Transformation theorems of abstract integrals on fuzzy sets

Fuzzy Sets and Systems 41 (1991) 175-185 North-Holland

175

Transformation theorems of abstract integrals on fuzzy sets Qiao Zhong Department of Basic Sciences, Hebei Institute of Architectural Engineering, Zhangjiakou, Hebei, China

Received October 1987 Revised May 1988 Abstract: The concept of abstract integrals on fuzzy sets is introduced, and some elementary

properties for such type of integrals are given. This paper also proves some transformation theorems for abstract integrals on fuzzy sets and presents some convergence theorems of a sequence of abstract integrals. Keywords: Fuzzy set; fuzzy measure; abstract integral; homeomorphic mapping.

1. Abstract integral on fuzzy set In this section we first introduce the concept of abstract integrals on fuzzy sets, next we discuss some properties of such type of integrals. We shall see that abstract integrals on fuzzy sets are both the generalization onto fuzzy sets of classical integrals and different f r o m Sugeno's fuzzy integrals studied in [7, 11]. We assume that X is a n o n e m p t y set, :~(X) = {A; A :X---~ [0, 1]} is the class of all fuzzy subsets of X. W e also m a k e the convention 0 • ~ = 0. Definition 1.1. A n o n e m p t y subclass ~ of ~ ( X ) is called a fuzzy o-algebra, if the following conditions are satisfied: (1) O, X E ~ ; (2) A e ~ implies ,~c ¢ # ; (3) If {,~,} c ~ , then U~=~ fi-~ ~ ~. It is clear that any classical o-algebra must be a fuzzy o-algebra.

Definition 1.2. A mapping fi : o~---~[0, ~] is called a fuzzy measure, if and only if (1) /~(0) = 0; (2) A c / ~ implies/~(A) ~< fi(/~) for any A, B e ~ ; (3) whenever {/i.} c ~:, A . c / i . + l , n = 1, 2 . . . . , then

0165-0114/91/$03.50 © 1991--Elsevier Science Publishers B.V. (North-Holland)

Qiao Zhong

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(4) whenever {fi,~} c ~:, .4~ ~fii~+l, n = 1, 2 . . . . . that/~(z~no) < ~, then # ( n ~ = l ~) = lim~_.~/~(fi~,).

and there exists no such

If a mapping only satisfies the above conditions (1)-(3), then it is called a semi-continuous fuzzy measure. Definition 1.3. A fuzzy measure/~ on ~: is said to be o-additive, if we have

n=l

whenever {/1~) c ~;, A i nfltj = O, i :/:j. Throughout this paper, ~: denotes a fuzzy o-algebra and/~ is a o-additive fuzzy measure on 4. We write ~: = ~ n ~ ( X ) , where ~ ( X ) = {E; E c X is a classical set}. It is easy to see that ~: is a classical o-algebra included by o~. A function f:X---> (-0% oo) is called measurable on 4 , if {x; f ( x ) I> re} • ~ for every a~ • (-0% o0). Clearly, a function f is measurable on o~, if and only if it is measurable on ~. Let us write M = {f; f is a measurable function on ~ } ,

M+={f;f

•M, 0~f}.

s • M is called a simple function on ~, if there exist sets El . . . . . En e (where Ei n Ej = O, i :/:j, Un=l Ei = X ) and real numbers trl . . . . , trn • ( - ~ , ~) such that n

s(x) = ~ o:i" El(X)

for any x • X,

i=1

where

if x • Ei, 1 Ei(x) = 0 if x ~ Ei,

i=1,2 .....

n.

Denote the set of all nonnegative simple functions on ~ by S ÷. Now we introduce the concept of abstract integrals on fuzzy sets. Definition 1.4. Let s = ~in=l O[i " E i • S +, 1~ • 4 . The abstract integral of s on .4 with respect to/fi is defined as follows:

f S d~ ~ ~ [0[i " ~(t~ r) Ei) ], JA i=1 To show that this definition is suitable, we give the following proposition.

Proposition

1.5. The value of f~i s dFt in Definition 1.4 /s independent of the expression of s, that is, if s • S + has two expressions:

i=1

j=l

Abstract integrals on fuzzy sets

177

then

[o<,O(A n E31 = ~ [,6jO(An F3I. i=1

j=l

Proof. For sets E~ and F~, the following relations are true: and E i n E~ = O, i =/=r, F j n Ft = O, j =/=t. This implies that E~ = @ (E~ n Fj)

and

U~'=~Ei = X

= U~%~Fi

Fj = 0 (E~ n Fj).

/'=1

i=1

Moreover, observe that if Ei n Fj ¢ 0, then o<~= flj, and if Ei n ~ = 0 , o(m n E i n e,) = o. So [0gi0(/Z~ N

Ei) ] : ~ '~

i=1

then

[O(iO(/d~('IE i ('1Fj)]

i=lj:l

/=1 i=1

/=1

1 . 6 . Suppose that sl, $2, s E S +, 1~, [~ E ~ . (1) 0(A) = 0 implies SA s dO = O. (2) sl <~s2 implies SASl dO <~S~iszdO. (3) If f t c B , then S A s d O ~ I ~ s d f i . (4) S~i (s~ + s2) dO = l~i sl dO + SA s2 dO. (5) IA as d o = a . I,~ s dO, where a e [0, ~). (6) Sa dO = 0(A)(7) S,inoS dO = f A s " O dO for any D • W', where (s . O)(x) = s ( x ) . D(x). (8) qv(A)= S/is dO is a o-additive function on ~, and hence qg* given by q0*(E) = q~(.4 n E) for any E ~ ~, is a classical measure on ~.

Theorem

Proof. We only prove (2), (4), (7), (8). Let us denote s--

s,: i-1

j-1

k=l

(2) We know that ~ = U~=I (Fjn Ilk), H~ = UT'=~ (~ n Ilk), and if Sl ~
and hence we have i:1 k=l

=

j=l

[,,0(A n F,)I + E [~kO(A n//k)l = k=l

s1 dO +

(7) By s • D = (Ei%~ o
sz dO.

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178

(8) Let us take (/)m) C S%, /5i n / ) / = ~, i =/=j. Thus

cP(O=~[)~)=i~=I [°60(O=I(D,,,AEi))] =

E m=l

[~(b~ e/)l E ~O(~m). n

=

i=1

m=l

The other conclusion is obvious. We have completed the proof. Definition 1.7. (1) I f f • M + and A • ~, then the abstract integral of f on .A with respect to 0 is defined by

~af dO ~-sup{ fa s dO; s • S+, s <~f }. (2) If f • M and ,4 • ~, and if J ' a f + d/2 < oo or f ~ i f - d/~ < ~, then we say the abstract integral o f f on ,4 with respect to/~ exists, and it is defined as follows:

f fdo&ff +dO-far-dO, where f + = m a x ( f , 0), f - = m a x ( - f , integrable on A.

0). If ISAfdOI<~,

then we say f is

Remark. Obviously, SAf dO is the classical integral when ~ is a classical o-algebra (cf. [9]). We obtain immediately the following proposition by Theorem 1.6.

Proposition 1.8. Let f, g • M +, A, B • ~. (1) If_f <-g, then SAf dO <~IAg dO. (2) A c B implies ~Af dO <~~hf dO. Now we can give a monotone convergence theorem for a sequence of abstract integrals. Theorem 1.9 (Monotone Convergence Theorem). Let f • M +, and suppose that {fn} c M + is an increasing sequence. If fn,~f on X, then

lim faL dO = f,if dO

for eaeh A • ~.

Proof. From Proposition 1.8 we know that {f,if. dO} is an increasing sequence and fA fn dO ~< f,~ f dO for every n = 1, 2 . . . . . We may assume limn__.= fAf, dO = a, and hence a ~< f,,i f d0. To establish the reverse inequality, take b • (0, 1), and let s • S +, s ~
E.={x;f,(x)>~b's(x)}

(n=l,

2 . . . . ).

Abstract integrals on f u z z y sets

179

Then {En} c ~ and E n c E,+I for all n, and [,_J~=l En = X . In fact, for every x • X, we have x • E1 when f(x) = 0, and if f(x) > 0, then b. s(x) < f ( x ) since b < 1. So there exists f,o(X) such that b • s(x) <~f,o(X), that is, x • E, 0. It follows from Proposition 1.8 that

NE

n

Next, let us take q0*(E) = SAnE s dO for any E • ~. According to Theorem 1.6(8), we know that qo* is a classical measure on o~, and hence lira ~p*(En) = q0*

E,

= ~p*(X).

Applying this result and letting n - - ~ in the inequality ( * ) , we obtain b . ~ A s d O ~ a . If we assume b--*l, then [.,isd¢~a for any s ~ f , and hence f~i f dO ~
where lira infn_-,~an = supn (infi~>nai). The following theorem presents some elementary properties for abstract integrals.

Theorem 1.U. Let f, g • M, fi, B • ~, and suppose that integrals o f f and g on fl

and B exist. (1) f <-g implies SAf dO <- S~ig dO. (2) fA af dO = a. f.~ f dO, where a • (-0% oo). (3) If g is integrable on .71, then ~A (f + g) dO exists, and f (f + g) dO = f f dO + f g dO. (4) If ~Ao~f dO exists, then

f~u f dO = f: f dO + f f dO when A n B = O. (5) If~f dot ~< S~i Ifl dO.

Proof. (1) By Proposition 1.8(1). (2) Suppose first that a/> 0 and f 1> 0. There exists {sn } c S ÷, such that s , / ~ f on X. By using Theorem 1.9 and 1.6(5), we have

f~ af dO =lim,,~ .JA ~ as,, dO = a • limn_~~sn dO=a • f,i fdO.

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Qiao Zhong

If f • M, since (af) ÷ = a .f+, ( a f ) - = a . f - , we also obtain the desired result. For the case a < 0 , it is easy to see .f.4afdO =aS.4fdO from (af) ÷ = - a f - and

(af)- = - a f +. (3) Suppose that f, g • M +. There exist {s~} c S + and {h~} c S + such that sn/~f on X, h~/~g on X. Observe s~ + h~/~f +g on X; it follows from Theorem 1.9 that

f.4(f+g)dO=lim(-(s'+h")dO=limf.4s"dO+limf.4 JA

n--*~

n-...-~c~

In general, since ( f + g)+ ~ f + + g+ and (f + g)- ~ f - + g-, according to the result just obtained and Proposition 1.8(1), we know that .f.4 ( f + g)+ d0 < ~ or J'.4 ( f + g ) - dO < ~ when S.4 g± dO < ~, that is, 1.4 ( f + g) dO exists. Furthermore, it follows, from ( f + g ) + + f - + g - =f+ +g+ + (f +g)-, that

f.4(/+g)+dO+f.4/-dO+f.4g-dO=[.4f+dO+f.4g+dO+f.4(Z+g)-dO, and hence S.4 ( f + g) dO = J'.4f dO + J'.4 g dO. (4) Similar to the proof of (3), it is easy to verify this conclusion. (5) Apply (2) and (1) to : I f ~< Ifl. We have completed the proof.

Let {fn, f } c M; if for any given e > 0, there is a natural number N such that Ifn(x) - f ( x ) J < e for all x • X , n > N , then we denote it byf~ -~ f o n X. Theorem 1.12. Let {fn, f } c M, file 4, and O(fi~)< oo. If fn z_~ f on X, then lim,__.~ J'.4f, dO = J'.4f d0. (We assume 1.4f, dO and 1.4f dO exist.) Proof. For any given e > 0, there exists N such that

If,-fl
onX, asn>N.

By Theorem 1.11, we have

f.4f~ d# - f.J d# = f (f~ - f) d# <<"fA If,, --fl dO <~e . O(A)

as n > N,

and we have completed the proof.

2. Transformation theorems of abstract integrals In this section we shall prove some transformation theorems of abstract integrals on fuzzy sets. These results reveal relations between abstract integrals on fuzzy sets and on classical sets.

Abstract integrals on fuzzy sets

181

In the first place, we shall give some formulas which transform abstract integrals in (X, 4, /2) into the classical integrals in (X, ~,/a*). Theorem 2.1. Fix A • ~. If we define/a* as follows:

/a*(E) a__/2(AO E) for any E • ~, then #* is a classical measure on (X, ~:). It follows easily from the cr-additivity of/2 that this conclusion is true. Theorem 2.2 (Transformation Theorem I). Let f • M and fix A • ,~. Then

S,~nef d/2 = f e f d/a * for any E • ~, where .fef d/a* is the classical integral given in [9]. Proof. For s = ~i~1 o:iEi • S +, we have

f,4f3Esd/2=

~ [°~i/2(ftngngi)]= i~= 1 [cci/a*(gnEi)]=

When f • M ÷, we see that

f,inef d/2 = sup {f, inEs d/2; s • S+, s <-f } =sup[fesd/a*; s • S + ,

fef d/a*.

So Sanef d/2 = [.el d/a* when f • M, too. Corollary 2.3. Fix fl • ,~, and let f • M. If supp fi~ • ~, then

f ~ f d/2 = fsuppAf d/a*' where supp fi~ = {x; A(x) > 0} and fsuppAf dta* is the classical integral in [9]. Next, we study some other transformation theorems. Let ~ be the class of all Borel sets on R = ( - ~ , ~). Definition 2.4. f • M is called a homeomorphic function from (X, ~:) onto (R, ~), if f is a bijection and f ( E ) • ~ whenever E • ,~. Theorem 2.5. FIX f • M and .71 • J;. If we define

/a(B) A_/2(/i O f - l ( B ) )

for any B c ~,

then/a is a classical measure on (R, ~).

Qiao Zhong

182

Proof. First, /~(~) =/~(/l nf-l(J~)) =/~(~) = 0. Moreover, taking {Bn} c ~ with B i A B j = ~ , i~=j, then

= Z /~(A n f - l ( B , , ) ) = n=l

Z/~(Bn). n=l

That is,/~ is a measure on (R, ~). Let/z* be the measure defined in Theorem 2.1. We see that /~(B) = ~ * ( f - l ( B ) )

for any B • ~.

From the result proved in [12, p. 193], we may give the following conclusion. Theorem 2.6. Let A • ~, and suppose that f • M is a homeomorphic function from (X, ~, Iz*) onto (R, ~, I~). f i g is a measurable function on (R, ~), then

fr ,~B)g°f d#* = fBg d # for any B • ~, where the integrals in the above equality are classical and g of is the composition of f and g. Using Theorem 2.2 and 2.6, we prove easily the following theorem. Theorem 2.7 (Transformation Theorem II). Let ,4 • ~;, f • M be a homeomorphic function from (X, ~, It*) onto (R, ~, ~), and suppose that g is a measurable function on (R, ~). Then

S,~

n/-~(B)

g o f d# = SBg dlt for any B e ~,

where fB g du is a classical integral in [9, 12]. If we take B = R in Theorem 2.7, then the following corollary is obtained. Coronary 2.8. Let A • ~, f • M be a homeomorphic function from (X, .~, I~*)

onto (R, ~, t~), and let g be a measurable function on (R, ~). Then fAg of dft = fR gdl z. Finally, we shall establish a transformation theorem which transforms abstract integrals in (X, ~:,/~) into abstract integrals in another (X1, ~ ) . We assume that X1 is a nonempty set, ~ is a fuzzy o-algebra on X1, and M1 = {f; f is a measurable function on ~ } .

Abstract integrals on f u z z y sets

Definition 2.9. A mapping f :X--> X1 (X, ~') onto (X1, ,~1), if it satisfies the (1) f is a bijAection; (2) f - l ( ~ 2 = {x;x • X , B(f(x)) > 0 } (3) f(,4) = {f(x); x • X , A(x) > 0 } •

183

is called a homeomorphic mapping from conditions: • ,~for any/~ • ,°~1; ~1 for any A • ~.

Proposition 2.10. Suppose that f is a homeomorphic mapping from (X, ~) onto

(Xl, #a). Then: (1) For any {Bn} c ~l, we have f-l([..f~=l /~n) = [..)~=af-l(/~n). (2) For any B1, IBz• ~ , we havef-l(B1 n Bz) = f - l ( / ~ l ) f')f-l(/~2). Proof. (1) x e f - l ( U ~ = l / ~ . ) iffx • X, V~=l B.(f(x)) > 0 iffx • X and there exists /~.o such that B.o(f(x))>O iff there exists /~.0 such that x •f-~(B.o) iff x • UT=,f-'(B.). (2) x • I - ' ( B , n/~z) iff x • X and nl(f(x)) A Bz(f(x)) > 0 iff x • X, nl(f(x)) ;> 0 and Bz(f(x)) > 0 iff x • f - l ( / ~ l ) nf-l(/~2). Theorem 2.11. Let f be a homeomorphic mapping from (X, ~) onto (X~, ~),

and suappose that 0 is a aadditive fuzzy measure on (X, #-). If we define 0a(/~) = 0(f-a(/~)) for any B • ~ , then 01 is a o-additive semi-continuous fuzzy measure on (Xi, ~1). Proof. (1) 0~(0) = 0 ( f - ~ ( ~ ) ) = 0(0) = 0. (2) For any B1, /~z • ,~, if/~1 c/~2, then f-a(/~a) cf-x(/~z), hence 0,(B,) = 0(I-'(B,))

=

(3) Taking {/~,} ~ ,~, /3, c/~n+l for all n, it follows from Proposition 2.10 that

= lim O(f-l(/~n))= lim Ox(B.). n---~

n--~oo

(4) Whenever {/~.} = ~ , /~i n/~j = O, i :/:j, then f-i(/~i) n f - l ( / ~ j ) = O, by Proposition 2.10, this implies that

n=l

n=l

That is, ~ is a a-additive semi-continuous fuzzy measure on (X~, ~ ) . Theorem 2.12 (Transformation Theorem III). Let f be a homeomorphic mapping from (X, ~, 0) onto (X, ~1, 01), and suppose that g is a measurable function on (Xl,

Then

ff_,(~)g of d# = f g d01 for any B • ~ ,

184

Qiao Zhong

where f o g d(t~ is the abstract integral of g by way of Definition 1.7. (We define (g of) + = g + of, (g of)- = g - of. ) Proof. (1) Let us consider g ~ O, and write

H(g of) = {s; s e S +, s ~ g of }, H~(g) = {g; g e M~ is a simple function such that 0 ~
= sup{f0 g dO,; g e Hi(g)}.

In fact, observe that f is a homeomorphic mapping from (X, ~-) onto (X~, ~ ) . We have g of e M +, and if g = 2'~=l fljFi e Hi(g), we may take s = E~=I fljf-l(Fj). Then s ~ H(g of), and by using Proposition 2.10, we have

aB

/=1

j=l

j=t

-~(O)

~
-~(O)

and hence, sup

g d/~l; g e H~(g) <~sup

-,(0)

That is,

fog d01 ~
-I(0)

s d 0 = ~ [a~d~(f-'(/~) N E,)] = i=l

i~l

[ad~(f-l(/~ nf(E,)))]

~- i=l ~ [O{i~l(n(qf(Ei))]-~" ~O~ d ~ l and therefore, fi_~(O) g of dfi ~
~ sup{fo s d01; s E Hi(g)} '

Abstract integrals on fuzzy sets

185

(2) If g e M1, then

fi_~(B)gofdf~=f1_,(~)(gof)+dfa-£ ,~) (g of)- d/~ = ff_,(~)g+ofd(L- ff_,(~)g-of dfz = f g+ d~l- ~ g- d~, = f g- d~l. The proof of the theorem is complete.

Acknowledgement The author is very much indebted to Prof. H.-J. Zimmermann and referees for their valuable suggestions for improvements.

References [1] D. Dubois and H. Prade, A class of fuzzy measures based on triangular norms, Internat. J. General Systems 8 (1982) 43-61. [2] E.P. Klement, Fuzzy o-algebras and fuzzy measurable functions, Fuzzy Sets and Systems 4 (1980) 83-93. [3] M. Purl and D. Ralescu, Integration on fuzzy sets, Adv. Appl. Math. 3 (1982) 430-434. [4] Qiao Zhong, Abstract integrals on fuzzy sets, BUSEFAL 32 (1987) 67-74. [5] Qiao Zhong, Product fuzzy measure space and Fubini's theorems of abstract integrals on fuzzy sets, Proc. NAFIPS'88, San Francisco, CA (1988) 187-191. [6] Qiao Zhong, On the extension of possibility measures, Fuzzy Sets and Systems 32 (1989) 315-320. [7] Qiao Zhong, On fuzzy measure and fuzzy integral on fuzzy set, Fuzzy Sets and Systems 37 (1990) 77-92. [8] Qiao Zhong, Fuzzy integrals on L-fuzzy sets, Fuzzy Sets and Systems 38 (1990) 61-79. [9] W. Rudin, Real and Complex Analysis (McGraw-Hill, New York, 1974). [10] M. Sugeno and T. Murofushi, Pseudo-additive measures and integrals, J. Math. Anal. Appl. 122 (1987) 197-222. [11] Wang Zhenyuan and Qiao Zhong, Transformation theorems for fuzzy integrals on fuzzy sets, Fuzzy Sets and Systems 34 (1990) 355-364. [12] Yan Shijian et al., Basis of Probability Theory (in Chinese) (Science Press, Beijing, 1985). [13] L. A. Zadeh, Fuzzy sets, Inform. and Control g (1965) 338-353.