Fuzzy integrals of fuzzy-valued functions

Fuzzy Sets and Systems 54 (1993) 63-67 North-Holland

63

Fuzzy integrals of fuzzy-valued functions Deli Zhang and Zixiao Wang Department of Mathematics, Jilin Provincial Institute of Education, Changchun, Jilin, 130022, China Received June 1991 Revised December 1991

Abstract: This paper, based on the fuzzy measures and integrals given by Sugeno, first defines fuzzy integrals of set-valued functions, and then the fuzzy integral of fuzzy-valued functions is obtained and discussed. Next, it defines the fuzzy integral of fuzzy mappings, then pays much effort to the fuzzy integral of interval-valued functions. The properties and convergence theorems are obtained. All these are generalizations of earlier results. Keywords: Interval number; fuzzy number; fuzzy integral; fuzzy-valued function.

1. Introduction

Since Zadeh [12] introduced the concept of fuzzy sets in 1965, fuzzy mathematics which deals with the fuzzy phenomena in reality has greatly developed and has been widely used in engineering. The theory of integrals of fuzzyvalued functions is an important field. The earliest work on the field can be found in Dubois and Prade [2], and then He [4], Klement [5], Luo [6], Matloka [7], Puri [8] and many others have made investigations. The results are beautiful, but are all based on classical integral (integration). Sugeno [10] brought out the theory of fuzzy integrals in 1974, It is another way, different from Zadeh's, to represent fuzzy sets. This made it possible to establish the corresponding theory about fuzzy-valued functions. This will be our main purpose. The paper consists of three parts. After the preliminaries in Section 2, the definition of fuzzy integrals of set-valued functions is given in Section 3, and then the fuzzy integral of interval-valued functions is obtained. In Section Correspondence to: Dr. Deli Zhang, Department of Mathematics, Jilin Provincial Institute of Education, Changchun, Jilin, 130022, China.

4, the fuzzy integral of fuzzy-valued mappings is defined at first, and then the fuzzy integral of fuzzy-valued functions is obtained and discussed. Various kinds of properties and convergence theorems are given at the same time. These results generalize those of [9, 10, 11]. In the paper, the following notations will be used. R + denotes [0, ~), P(R +) is the power set of R ÷, F(R ÷) is the fuzzy power set of R ÷, X is an arbitrary fixed set, ~¢ is a a-algebra formed by the subsets of X, (X, M) is a measureable space, g:M---~R ÷ is a fuzzy measure defined by Sugeno. For a measurable function f :X---~ R +, let Smfdg be the resulting fuzzy integral. Operations * e { + , . , v, A }.

2. Preliminaries

In this section, we shall summarize some concepts about fuzzy numbers and fuzzy-valued functions, which will be used in the rest of the paper. They can be found in [2, 4, 6, 13]. Let

I(R +) = {f: ? = [r-, r +1 c R+}. Then the elements in the set I(R +) are called interval numbers. On the interval numbers set, we make following definitions:

?*p = [r-*p-, r+*p+], k- ~= [k" r-, k . r+], f~
0165-0114/93/$06.00 © 1993--Elsevier Science Publishers B.V. All rights reserved

Deli Zhang, Zixiao Wang / Fuzzy integralsof fuzzy-valuedfunctions

64

that f~ ---->~, if d(f,, ~)--->O. Obviously, ?n ~ ~ iff r~--->r-, r+~--* r +. A fuzzy number is a fuzzy set f in F(R +) satisfying the following conditions, (i) (Normality) f (r) = 1 for some r ~ R +. (ii) (Closed convexity) For every Z ~ (0, 1],

~ = {r e R+: f(r)I>/~.}e I(R+). Let R+ denote the set of fuzzy numbers. W e make following definitions on it,

(f + p)~ = ~ + p~, (k- f)x = k . e~, f ~ / ~ iff ?~
for all )~ e (0, 1],

D(e, p) = sup(d(ex, p~), Z ~ (0, 1]}.

Then (/~+, 4 ) is a partially ordered set, and D is a metric on R +. For a sequence of fuzzy numbers {f,} c / ~ + , we say: (i) it is convergent iff {(f,)x} is a convergent sequence of interval numbers for all Z e (0, 1], and the limit is defined as f(r) = sup{Z: r ~ lim(f,),}, simply write as fn---> f; (ii) it is strongly convergent iff there exist an ~/~+ such that D(f,, f)---> 0, or simply said {fn} is strongly convergent to f, denoted with f , - ~ f. Obviously, ?, -~ f implies rn---> r. A set-valued function is a mapping F :X---> P(R +) \ {0}. F is said to be closed valued iff F(x) is closed for every x • X, and F is said to be measurable iff

FW(B) = {x ~ X, F(x) :'1B 4=0} belongs to ~¢ for every B e Borel(R+). An interval-valued function is a special closed-valued set-valued function f :X---> I(R +). It is usually written as f ( x ) = [ f - ( x ) , f + ( x ) ] , where

f+(x)= s u p f ( x ) . A fuzzy-valued function is a mapping f : X - - * /~+. It is said to be measurable iff fx is measurable for each ~,e (0, 1], where fx(x)--

For example, we define f ~<~ iff f(x) <<-g(x)

for every x E X,

f ~ g iff f(x) <
for every x e X.

Lemma 2.1. Let f(x) = [f-(x),f+(x)]. Then f is measurable iff f - and f+ are measurable.

Proof. If f is measurable, then by the Castaing representation, there exists a sequence of measurable selections of f, denoted with {a~}, such that )~(x) = cl{an(X)}. Then we have

f - ( x ) = inf{a~(x)},

f+(x) = sup{an(x)}.

Consequently, f - and f + are measurable. The necessity is proved. To prove the sufficience, let B e Borel(R÷), and for b e B, assume I(b) is the connected branch including b. Then I(b) is a single point set or an interval. Further, let B 1 = {b E B: I(b) is a single point set}. Then B - B 1 can be represented by the set LAn (pn, qn) where the interval may be open, closed, half open or half closed, and the union is at most countable. Since Gw(Bc)) c = (x ~ x: ](x) n B c ~ O} c

= (x ~ x : : ( x ) n B ° = 0} = (x ~ X : : ( x ) = B }

= (x ~ X : [ f - ( x ) , / + ( x ) ] = B}

(x ~X: [f-(x),f+(x)] c BI U [,..J(P,,qn)l, n

= U ( x e X : f - ( x ) = f + ( x ) =r} fEB 1

U ([.h] {x e X : [ f - ( x ) , f+(x)] c (p,, qn)}) = ((x ~ x : f - ( x ) ~ B,} n ( f + - f - ) - l ( 0 ) )

f - ( x ) = inff(x),

(t(x)). In the ordinary way (pointwise), we can define the operations, orders, convergences of interval-valued functions and fuzzy-valued functions.

Since

B, = B\t_J (p., q.), n

and {0} are in Borel (R+), f - and f + are measurable. This implies that (fW(BC))c~zg. Further fW(B c) e ~¢. As B ~ can be viewed as arbitrary, therefore f is measurable. []

Deli Zhang, Zixiao Wang / Fuzzy integralsof fuzzy-valued functions The lemma shows us that the measurability of interval-valued functions is closed under any operations.

assume that r • (a, b ). Let

f-(x), f f(x)=~r, f+(x),

3. Fuzzy integrals of interval-valued functions In this section, we first define the fuzzy integral of set-valued functions in a similar way to Aumann, and then we mainly discuss the fuzzy integral of interval-valued functions. Extended results corresponding to Sugeno's fuzzy integral are obtained.

65

x • ( f - > r), x•(f+>~r)N(f-<~r), x • ( f +
where ( f + > I r ) = { x • X , f + ( x ) > t r } , and the other sets are similar. It is clear that f is a measurable selection of f. Since

g(f >~r) =g(f+ >Ir)>lg(f+ >>-b)>>-b> r, g ( f > r) = g ( f - > r) <-g ( f - > a ) ~
Ddinition 3.1. Let H:X--+P(R+)\{~} be a set-valued function. Then the s e t (~A f d g ' f is a measurable selection of H} is called the fuzzy integral of H on A, where A • ~ . H is said to be integrable on A, if J'A H dg ~ I~.

Proposition 3.1. ( i ) I f H is a closed-valued measurable set-valued function, then H is integrable on every A • ~t. (ii) If Ill c HE, then S,a H1 dg c Sa HE dg. (iii) ~a H dg = ~x IA "H dg. Here A • ~t, and la is the characteristic function of A.

g(f >- r) > r > g(f > r). Consequently, r = ~ f dg, and then r • J"f dg. [] Next we give other two representations of the fuzzy integral of interval-valued functions. Let

k=l

where rk•I(R+), E k • ~ , for k = l , 2 . . . . ,n, and [._J7=1Ek = X, ri :/: ~ (i :/:j). Further we write

Q(s) = ~/ (rk ^ g(Ek)). k=l

Remark 3.1. Because of this property, without loss of generality, sometimes we only discuss the fuzzy integral on X, and .[x is simply written as

f.

Theorem 3.1. Let f be a measurable interval-

valued function. Then (i) ( f dg = sup Q(g),

Proposition 3.2. Let f be a measurable interval-

valued function. Then f is integrable, and

(ii) f]" d g = rY~ ( x ~ j ( x ) ^ g(F)). The proof of the theorem can be obtained by Proposition 3.2 and corresponding results in [9].

Proof. The integrability becomes obvious by Proposition 3.1(i). Let r • J'] dg. Then there exists a measurable selection f of )r such that r = f f d g , but f - ~
Remark 3.2. The infinite operation on I(R ÷) can be defined similarly as in Section 2.

ff-dg< ff dg<~ff+ dg,

This theorem shows that if we view an interval-valued function as function, and define its fuzzy integral as in [9], then the two definitions are equivalent.

i.e. r • [J'f- dg, f f + dg]. We write a = S f - d g , b = J ' f + d g , for r • [a, b]. If r is equal to a or b then r • J'f dg. Next

Theorem 3.2. Fuzzy

integrals of measurable interval-valued function have the following

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Deli Zhang, Zixiao Wang / Fuzzy integrals of fuzzy-valued functions

properties: (i) fl
an fuzzy integral theory with respect to fuzzy-valued functions. It is an extension of Section 3. Definition 4.1. Let G:X--*F(R +) be a fuzzy mapping. Then the fuzzy set

fAf dg<- fBf dg" (iii) ff A ~ ~, e e I(R+), then

O.(r)=sup{Z:refAGXdg } a ~ dg

^ g(A ).

(iv) Ire ~ I(R+), then

f(f +Odg<~ff dg+f~dg. (v) Let r e g +. If d(fl, fE)(X) < r on X, then

d(ff, dg,ff2dg)
Corollary 3.1. Let {f~} be a sequence of measurable interval-valued functions. (i) If fl c f2 c ' " ", i.e. fl(x) cf2(x) c ' " for each x e X, and [._J~f~ is an interval-valued function, where

is called the fuzzy integral of G on A, where A e .if, ¢~x(x)= (¢~(x))x. It is simply written as fA adg. Clearly, fA G dg = f x (It" G) dg.

Theorem 4.1. Let f be a measurable fuzzyvalued function. Then f f dg e 1~+, and

where 3. e (0, 1]. Proof. The normality is obvious. To prove the closed convexity, it is sufficient to verify the equation. For 2, e (0, 1]. By the representation theorem of fuzzy sets [12] we have

(? Further, if we assume An = (1 - 1/(n + 1))L then )-n-'~Z. It is easy to see that

n fy, g=@fy,. g

n

then

3.'<~,

f(u;o) g=yfio g.

Since f,, ~fa2 ~ " " ' , by Corollary 3.100, we can obtain

(ii) I f ~ ~f2 = ' " ,

then

f(o;O g=Ofio g where

Put fx = N.L. and therefore the conclusion is proved. []

4. Fuzzy integrals of fuzzy-valued functions

Theorem 4.2. Fuzzy integrals of fuzzy-valued functions have the following properties: (i) fl <<-f2 implies f f~ dg ~ f f2 dg. (ii) If A, B ~ M, then A c B implies

In this section, after giving the definition of fuzzy integrals of fuzzy mappings, we establish

f A f dg ~ f j

. .

dg •

Deli Zhang, Zixiao Wang / Fuzzy integrals of fuzzy-valued functions

Corollary 4.2 (Fatou's lemma). Let {fn} be a sequence of measurable fuzzy-valued functions, and assume lim inf, f, is a fuzzy-valued function. Then

(iii) If A ~ d, f ~ 1~+, then

f, a f dg

/x g(A )

(iv) If f ~ R +, then

f liminf f~ dg ~ lim inf ff. dg.

f ff +Odg ff dg+f dg.

Theorem 4.4. Let {fn} be a sequence of

(v) Let a ~ R +. If D(fl, f2)(x) < a on X, then

D(ffldg, ff dg)
0(r)=

1, 0,

r=O, r:/:0.

Theorem 4.3. Let {f~} be a sequence of measurable fuzzy-valued functions. If f,--->f, then ff,, dg--->f f dg.

Proof. By the measurability off., see that f is measurable. Hence defined. Further assume Z e (0, 1 / ( k + l ) ) L Then Zk/~Z, SO representation theorem we have

and f,--+f, we S f dg is well 1], Zk = (1 -that by the

= ("] lim (f,,)a, = ('] lim (f,,)a, )C<~,

n

k

n

= lim lim (f,)xk. k

n

Therefore, by Theorem 3.3 we obtain

ffadg=limlimf(f,,)akdg=(limff,,dg)/ Consequently,

measurable fuzzy-valued functions. If fn--~f uniformly on X, then

fi.

Corollary 4.1. (i) J"fl dg v J"f2 dg ~< J"ql ~ f2) dg. (ii) (iii) (iv) (v)

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fi dg.

It is easy to get the proof by Theorem 4.2(v).

References [1] J. Aumann, Integrals of set-valued functions, J. Math. Anal. Appl. 12 (1965) 1-12. [2] D. Dubois and H. Prade, Fuzzy Sets and SystemsTheory and Applications (Academic Press, New York, 1980). [3] D. Dubois and H. Prade, Towards fuzzy differential calculus, Part 1: Integration of fuzzy mappings, Fuzzy Sets and Systems 8 (1982) 1-17. [4] Jiaru He, The Lebesgue integral of fuzzy-valued functions, Sichun Shida Xuebao 4 (1985) 31-40 (in Chinese). [5] E.P. Klement, Integration of fuzzy-valued functions, Rev. Roumaine Math. Pures Appl. 30 (1985) 375-384. [6] Chengzhong Luo and Deme Wang, Extension of the integral of interval-valued functions and integrals of fuzzy-valued functions, Fuzzy Math. 3 (1983) 45-52 (in Chinese). [7] M. Matloka, On fuzzy integral, Proc. Polish Syrup. Interval and Fuzzy Mathematics 4-7 (1986) 163-170. [8] M. Purl and D. Ralescu, Fuzzy random variables, J. Math. Anal. Appl. 144 (1986) 409-422. [9] D. Ralescu and G. Adams, The fuzzy integral, J. Math. Anal. Appl. 75 (1980) 562-570. [10] M. Sugeno, Theory of fuzzy integrals and its applications, Ph.D. Dissertation, Tokyo Institute of Technology (1974). [11] Zhenyuan Wang, The autocontinuity of set-function and the fuzzy integral, J. Math. Anal. Appl. 99 (1984) 195-218. [12] L.A. Zadeh, Fuzzy sets, Inform. and Control 8 (1965) 338-353. [13] Wenxiu Zhang, Basics of Fuzzy Mathematics (Xi'an Jiaotong Univ. Press, 1984).