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Fuzzy integrals of set-valued mappings and fuzzy mappings
sots and sVstams ELSEVIER
Fuzzy Sets and Systems 75 (1995) 103-109
~
,
Fuzzy integrals of set-valued mappings and fuzzy mappings Deli Zhang a'*, Caimei Guo b "Department of Mathematics, Jilin Prov. Institute of Education, Changchun, Jilin 130022, People "s Republic of China bDepartment of Basic Science, Changchun University, Changchun, Jilin 130022, People's Republic of China
Received November 1992; revised August 1994
Abstract
The paper based on (Sugeno, 1974) first defines the concept of fuzzy integrals of set-valued mappings by the similar way of Aumann (1965). Then it is discussed, some results similar to Aumann integral are obtained, these include convexity, closedness, Fatou's lemmas and the Lebesgue convergence theorem, etc. At last, the fuzzy integral of fuzzy mappings is defined, the extended results corresponding with fuzzy integrals of set-valued mappings are showed. Keywords: Fuzzy integral; Set-valued mapping; Fuzzy integral of set-valued mapping; Fuzzy integral of fuzzy mapping
1. Introduction
Since Prof. Zadeh [161 gave the concept of fuzzy set in 1965, fuzzy mathematics has been developing rapidly. The theory of fuzzy-valued integrals is an important field, and it has been shown very useful in other branch of fuzzy theory and in reality. The good work can be found in [3, 6-9, 111, but these fuzzy-valued integrals are all based on classical integrals (integrations). Sugeno had brought out the theory of fuzzy integrals in 1974 [13], and it was made much deeper by the authors in [12, 14, 151, but their integrands are all functions (point-valued), there is no set-valued mapping (set-valued function) or fuzzy mapping. In Ref. [17], we first introduced the fuzzy integral of fuzzy-valued functions, where the fuzzy-valued function was valued in fuzzy numbers on [0, + ~ 1. In Ref. [181, we have discussed the fuzzy integral of set-valued functions. This paper can be viewed as the continuing work of above, a thorough investigation about the fuzzy integrals of set-valued mappings will be given, these results will be extended to the circumstance of fuzzy mappings. It will show that Sugeno's fuzzy integral is a special case of our work to some extent. In the paper, the following notations will be used. Symbol I denotes the unite interval [0, 11, P(I) denotes the power set of I, ~(I) denotes the fuzzy power set of I. The triplet (X,d,m) is a classical complete
* Corresponding author. 0165-0114/95/$09.50 © 1995 - Elsevier Science B.V. All rights reserved SSDI 01 65-0 1 1 4 ( 9 4 ) 0 0 3 4 2 - 4
104
D. Zhang, C. Guo / Fuzzy Sets and Systems 75 (1995) 103-109
probability space (non-fuzzy), # :~¢ ~ I is a fuzzy measure of Sugeno's sense, and in addition, p satisfies the following two conditions: (i) p is null-additive, i.e. #(A) = 0 implies #(A w B) = p(B); (ii) # ,~ m, i.e. re(A) = 0 implies #(A) = 0. Throughout the paper, our discussion is supposed to be carried on (X, d , m, #). The fuzzy integral of a measurable function f : X ~ I on X is defined as
ff
d# = V [~x A #(f(x) >1~)],
where (f>~ ~) = (x e X:f(x) >~~}. Lemma. Let fl and f2 be two measurable functions. Then fl(x)=f2(x) for x e X, m-a.e, implies if1 d# =
If2d . A set-valued mapping is a mapping F:X ~ P(l)\{(p}, and it is said to be measurable if its graph is measurable, i.e. G r F = {(x,r) ~ X × 1: r e F(x)} belongs to d ® Borel (I). A fuzzy mapping is a mapping/~: X ~ ~ ( I ) \ {~o}, and it is said to be measurable if its 2-cutting set-valued mapping Fz is measurable for every 2 ~ (0, 1], where Fz(x) = [ff(x)]z = {r • I: ff(x)(r) >>.2}. It is clear that a set-valued mapping is a special case of a fuzzy mapping. Since a fuzzy mapping is a set-valued mapping when its values are crisp sets (classical sets, non-fuzzy). The rest of this paper consists of two parts. In Section 2, we discuss the fuzzy integral of set-valued mappings. Since a fuzzy mapping is essentially a family of set-valued mappings, this section is the necessary preparation of next part. In Section 3, we have a study of fuzzy integrals of fuzzy mappings, it is a good extension of the fuzzy integral of set-valued mappings.
2. Fuzzy integrals of set-valued mappings In this section, the definition of fuzzy integrals of set-valued mappings is first given, and then its various kinds of properties are shown.
Definition 2.1. Let F be a set-valued mapping. Then the fuzzy integral of F on X is defined as,
f F d ~ = ( f f d~: f e S ( F ) } , where S(F) = { f : f i s measurable, f(x) • F(x) for x e X, m-a.e.}, i.e. the family of m-a.e, measurable selections of F. Obviously, ~ F d# may be empty. A set-valued mapping F is said to be integrable if ~ F d# ~ ~o.
Proposition 2.1. If F is a measurable set-valued mapping, then F is integrable. It is clear that Definition 2.1 (of course, # may be arbitrary) is a natural extension of Sugeno's fuzzy integral, as a measurable function can be viewed as a special set-valued mapping. A set-valued mapping F is said to be convex-valued, if F(x) is convex for all x e X, m-a.e. The following lemma will show that the fuzzy integral of a convex-valued mapping is convex.
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105
Lemma 2.1. l f a measurable set-valued mapping F is convex-valued, then ~ F dlz is convex. Proof. I f ~ F d p is a single point set, then it is convex. Otherwise, let Yl,Y2 E ~Fd#, and yl < Y2, then there exist fl,f2eS(F), s.t. y~ = ~ f l d # , Y2 = j f z d # . y = ~fd/z. Define
f(x) =
Further, let y~(Yl,Y2), we need to find f e S ( F ) , s.t.
y,
x e ( f ~ <~y<<.f2),
fl(x),
x e ((fj Afz) > y ) u ( f l > y >fz),
f2(x),
xe((f, Vf2)
It is easy to see that f e S(F), and furthermore tt(f~> y)/> #(f2 1> Y)/> P(f2 >~ Y2) ~> Y2/> Y, P ( f > Y) = #(f~ > Y) ~< P(f~ > Yl) ~< Yl ~< Y. Then we obtain y = ~ f d # . By the arbitrary of y, we know that {y: Yl < Y < Y2} is an interval. Further, by the arbitrary of yx and y2, it shows that ~ F d p is also an interval. Consequently, f F d p is convex. [] For a set-valued mapping F, let co F(x) denote the convex hull of F(x) for x ~ X, m-a.e, then holds the following.
Proposition 2.2. If F is a measurable set-valued mapping, then co ~ F d# = ~ co F dp. Proof. The relation c is obvious. To obtain the converse relation ~ , we assume y e S co F d#, then there exists f e S(co F), s.t. y = ~ f dp. If f is in S(F), then y E co ~ F dp. Otherwise, we assume y is not S(F). Let us set
G r E = G r F n { ( X x l : f ( x ) <~r <~ 1}. Then Gr E is non-empty, measurable, further it defines a measurable set-valued mapping E, then there exists a m-a.e, measurable selection fl of E. Of course ./'1 e S(F), and ~ f d p ~< ~fl dp. Similarly, we can find a f2 e S(F), s.t. S f d # / > ~f2 dp. Hence y ~ co ~ F d#, relation ~ is proved. [] A set-valued function F is said to be closed-valued if F(x) is closed for x e X, m-a.e.
Proposition 2.3. If a measurable set-valued function F is closed-valued, then S F dla is closed. The proof will be given after Theorem 2.2.
Corollary 2.1. Let F be a measurable interval-valued function, i.e. F(x)= [ f-(x), f+ (x)]. Then ~ F dlt = [ ~ f - dp, ~ f + dp]. Let {A.} c P(I) be a sequence. We define
limsupA" = {x: x = lim x"k' x"~ A" (n >>"l)} liminfA, = {x: x = limx,, x, e A, (n >/1)}.
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D. Zhang, C. Guo / Fuzzy Sets and Systems 75 (1995) 103-109
It had been shown that lim sup A, and lim infA. are closed sets [5]. If lim sup A, = lim infA, = A, then we say {A.} is convergent to A, simply written by limA. = A or A, -* A. Using above definition, let {/7.} be a sequence of set-valued mappings, we can define lim sup F,, lim infF., lim F,, by pointwise way. For example: (limsupG)(x) = limsupF,(x)
for x ~ X, m-a.e.
Theorem 2.1 (Fatou's lemma). If {F.} is a sequence of measurable set-valued mappings, then (i) limsup~F~d# c ~limsupF.d/a; (ii) ~liminfF.d#, ~ liminf~F.d#. Proof. (i) If the left is empty, then c is obvious. Otherwise, let y e lim sup ~ F. d/~, then from the definition of limsup we have that y =limk~o~y.k, and y , e ~ F , dlt (n >~ 1). So there exists f.keS(F.~) (k >~ 1), s.t. Y.k = ~f.~ dl~ ~ y(k --* ~). Since { f.~} is included in the compact space I x (with the usual product topology), we can find a subsequence {fro} of {f,k}, s.t. {fro} is convergent. So ~fmd#--+y~(m--+~). That is, y = lim Sfm d/~ = ~ lim f,~ d# e ~lim sup F, d#. (ii) If the left is empty, then ~ is clear. Otherwise, let y e ~lim inf/7, d/~, then there exists a f e S (lim inf F,), s.t. y = ~ f d # . Write I ~ = I x I x .--, then I ~ is a complete metric space (with the metric induced by the usual product topology). For each x ~ X, define a subset G(x) of I ~ by
G(x) = {(Yl,Y2 .... ): y , ~ F , ( n >7 1), limy, = f ( x ) } then the statement f e S(lim infF.) is equivalent to the statement that G is a set-valued mapping. Further, we can prove that G is measurable (for a metric the measurability is same defined) [1]. Then there exists a m-a.e. measurable selection g of G, that is a sequence of measurable functions { f , }, s.t. f, ~ S(F~) (n ~> 1), lim f, = f This implies ~f . d# --+ ~f d# = y(n --, oo). Consequently, y ~ lim inf~ F. d/~. [] From the above theorem, we can easily obtain the following Lebesgue convergence theorem.
Theorem 2.2. Let {F,} be a sequence of measurable set-valued mappings, and F be a set-valued mapping. If lim F, = F, then lim ~ F, d# = ~ F d/~. Proof. Since F = liminfF, = limsupF., then
fFdp=fliminfF.d~,~liminffF.d#~limsupfF.d~,~flimsupF.d~,=fFdp. Hence, lim S/7, d/a exists and equals to ~ F d#.
[]
Proof of Proposition 2.3. By Fatou's lemma, we can easily get the proof of Proposition 2.3. That is, if we set F. = F, then lim sup F. = F. Further,
limsupfF,,di,~flimsupF,,d#=fFd~. This implies S F d/a is closed.
[]
3. Fuzzy integrals of fuzzy mappings In the section, we will extend the results in Section 2 to the integral of fuzzy mappings. For this aim, we begin with the definition.
D. Zhang, C Guo / Fuzzy Sets and Systems 75 (1995) 103-109
107
Defiaitiou 3.1. Let P be a fuzzy mapping. Then the fuzzy integral of P on X is defined as
( f F d p ) ( r ) = s u p { 2 c ( O , 1]: rc ;F~dp} • By representive theorem of fuzzy sets, we directly have the following proposition. Proposition 3.1. If P is a measurable fuzzy mapping, then S ffdp is a non-empty fuzzy set, and (~Pd#)a =
A fuzzy set f e ~ ( I ) is said to be fuzzy convex (fuzzy closed, resp.) if rz = {r e (0, 1]: f(r)/> 2} is convex (closed, resp.) for each 2 e (0, 1]. A fuzzy mapping F is said to be fuzzy convex-valued (fuzzy closed-valued, resp.) if/V(x) is fuzzy convex (fuzzy closed, resp.) for x c X, m-a.e. Proposition 3.2. If ff is a measurable fuzzy mapping, then (i) ff is fuzzy convex-valued implies ~ F d# is fuzzy convex; (ii) F is fuzzy closed-valued implies ~ F dp is fuzzy closed, and (~ F dp)a = ~ Fa d# for 2 e (0, 1]. Proof. By Proposition 2.3, (i) is obvious. In order to prove (ii), it is sufficient to prove the equation. For 2 ~ (0, 1], let 2k = (1 - 1/(k + 1))2, then 2~/~2, we can easily obtain
= lim fF~, dp
( by closedness)
= flim F~, dp
(by Theorem 2.2) (k~l F~, is defined by pointwise way)
= f F ~ dp. Consequently, the equation is proved.
[]
A fuzzy set ~c ~ ( I ) is said to be a fuzzy number, if r~ is a closed convex set for each 2 c (0, 1]. A fuzzy mapping F is said to be a fuzzy number mapping, if/V(x) is a fuzzy number for each x e X, m-a.e. Corollary 3.1. lf a measurable fuzzy mapping ff is a fuzzy number mapping, then S ff dp is a fuzzy number. Next, we will show the convergence theorem. At first, we make following definitions. Let {~n} be a sequence of fuzzy sets. Its limit superior and limit inferior are defined as (lim sup ~n)(r) = sup {2 c (0, 1]: r ~ lim sup (~,)~ }, (lim inf~)(r) = sup {2 e (0, 1]: r c lim inf (~)~}. Iflim sup ~ = lim inf~n = ~, then we say that {f~} is convergent to F, simply written by lim ~ ~ ~, or F~ ~ f.
D. Zhang, C. Guo / Fuzzy Sets and Systems 75 (1995) 103-109
108
Similarly, for a sequence of fuzzy mappings {ft. }, lim sup if'., lim infff,, and lim F, are just defined with pointwise way (m-a.e.). For instance: (lim sup ff.)(x) = lim sup/~.(x) for x e X, m-a.e. Theorem 3.1 (Fatou's lemma). Let {F.} be a sequence of measurable fuzzy mappinos. Then
(i) limsupf/~d# c flimsup/~d#; (ii) lf F. (n >11) is fuzzy closed-valued, then ~liminf/~.d# ~ liminf$/T.d#. Proof. For 2 e (0, 1], we have (lira sup fF, d # ) = 2
0 lim sup ( f P . d/~) 2'<).
2"
= ~'<~Nlim sup k@l
f(p.)2kd#
2,<~k~ ~ lim flimsup(P.)2~d/~
(2k = (1 -- 1/(k + 1))2')
(by Theorem 2.1)
= 0 f ~ hm ' sup(F.) - d#2~ (by Theorem 2.2) 2' < ) . ,
k
1
= 2'<2. ('] ( (limsup~")a'd/~
= (flim supP.d#)2. Then (i) is proved. (ii) can be proved in the same way, that is -
f (lim inf/~.)2 d# fk=~-]llira inf(ff.)2kd#
(~ = 1 - (1/(k + l))x)
= k=l (~ fliminf (P.)2, d# c ~01 lim in
rf(r.)2,d#
= (] lim in k= 1
\d
= (lim inf fF. d#)2-
/2k
[]
D. Zhang, C Guo / Fuzzy Sets and Systems 75 (1995) 103-109
109
Theorem 3.2 (Lebesgue convergence theorem). If {Fn} is a sequence of measurable fuzzy closed-valued mappings, then {/~n} is convergent implies ~ Fn dg is convergent, and ~ limF. dg = lira j"Pn dg. By Fatou's lemmas, the proof is easy.
4. Concluding remark Up to here, we have built up a theory of fuzzy integrals of set-valued mappings and fuzzy mappings. The result are very similar to Aumann's integral [1]. It is well known that Aumann's integral comes from the study of Economics, and plays important roles in it. So we can expect that fuzzy integrals of fuzzy mappings can show the same application. That is our further task of study.
References i-I] [2] [3] [4] [5] [6] [7] [8] [9] [ 10] [11] [12] [13] [14] [15] [16] [17] [18]
R.J. Aumann, Integrals of set-valued functions, d. Math. Anal. Appl. 12 (1965) 1-12. C. Castaing and M. Valadier, Convex Analysis and Measurable Multi function (Springer, Berlin, 1977). D. Dubois and H. Pride, Towards fuzzy differential calculus. Part I, Fuzzy Sets and Systems 24 (1982) 1-17. D. Dubois and H. Pride, Fuzzy Sets and Systems - Theory and Applications (Academic Press, New York, 1980). E. Klein and A. Thompson, Theory of Correspondences (Wiley lnterscienc~, New York, 1984). E.P. Klement, Integration of fuzzy-valued functions, Rev. Roumaine Math. Purels Appl. 30 (1985) 375-384. O. Kaleva, Fuzzy differential equations, Fuzzy Sets and Systems 24 (1987) 301-317. M. Matloka, On fuzzy integral, Prec. Polish Syrup. Interval & Fuzzy Math. 4-7 (1985) 163-170. S. Nanda, On integration of fuzzy mapping, Fuzzy Sets and Systems 32 (1989) 95-101. N.S. Papageorgiou, On the theory of Banach space valued multifunctions. 1. Integration and conditional exl~ction, d. Multivariate Anal. 17 (1985) 185-206. M.L. Puri and D. Ralescu, Fuzzy random variables, J. Math. Anal. Appl. 114 (1986) 409-422. D. Ralescu and G. Adams, The fuzzy integral, J. Math. Anal. Appl. 75 (1980) 562-570. M. Sugeno, Theory of fuzzy integrals and its applications, Ph.D. Dissertation, Tokyo Inst. of Tech. 1974. Z.X: Wang, Fuzzy measures and measures of fuzziness, J. Math. Anal. Appl. 104 (1984) 589-601. Z.Y. Wang, The autocontinuity of set-function and the fuzzy integral, J. Math. Anal. Appl. 99 (1984) 195-218. L.A. Zadeh, Fuzzy sets, Inform. Control g (1965) 338-353. D.L. Zhang and Z.X. Wang, Fuzzy integrals of fuzzy-valued functions, Fuzzy Sets and Systems 54 (1993) 63-67. D.L. Zhang and Z.X. Wang, Fuzzy integrals of set-valued functions, Busefal 46 (1991) 56-64.
Fuzzy Sets and Systems 75 (1995) 103-109
~
,
Fuzzy integrals of set-valued mappings and fuzzy mappings Deli Zhang a'*, Caimei Guo b "Department of Mathematics, Jilin Prov. Institute of Education, Changchun, Jilin 130022, People "s Republic of China bDepartment of Basic Science, Changchun University, Changchun, Jilin 130022, People's Republic of China
Received November 1992; revised August 1994
Abstract
The paper based on (Sugeno, 1974) first defines the concept of fuzzy integrals of set-valued mappings by the similar way of Aumann (1965). Then it is discussed, some results similar to Aumann integral are obtained, these include convexity, closedness, Fatou's lemmas and the Lebesgue convergence theorem, etc. At last, the fuzzy integral of fuzzy mappings is defined, the extended results corresponding with fuzzy integrals of set-valued mappings are showed. Keywords: Fuzzy integral; Set-valued mapping; Fuzzy integral of set-valued mapping; Fuzzy integral of fuzzy mapping
1. Introduction
Since Prof. Zadeh [161 gave the concept of fuzzy set in 1965, fuzzy mathematics has been developing rapidly. The theory of fuzzy-valued integrals is an important field, and it has been shown very useful in other branch of fuzzy theory and in reality. The good work can be found in [3, 6-9, 111, but these fuzzy-valued integrals are all based on classical integrals (integrations). Sugeno had brought out the theory of fuzzy integrals in 1974 [13], and it was made much deeper by the authors in [12, 14, 151, but their integrands are all functions (point-valued), there is no set-valued mapping (set-valued function) or fuzzy mapping. In Ref. [17], we first introduced the fuzzy integral of fuzzy-valued functions, where the fuzzy-valued function was valued in fuzzy numbers on [0, + ~ 1. In Ref. [181, we have discussed the fuzzy integral of set-valued functions. This paper can be viewed as the continuing work of above, a thorough investigation about the fuzzy integrals of set-valued mappings will be given, these results will be extended to the circumstance of fuzzy mappings. It will show that Sugeno's fuzzy integral is a special case of our work to some extent. In the paper, the following notations will be used. Symbol I denotes the unite interval [0, 11, P(I) denotes the power set of I, ~(I) denotes the fuzzy power set of I. The triplet (X,d,m) is a classical complete
* Corresponding author. 0165-0114/95/$09.50 © 1995 - Elsevier Science B.V. All rights reserved SSDI 01 65-0 1 1 4 ( 9 4 ) 0 0 3 4 2 - 4
104
D. Zhang, C. Guo / Fuzzy Sets and Systems 75 (1995) 103-109
probability space (non-fuzzy), # :~¢ ~ I is a fuzzy measure of Sugeno's sense, and in addition, p satisfies the following two conditions: (i) p is null-additive, i.e. #(A) = 0 implies #(A w B) = p(B); (ii) # ,~ m, i.e. re(A) = 0 implies #(A) = 0. Throughout the paper, our discussion is supposed to be carried on (X, d , m, #). The fuzzy integral of a measurable function f : X ~ I on X is defined as
ff
d# = V [~x A #(f(x) >1~)],
where (f>~ ~) = (x e X:f(x) >~~}. Lemma. Let fl and f2 be two measurable functions. Then fl(x)=f2(x) for x e X, m-a.e, implies if1 d# =
If2d . A set-valued mapping is a mapping F:X ~ P(l)\{(p}, and it is said to be measurable if its graph is measurable, i.e. G r F = {(x,r) ~ X × 1: r e F(x)} belongs to d ® Borel (I). A fuzzy mapping is a mapping/~: X ~ ~ ( I ) \ {~o}, and it is said to be measurable if its 2-cutting set-valued mapping Fz is measurable for every 2 ~ (0, 1], where Fz(x) = [ff(x)]z = {r • I: ff(x)(r) >>.2}. It is clear that a set-valued mapping is a special case of a fuzzy mapping. Since a fuzzy mapping is a set-valued mapping when its values are crisp sets (classical sets, non-fuzzy). The rest of this paper consists of two parts. In Section 2, we discuss the fuzzy integral of set-valued mappings. Since a fuzzy mapping is essentially a family of set-valued mappings, this section is the necessary preparation of next part. In Section 3, we have a study of fuzzy integrals of fuzzy mappings, it is a good extension of the fuzzy integral of set-valued mappings.
2. Fuzzy integrals of set-valued mappings In this section, the definition of fuzzy integrals of set-valued mappings is first given, and then its various kinds of properties are shown.
Definition 2.1. Let F be a set-valued mapping. Then the fuzzy integral of F on X is defined as,
f F d ~ = ( f f d~: f e S ( F ) } , where S(F) = { f : f i s measurable, f(x) • F(x) for x e X, m-a.e.}, i.e. the family of m-a.e, measurable selections of F. Obviously, ~ F d# may be empty. A set-valued mapping F is said to be integrable if ~ F d# ~ ~o.
Proposition 2.1. If F is a measurable set-valued mapping, then F is integrable. It is clear that Definition 2.1 (of course, # may be arbitrary) is a natural extension of Sugeno's fuzzy integral, as a measurable function can be viewed as a special set-valued mapping. A set-valued mapping F is said to be convex-valued, if F(x) is convex for all x e X, m-a.e. The following lemma will show that the fuzzy integral of a convex-valued mapping is convex.
D. Zhang, C Guo / Fuzzy Sets and Systems 75 (1995) 103-109
105
Lemma 2.1. l f a measurable set-valued mapping F is convex-valued, then ~ F dlz is convex. Proof. I f ~ F d p is a single point set, then it is convex. Otherwise, let Yl,Y2 E ~Fd#, and yl < Y2, then there exist fl,f2eS(F), s.t. y~ = ~ f l d # , Y2 = j f z d # . y = ~fd/z. Define
f(x) =
Further, let y~(Yl,Y2), we need to find f e S ( F ) , s.t.
y,
x e ( f ~ <~y<<.f2),
fl(x),
x e ((fj Afz) > y ) u ( f l > y >fz),
f2(x),
xe((f, Vf2)
It is easy to see that f e S(F), and furthermore tt(f~> y)/> #(f2 1> Y)/> P(f2 >~ Y2) ~> Y2/> Y, P ( f > Y) = #(f~ > Y) ~< P(f~ > Yl) ~< Yl ~< Y. Then we obtain y = ~ f d # . By the arbitrary of y, we know that {y: Yl < Y < Y2} is an interval. Further, by the arbitrary of yx and y2, it shows that ~ F d p is also an interval. Consequently, f F d p is convex. [] For a set-valued mapping F, let co F(x) denote the convex hull of F(x) for x ~ X, m-a.e, then holds the following.
Proposition 2.2. If F is a measurable set-valued mapping, then co ~ F d# = ~ co F dp. Proof. The relation c is obvious. To obtain the converse relation ~ , we assume y e S co F d#, then there exists f e S(co F), s.t. y = ~ f dp. If f is in S(F), then y E co ~ F dp. Otherwise, we assume y is not S(F). Let us set
G r E = G r F n { ( X x l : f ( x ) <~r <~ 1}. Then Gr E is non-empty, measurable, further it defines a measurable set-valued mapping E, then there exists a m-a.e, measurable selection fl of E. Of course ./'1 e S(F), and ~ f d p ~< ~fl dp. Similarly, we can find a f2 e S(F), s.t. S f d # / > ~f2 dp. Hence y ~ co ~ F d#, relation ~ is proved. [] A set-valued function F is said to be closed-valued if F(x) is closed for x e X, m-a.e.
Proposition 2.3. If a measurable set-valued function F is closed-valued, then S F dla is closed. The proof will be given after Theorem 2.2.
Corollary 2.1. Let F be a measurable interval-valued function, i.e. F(x)= [ f-(x), f+ (x)]. Then ~ F dlt = [ ~ f - dp, ~ f + dp]. Let {A.} c P(I) be a sequence. We define
limsupA" = {x: x = lim x"k' x"~ A" (n >>"l)} liminfA, = {x: x = limx,, x, e A, (n >/1)}.
106
D. Zhang, C. Guo / Fuzzy Sets and Systems 75 (1995) 103-109
It had been shown that lim sup A, and lim infA. are closed sets [5]. If lim sup A, = lim infA, = A, then we say {A.} is convergent to A, simply written by limA. = A or A, -* A. Using above definition, let {/7.} be a sequence of set-valued mappings, we can define lim sup F,, lim infF., lim F,, by pointwise way. For example: (limsupG)(x) = limsupF,(x)
for x ~ X, m-a.e.
Theorem 2.1 (Fatou's lemma). If {F.} is a sequence of measurable set-valued mappings, then (i) limsup~F~d# c ~limsupF.d/a; (ii) ~liminfF.d#, ~ liminf~F.d#. Proof. (i) If the left is empty, then c is obvious. Otherwise, let y e lim sup ~ F. d/~, then from the definition of limsup we have that y =limk~o~y.k, and y , e ~ F , dlt (n >~ 1). So there exists f.keS(F.~) (k >~ 1), s.t. Y.k = ~f.~ dl~ ~ y(k --* ~). Since { f.~} is included in the compact space I x (with the usual product topology), we can find a subsequence {fro} of {f,k}, s.t. {fro} is convergent. So ~fmd#--+y~(m--+~). That is, y = lim Sfm d/~ = ~ lim f,~ d# e ~lim sup F, d#. (ii) If the left is empty, then ~ is clear. Otherwise, let y e ~lim inf/7, d/~, then there exists a f e S (lim inf F,), s.t. y = ~ f d # . Write I ~ = I x I x .--, then I ~ is a complete metric space (with the metric induced by the usual product topology). For each x ~ X, define a subset G(x) of I ~ by
G(x) = {(Yl,Y2 .... ): y , ~ F , ( n >7 1), limy, = f ( x ) } then the statement f e S(lim infF.) is equivalent to the statement that G is a set-valued mapping. Further, we can prove that G is measurable (for a metric the measurability is same defined) [1]. Then there exists a m-a.e. measurable selection g of G, that is a sequence of measurable functions { f , }, s.t. f, ~ S(F~) (n ~> 1), lim f, = f This implies ~f . d# --+ ~f d# = y(n --, oo). Consequently, y ~ lim inf~ F. d/~. [] From the above theorem, we can easily obtain the following Lebesgue convergence theorem.
Theorem 2.2. Let {F,} be a sequence of measurable set-valued mappings, and F be a set-valued mapping. If lim F, = F, then lim ~ F, d# = ~ F d/~. Proof. Since F = liminfF, = limsupF., then
fFdp=fliminfF.d~,~liminffF.d#~limsupfF.d~,~flimsupF.d~,=fFdp. Hence, lim S/7, d/a exists and equals to ~ F d#.
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Proof of Proposition 2.3. By Fatou's lemma, we can easily get the proof of Proposition 2.3. That is, if we set F. = F, then lim sup F. = F. Further,
limsupfF,,di,~flimsupF,,d#=fFd~. This implies S F d/a is closed.
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3. Fuzzy integrals of fuzzy mappings In the section, we will extend the results in Section 2 to the integral of fuzzy mappings. For this aim, we begin with the definition.
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Defiaitiou 3.1. Let P be a fuzzy mapping. Then the fuzzy integral of P on X is defined as
( f F d p ) ( r ) = s u p { 2 c ( O , 1]: rc ;F~dp} • By representive theorem of fuzzy sets, we directly have the following proposition. Proposition 3.1. If P is a measurable fuzzy mapping, then S ffdp is a non-empty fuzzy set, and (~Pd#)a =
A fuzzy set f e ~ ( I ) is said to be fuzzy convex (fuzzy closed, resp.) if rz = {r e (0, 1]: f(r)/> 2} is convex (closed, resp.) for each 2 e (0, 1]. A fuzzy mapping F is said to be fuzzy convex-valued (fuzzy closed-valued, resp.) if/V(x) is fuzzy convex (fuzzy closed, resp.) for x c X, m-a.e. Proposition 3.2. If ff is a measurable fuzzy mapping, then (i) ff is fuzzy convex-valued implies ~ F d# is fuzzy convex; (ii) F is fuzzy closed-valued implies ~ F dp is fuzzy closed, and (~ F dp)a = ~ Fa d# for 2 e (0, 1]. Proof. By Proposition 2.3, (i) is obvious. In order to prove (ii), it is sufficient to prove the equation. For 2 ~ (0, 1], let 2k = (1 - 1/(k + 1))2, then 2~/~2, we can easily obtain
= lim fF~, dp
( by closedness)
= flim F~, dp
(by Theorem 2.2) (k~l F~, is defined by pointwise way)
= f F ~ dp. Consequently, the equation is proved.
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A fuzzy set ~c ~ ( I ) is said to be a fuzzy number, if r~ is a closed convex set for each 2 c (0, 1]. A fuzzy mapping F is said to be a fuzzy number mapping, if/V(x) is a fuzzy number for each x e X, m-a.e. Corollary 3.1. lf a measurable fuzzy mapping ff is a fuzzy number mapping, then S ff dp is a fuzzy number. Next, we will show the convergence theorem. At first, we make following definitions. Let {~n} be a sequence of fuzzy sets. Its limit superior and limit inferior are defined as (lim sup ~n)(r) = sup {2 c (0, 1]: r ~ lim sup (~,)~ }, (lim inf~)(r) = sup {2 e (0, 1]: r c lim inf (~)~}. Iflim sup ~ = lim inf~n = ~, then we say that {f~} is convergent to F, simply written by lim ~ ~ ~, or F~ ~ f.
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108
Similarly, for a sequence of fuzzy mappings {ft. }, lim sup if'., lim infff,, and lim F, are just defined with pointwise way (m-a.e.). For instance: (lim sup ff.)(x) = lim sup/~.(x) for x e X, m-a.e. Theorem 3.1 (Fatou's lemma). Let {F.} be a sequence of measurable fuzzy mappinos. Then
(i) limsupf/~d# c flimsup/~d#; (ii) lf F. (n >11) is fuzzy closed-valued, then ~liminf/~.d# ~ liminf$/T.d#. Proof. For 2 e (0, 1], we have (lira sup fF, d # ) = 2
0 lim sup ( f P . d/~) 2'<).
2"
= ~'<~Nlim sup k@l
f(p.)2kd#
2,<~k~ ~ lim flimsup(P.)2~d/~
(2k = (1 -- 1/(k + 1))2')
(by Theorem 2.1)
= 0 f ~ hm ' sup(F.) - d#2~ (by Theorem 2.2) 2' < ) . ,
k
1
= 2'<2. ('] ( (limsup~")a'd/~
= (flim supP.d#)2. Then (i) is proved. (ii) can be proved in the same way, that is -
f (lim inf/~.)2 d# fk=~-]llira inf(ff.)2kd#
(~ = 1 - (1/(k + l))x)
= k=l (~ fliminf (P.)2, d# c ~01 lim in
rf(r.)2,d#
= (] lim in k= 1
\d
= (lim inf fF. d#)2-
/2k
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Theorem 3.2 (Lebesgue convergence theorem). If {Fn} is a sequence of measurable fuzzy closed-valued mappings, then {/~n} is convergent implies ~ Fn dg is convergent, and ~ limF. dg = lira j"Pn dg. By Fatou's lemmas, the proof is easy.
4. Concluding remark Up to here, we have built up a theory of fuzzy integrals of set-valued mappings and fuzzy mappings. The result are very similar to Aumann's integral [1]. It is well known that Aumann's integral comes from the study of Economics, and plays important roles in it. So we can expect that fuzzy integrals of fuzzy mappings can show the same application. That is our further task of study.
References i-I] [2] [3] [4] [5] [6] [7] [8] [9] [ 10] [11] [12] [13] [14] [15] [16] [17] [18]
R.J. Aumann, Integrals of set-valued functions, d. Math. Anal. Appl. 12 (1965) 1-12. C. Castaing and M. Valadier, Convex Analysis and Measurable Multi function (Springer, Berlin, 1977). D. Dubois and H. Pride, Towards fuzzy differential calculus. Part I, Fuzzy Sets and Systems 24 (1982) 1-17. D. Dubois and H. Pride, Fuzzy Sets and Systems - Theory and Applications (Academic Press, New York, 1980). E. Klein and A. Thompson, Theory of Correspondences (Wiley lnterscienc~, New York, 1984). E.P. Klement, Integration of fuzzy-valued functions, Rev. Roumaine Math. Purels Appl. 30 (1985) 375-384. O. Kaleva, Fuzzy differential equations, Fuzzy Sets and Systems 24 (1987) 301-317. M. Matloka, On fuzzy integral, Prec. Polish Syrup. Interval & Fuzzy Math. 4-7 (1985) 163-170. S. Nanda, On integration of fuzzy mapping, Fuzzy Sets and Systems 32 (1989) 95-101. N.S. Papageorgiou, On the theory of Banach space valued multifunctions. 1. Integration and conditional exl~ction, d. Multivariate Anal. 17 (1985) 185-206. M.L. Puri and D. Ralescu, Fuzzy random variables, J. Math. Anal. Appl. 114 (1986) 409-422. D. Ralescu and G. Adams, The fuzzy integral, J. Math. Anal. Appl. 75 (1980) 562-570. M. Sugeno, Theory of fuzzy integrals and its applications, Ph.D. Dissertation, Tokyo Inst. of Tech. 1974. Z.X: Wang, Fuzzy measures and measures of fuzziness, J. Math. Anal. Appl. 104 (1984) 589-601. Z.Y. Wang, The autocontinuity of set-function and the fuzzy integral, J. Math. Anal. Appl. 99 (1984) 195-218. L.A. Zadeh, Fuzzy sets, Inform. Control g (1965) 338-353. D.L. Zhang and Z.X. Wang, Fuzzy integrals of fuzzy-valued functions, Fuzzy Sets and Systems 54 (1993) 63-67. D.L. Zhang and Z.X. Wang, Fuzzy integrals of set-valued functions, Busefal 46 (1991) 56-64.