the set of all integrably bounded fuzzy random variables. For the convenience of the reader we state here one result that we shall need in the sequel. Lemma 1 ([12]). Let M be a set and let {M~:cte[O, 1]} be a family of subsets of M such that (1) Mo = M (2) o t ~ < f l ~ M a~_M~ (3) Otl ~< 0~2 ~....lim.+ooct, = o t ~ = M~ = 0 . = 1 M ~,. Then, the function dp:M~ [0, 1] defined by ~(x) = sup{ere[O, 1]:xeM~} has the property {xe M : ck(x) >~or} = M~for every ore[O, 1]. For all
X, YeA we can define the function @ : A x A --* R
@(X, Y) = sup A (X~, Y~). ~>0
Two fuzzy variables X, YeA are considered to be identical if ~ ( X , Y) = 0. (A, 9 ) is a complete metric space. [17-1 In [18] the integral Sa Xd# of XeA is introduced by
(fxd,)=fox d.,
ae(O, 1].
and
(f Xdp)(x)=
sup
{xefaX~dl.~ }.
• e(O, 1]
If A e~¢, SAX dp is the integral of the restriction
X IA.
that
100
M . S t o j a k o v i k / F u z z y S e t s a n d S y s t e m s 65 (1994) 9 5 - 1 0 4
A fuzzy valued measure d¢:~¢ ~ ~-c(Z) is of bounded variation if ~/~ :~¢ ~ ~c(Z) is a set valued measure of bounded variation for all 0re(0, 1] and ,/¢ is p-continuous if A e ~ with #(A) = 0 implies that JC(A) = I~o~ where I~0~ is the indicator function of {0}. Theorem 3. ,./¢(A) =
If XeA(f2, d, p, Z), then f a Xda
is a p-continuous fuzzy valued measure of bounded variation. Proof. We shall show that Jr' satisfies the condition of countable additivity. If {A~}~N is the sequence of pairwise disjoint subsets of Z and A = Ui°°__~A~, then, using results from [18]
•gI~(A)=[JC(A)]==(fAXdp)=fAX~d# • Since X,,e..W, we have
i=1
. Ai
i=1
From [18] we know that JCa(A)cJI~(A) for all 0 < ct < fl ~< 1 and all Ae~q¢. Now, we can use Theorem 2. In order to prove that ¢g is of bounded variation we proceed as follows: Since XeA, there exists integrably bounded function h~ : D ~ R such that IX~I(~) : IX=(~)I :
sup IIx[I ~< h,(co) for all coet2. x~Xo(~O)
Further, according to Theorem 2.2 [4], we obtain I~t',lff2)= sup ~. I~¢,(Ai)I = sup ~. i=1
=sup~ i=1
sup Ilxll
i= 1 xe,ll,(AO
fAfdP
sup f¢Sx
= sup
~
supf feSx
sup ]lxlld# = sup ~ i= 1
~
A i xeX,(o~)
i= 1
Ilflldp A~
IX~(co)ld# Ai
h~(og)dp=fah~(og, dll
f , Ai
So, we have proved that if XeA then J l : d ~ ~c(Z) defined by ¢g(A) = ~a X dp is a fuzzy valued measure. Let ..¢t' : d ~ ~c(;t) be a fuzzy valued measure and let XeA. We call X a Radon-Nikod~,m derivative of ./¢ with respect to p if ..¢t'(A)= ~ Xdp, JA
Aed.
In the next theorem we present the existence of Radon-Nikod~m derivative of fuzzy valued measure of bounded variation.
M. StojakoviO / Fuzzy Sets and Systems 65 (1994) 95-104
101
Theorem 4. If z has the Radon-Nikodj~m property, Z* is separable and J [ : s l ~ ~¢(Z) is a #-continuous fuzzy
valued measure of bounded variation, then ,At has Radon-Nikod~m derivative contained in A. ProoL The function ,At : ~¢ ~ ~-¢(X) is a/~-continuous fuzzy valued measure of bounded variation which means that ~g, : d ~ ~ ( g ) defined by
is a/~-continuous set valued measure of bounded variation. According to Theorem 4.5 and Remark after Theorem 4.6 [6], Jr', has a unique Radon-Nikod#m derivative X,~£~'. We need to prove that the family {X, },~¢o.1~ define one and only one fuzzy set X defined by X(o2)(x)= sup {xeX,(o2)}, ~t~(0.1]
that is, we shall show that all the conditions of Lemma 1 are satisfied. The construction of Radon--Nikod~,m derivative X, of set valued measure ~t', is given in [-6] (Theorem 4.3, Theorem 4.4). The sequence {m~}~N of selections of Jr', is such that JI,(A) = cl{m~(A)}~N for all A ~ and X,((~) = C--o{f~(o2)}~N, wheref~ are Radon-Nikod~,m derivative of m~. It is obvious that relation
< fl=~Cta(A)=~I,(A)
for all A6~¢
implies that
=:~-'~{ f ~(o)) } j e N = ~ { f J(oJ) } j~N =~ X#(~)= X~t(OO) for all ~o~0. If we consider the sequence ~ < ~2 <
"'"
,lim,~t.
= ~ > O,~ie[O, 1], we have to prove that
x:(o~) = (~ x:,(~o). i=l
Let Ta be the set of selection of ~t'a, fl~{ct, ~ , ..., ~ .... }. Since JI,(A)=,/t',,(A) for all A ~ ¢ it is obvious that T , ~ T,, for all i~N, that is T , ~ n~=l T,,. If we suppose that there exists m from N ~ T,, such that me T,, then
m6T,~ ~m(A)~Jg,,(A) for all i~N and all A6~¢, and there exists Aez¢ such that m(A)¢ ,I,[,(A). It is a contradiction with ~¢/,(A) = n i ~ l ~t',,(A). So, we have proved that T, = N,~l T,,. If we denote by Sa the set of measurable selections of Xa, fie{or, ~1 .... }, then from the relation X,(o))cX,,(to) for all ieN and all o2~t2, we get that S, c N I ~ I S , . In order to prove that S, = N/~lS,, we assume that there exists fE L such that f e N i~ 1s,, and f ¢ S,. Let the sequence {J~ },~Nc N ~ ~S,, be such that l i m , ~ J ~ = f , J~ is a Radon-Nikod~m derivative for some m,~ni~iT~,neN. This means that l i m . . ~ IIL - f IIL = 0 ~ lim._~~ ~a II~(tn) - f(co)II d/~ = 0..~ lim._~~ j~(~o) = f(o~) g-a.e. If B = ~, /~(B) = 0 is such that l i m . ~ ( c o ) =f(o~)for all o~¢P\B, then let the sequence {f.}.~N be defined by f,(to) = j~(og), o 2 ~ \ B , f.(¢o) = f(aO,
co~B.
102
M. Stojakovi5 / Fuzzy Sets and Systems 65 (1994) 95-104
It is obvious that IIL--~IIL=0 for all neN, lim.~oof.=f (i.e. l i m . ~ [ I f . - - f l l L : 0 ) and lim,~ ~of. (~o) = f(~o) for all ~oe t2. From closeness of 0 ~= iS~, it follows that {f, }.~ ~ 0 i~S~,, • Now, we can take a countable field ~¢o = A such that {fn} ~ L(f2, .~t~, #, Z), where ~ 1 is the a-field generated by ~¢o ([6]). From compactness of ~t',(A)= ~I°°_I~,,(A), for all AeMo, since {~Af.d#},~u=JC~(A), the sequence { ~af, d/z}.~u has a convergent subsequence. Thus, by the Cantor diagonal process, there exists a subsequence {g.} of {f.} such that {~a g,d;t}.~u is convergent for every A=~¢o. Since IIg,(og)[I ~< IZ(og)l/~-a.e., where Z = Aa is defined by Z(~o)= cl{f(~o)}, f s 0 L I S~,, to el2, the sequence {~Ag, dkt}.~ is convergent and lim f g.dp = m(A) n~oo A
for all A ~ .
Z is a finitely additive set function and since IIm(A)II ~ I ~ I(A) for all A ~ ~¢1 it follows that m is countable additive and m is a selection of ~t',. From the extension of the Lebesgue dominated convergence theorem T h e n m:~q~ 1 ~
m(A) ~--lira f gnd]~ = f lim gnd~ n~ A a n~
and, since {g. }.,N is subsequence of {f. }.~N, we get
re(A) =
ff d#
which means that feS~, that is S~ = (]~=1S~,. To see that X~(o~) = (] i~ aX~(a~) for all oJ~O, we denote by Y(o~) the set (]i~ iX~,(°~), ¢oeY2. Since the family of sets {X~,(o~)}i~N has the finite intersection property, Y(~o)4= 0 and Y(o~) is a compact set. Also limi~ ~H(Y(o~), X~,(m)) = 0 for all ~ o ~ , which implies that limio ooX~,& Y and from completeness of (.L~, A ) we get that YeLlow. Further, we have the next implications.
> ~:::.X~(oJ)~X~,(o~)~X~(o~)~ N X~,, i=1
that is, X~(~o)~ Y(o~) for all oJ~g2. From Y ~ A a and Lemma 1.1. [-4] it follows that there exists the sequence {h.}.~N~L such that Y(o~) = cl{h.(~o)} for all o ~ O and
Y(og)~X.,(~)=:. {h.}.~N=S.,=~ {h.}.~u~ (~ S., = S~ i=1
~cl{h,(~o)}~cl{g(m):geS~}
for all wsg2,
which means that Y(~o)~ X~(oJ). So, we have proved
Y(o~) = N X~,(~o) = X~(~o) i=1
for all o~eQ.
Theorem 5. Let Z has the Radon-Nikod~m property, Z* be separable and ~ sub-a-algebra of d . If X e A then
there exists E(XI~)eA(f2, ~ ) such that A
Xd# = f
A
E(Xr~-)d#
for all A ~ .
M. Stojakovid / Fuzzy Sets and Systems 65 (1994) 95-104
103
Proof. Since X e A , from Theorem 3 there exists ./4 : ~¢ ~ ~c(X) such that
Jt'(A) = C Xd#, JA
A~¢.
The restriction of ~¢/on ~ is defined by ./t'(A) = f Xd#, JA
A~.
That function is fuzzy valued measure on ~ and from Theorem 4 there exists YeA(t2, ~ ) which is RadonNikod~m derivative of .A( such that J I ( A ) = ~ Ydp, dA
A~.
That Y we denote by E ( X I ~ ) and call conditional expectation of X. It is obvious (Theorem 2 [18,]) that
E(X~I~) = (E(XI~-))~. Further, uniqueness of the existence of conditional expectation E(XI °~) follows from the uniqueness of conditional expectation for random set X,, ~e(0, 1]. Theorem 6. Let ~t[ : d ~ ~c(Z) be a fuzzy valued measure. Then ~l[ has a unique Radon-Nikod~m derivative
contained in A if and only if the following conditions are satisfied : (1) ~t' is #-continuous, (2) Jr' is of bounded variation, (3) 9iven A e d , 0 < #(A) < o% there exists a B ~ A and a compact subset C of z such that #(B) > 0 and ,Ag'o(B')/p(B')cC for all B ' c B with #(B') > 0 (where ~¢[o(A ) = supp J4 (A ) = {x e z : Jg (A ) (x) > 0}). Proof. Since ,g~(B')cJCo(B') for all ~E(0, 1,] we get
.//~(B') - C #(B')
-
JCo(B') C C, #(B')
and all the conditions of Theorem 5.2 [6] are satisfied for ./¢~ : ~¢ ~ ~'c(Z). Further we can proceed as in the proof of Theorem 5. Theorem 7. If . X / " : d ~ ~c(X), n = 1, 2, are fuzzy valued measures of bounded variation which are #-
continuous and such that for all J I ~ ( A ) ~ ~¢[2(A) for all ~te(0, 11 and all A e ~ then (d~g~/d#)(og)c (djt'2/d#)(~o) p-a.e, where d~¢~ and d# are Radon-Nikodj~m derivative of ~g~ and p, ~e(0, 1,]. Proof. The proof is based on Proposition 5.1. [13].
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M. StojakoviO / Fuzzy Sets and Systems 65 (1994) 95-104
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