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Fuzzy integrals on L-fuzzy sets
Fuzzy Sets and Systems 38 (1990) 61-79 North-Holland
FUZZY
INTEGRALS
61
ON L-FUZZY
SETS
QIAO Zhong Department of Basic Sciences, Hebei Institute of Architectural Engineering, Zhangjiakou, Hebei, China Received September 1988 Revised December 1988
Abstract: The fuzzy measures and fuzzy integrals for Zadeh's fuzzy sets were studied by Qiao and Wang [5, 6, 7, 12]. In this paper we consider the real-valued fuzzy measures and fuzzy integrals for L-fuzzy sets (where L is a pseudocomplemented infinitely distributive complete lattice). On a fuzzy o-algebra of L-fuzzy sets, the concepts of fuzzy measure and fuzzy integral are introduced, some equivalent forms of fuzzy integrals on L-fuzzy sets are given. Moreover we discuss several properties of fuzzy integrals on L-fuzzy sets, and prove some convergence theorems for a sequence of fuzzy integrals on L-fuzzy sets.
Keywords: Lattice; L-fuzzy set; fuzzy measure; fuzzy integral on L-fuzzy set.
1. Introduction
The fuzzy measure and fuzzy integral, defined on a classical o-algebra, were studied by Sugeno [10], Ralescu and Adams [9] and Wang [11] by dropping the o-additivity property of classical measures and replacing it by monotonicity and continuity. Qiao and Wang [5, 6, 7, 12] generalized the concepts of the fuzzy measure and fuzzy integral to a fuzzy o-algebra of Zadeh's fuzzy sets (where a fuzzy set is a mapping from a nonempty set X to the interval [0, 1]). In this paper we shall establish a theory of fuzzy measure and fuzzy integral on a fuzzy o-algebra of L-fuzzy sets (where a L-fuzzy set is a mapping from X to the pseudocomplemented infinitely distributive complete lattice L). In Section 2 of this paper, we shall introduce the concepts of fuzzy o-algebra and fuzzy measure for L-fuzzy sets, and some structural characteristics of fuzzy measures will be given. In Section 3, fuzzy integrals on L-fuzzy sets will be defined, and some equivalent forms of such a type of fuzzy integrals will be proposed. In Section 4, we shall discuss several properties of fuzzy integrals on L-fuzzy sets. In Section 5, some convergences for a sequence of measurable functions will be defined, and the relations between these concepts will be discussed. In Section 6, we shall prove two monotone convergence theorems for a sequence of fuzzy integrals on L-fuzzy sets. In Section 7, an everywhere convergence theorem and a (pseudo-) almost everywhere convergence theorem for a sequence of fuzzy integrals on L-fuzzy sets will be studied. Finally, in Section 8, we shall give the convergence (pseudo) in fuzzy measure theorem for a sequence of fuzzy integrals. Throughout this paper, L denotes a pseudocomplemented infinitely distributive 0165-0114/90/$03.50 © 1990---Elsevier Science Publishers B.V. (North-Holland)
Qiao Zhong
62
complete lattice, namely the lattice L satisfies the following conditions: (1) For any H c L, /jheH h and VhEn h exist in L. (2) ForanyHcL, MEL,
(3) There exists a mapping N : L + L, such that N(N(a)) = a for any a E L, and if a S b, then N(b) s N(a) for any a, b E L. That is, N is a pseudocomplementation on L. Obviously, N(Z) = 8, N(0) = Z, where Z and 8 are respectively the greatest and least element of L. (By condition (l), Z and 8 exist.) In this paper, X denotes a nonempty set, 5$(X) = {A; A : X- L} is the class of all L-fuzzy sets on X, SPL(X) = {E’; Z? : X-+ {I, e}}; evidently, CPL(X)c &(X). We shall use the following definitions and notation: Let A, Z?E .9$(X), {A,; t E T} c 9=(X), where T is any index set. AcB
e
A(x)SB(x)
foranyxEX;
(A U B)(x) = A(x) v B(x)
for any x E X;
(A fl B)(x) = A(x) A B(x)
for any x E X;
A’(x) = A@(x))
for any x E X,
where N is a pseudo-complementation Observe that
on L.
(u A,>== I?,“‘,
(lP)‘=A,
IET
((FTAJC =,VT”‘.
Thus, 9”(X) has a lattice structure induced pointwise by L, namely 9=(X) is a pseudocomplemented infinitely distributive complete lattice. The greatest element of S=(X) is the L-fuzzy set X: X(x) = Z for any x E X. The least element of S=(X) is the L-fuzzy set 4%g(x) = 8 for any x E X. For a class SPt(X), we must observe the fact X, fl E SPJX),
and
Xc = I, 8’ =X.
ForanyEES)L(X), Z?flZ?=#, i?UE=X. We make the following conventions: ,‘I,‘*‘=& ini {a,; a,
l-$)=X, l
[O, 031)= w,
where 0 is the classical empty set.
o*,==o,
Fuzzy integrals on L-fuzzy sets
63
2. Fuzzy measure on fuzzy ~r-algebra of L-fuzzy sets In this section, we shall introduce the concepts of fuzzy o-algebra and fuzzy measure for L-fuzzy sets. Moreover, we shall give some structural characteristics of fuzzy measures. Definition 2.1. A nonempty subclass # of ,~L(X) is called a fuzzy o-algebra, if it satisfies the following conditions: (1) t~, X e ~ ; (2) if .4 • ~, then `4c • ,~; (3) if {/i.} m o~, then U~=~/i~ • ~. Evidently, ffL(X) and ~L(X) are fuzzy o-algebras. If ~ is a fuzzy o-algebra, then ~ = ~L(X) n ~ is a fuzzy o-algebra. Observe that the fuzzy o-algebra satisfies: (4) For any {/i~} ~ ~, n n = l / i n • ~" In this paper, .~ will always denote a fuzzy o-algebra. Definition 2.2. A mapping/~ : ~---> [0, oo] is said to be a fuzzy measure on o~, if and only if: (1) /i(0) = 0. (2) For any A, B • ~, if `4 ~/~, then/~(/i) ~
Definition 2.3. /i is called autocontinuous from above (resp. from below), if /~(/~)---~ 0 implies/i(/i U/3~)--~/~(/i) (resp./~(/i n/~)-->/fi(/i)), w h e n e v e r / i • ~, {/~} c ~. /fi is called autocontinuous, if it is both autocontinuous from above and from below. Definition 2.4. Let `4 • #, /](`4)< ~. /i is said to be pseudoautocontinuous from above with respect to /i (resp. from below with respect to /i), if for any {/~,} c ~, when /~(/~,, n/i)--, 9(`4), we have /~((/)~, n:i) u/~)~ ~(/~) (resp. /~(/~. n/~)--~/~(/~)) whenever E e A n ~. /i is called pseudo-autocontinuous with respect to `4, if it is both pseudo-autocontinuous from above with respect to `4 and from below with respect t o / i ( w h e r e / i n ~ = {.4 n / ) ; / ) • ~}). Definition 2.5. /~ is called null-subtractive, if we have /i(`4 n/~c) =/fi(`4) whene v e r / i , B • ~, /~(B) = O.
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Qiao Zhong
Definition 2.6. Let A e 4 , t~(.A) < oo. t~ is said to be pseudo-null-subtractive with respect t o i{, if for any E ~ A fq 4, we have #(/~ fl/~) =/~(/~) whenever/~ ~ 4,
O(B A A) =/~(A). It is clear that the following conclusions are true.
Proposition 2.7. If fz is autocontinuous from below, then it is null-subtractive. Proposition 2.8. If fz is pseudo-autocontinuous from below with respect to ,4, then it is pseudo-nuU-subtractive with respect to A.
3. Fuzzy integral on L-fuzzy set In this section, we shall introduce the concept of fuzzy integral on an L-fuzzy set, and some equivalent forms of such a type of fuzzy integral will be given. Definition 3.1. A mapping f :X--} ( - ~ , ~) is called a measurable function on 4, if F~ • 4 for every cr • [ - ~ , oo], where F~ is the L-fuzzy set such that
Fo,(x) =
I
i f f ( x ) I> or, for any x ~ X. i f f ( x ) < c~,
Denote h4 = {f; f is a measurable function on 4},
gft + = { f ; f e ift, f >~O} . Definition 3.2. Let ,,1 e 4, f ~ M+. The fuzzy integral of f on .4 with respect to/~ is defined by
f f d/~ = #e[0,SUp~o][c~ ^/~(A n F~)], where a ~ [0, oo], and F~ is as in Definition 3.1. Remark. When L = [0, 1], J'~f d~ is the fuzzy integral defined in [6, 7, 12]. If we take L = {0, 1}, then f ~ f dg is the fuzzy integral in [9, 11]. In the following, we shall give equivalent forms of the fuzzy integral on an L-fuzzy set.
Proposition 3.3.
f
fd#=
sup [ c ~ ^ / ~ ( A N F , ) ] =
ote[o,~)
sup [ c ~ ^ # ( A N F ~ ) ] .
oce(o,~)
Fuzzy integrals on L-fuzzy sets
65
Theorem 3.4.
fA
f d/2 = sup [o~ ^ / 2 ( 4 n F~)] = sup [o~ ^ ta(A n F~)], ae[O,~]
0~ (0, ~ )
where Fc~is the L-fuzzy set such that I
/ff(x)>o¢,
F~(x) = 0 iff(x) ~ o~,
foranyxeX.
Proof. We only prove SAf dO = sup~to,® 1 [a ^ / 2 ( 4 n F,)]. First of all, we observe that F , • ~ for every a~ • [0, ~]. In fact, for any natural number n, we have F~+l/n c F,, where {I
F~+v~(x) = . ~
iff(x)>~ o¢+ 1/n, if f (x) < a¢ + 1/n,
for any x • X,
and hence U : = l F~+l/n c FaOn the other hand, when f(x) > ~, x • X, there exists a natural number no such that f(x) >i oc + 1~no, x • X. Hence F , ~ F~+l/,o, and therefore F a ~ U:=~ F~+v~. Thus Fa = U : = l F~,+v, • J;. Furthermore, by using the monotonicity of/~, we h a v e / ~ ( 4 n F~,) I>/2(4 n Fa) for any o¢ • [0, ~], and therefore S.if d/~ I> sup~lo, o~1 [o¢ ^ / 2 ( 4 ¢q Fa)]. If we assume that
f f d/2 >
sup ,~1o,~1
[~ ^ / ~ ( 4 n
F~)I =
b,
then there exists ~ > 0, such that supo,~to ~oI [c~ ^ # ( 4 n F~)] > b + e, and thus there exists O¢o, such that O¢o^/2(A n F~_0)> b + E, namely Oro> b + e and /~(4nF~0)>b+e. It follows that / ~ ( A n F ~ ) t > / ~ ( 4 n F ~ o ) > b + e , and therefore sup [o~ ^f~(4nF~)]>~(b+ e ) ^ f ~ ( 4 n
~[o,~1
bF~+~)=b+e > b .
This is a contradiction. Theorem 3.5.
fAf d/2 = sup [ ( inf f ( x ) ) A /~(4CI/~)] E e ~ t. \t~(x)=l
/
where ~ = ~L(X) n ~. Proof. First, for any given cr e [0, ~], We have infF.tx)=~f(x) >! 0:. Since F~ • ~ , then
Qiao Zhong
and hence
f, i f d/~ ~
L \E(x)=I
¢q/~)].
/
Furthermore, observe that ~ c ~. It follows that sup[( i~f__f(x))^~(flNE)]<~sup[(
~'~
E
inf f ( x ) ) ^ / f i ( A N / ~ ) ] .
Ee,~ L \~'(x)> 0
Finally, for any given /~ • ~-, if we take o~'= therefore
I
inf~(x)>of(x),
then /~ c F~., and
It follows that sup[(
inf f ( x ) ) ^ f t ( , A f q l E , ) J < ~ f f d f t .
/~e~: L \~'(x)>0
/
The proof of this theorem is complete. Definition 3.6. s • M+ is called a nonnegative simple function on ~, if there exist E1 . . . . . En•~ (where ~ = 3 a L ( X ) f q ~ :, /~i#~, i = 1 , 2 . . . . . n, l~,i[")JEj~'~ , i # j , I,_,J~=l/~i= X) and real numbers o:1. . . . . or, • [0, oo) (where tri # o:j, i # j ) such that for any x • X, s(x)=~
ifL(x)=I,i=l,
2. . . . .
n.
Denote the set of all nonnegative simple functions on ~ by H. The following conclusion shows that the nonnegative function given in Definition 3.6 has unique representation. Proposition 3.7. I f s • H
S(X)=Offi
has two representations:
ifE, i(X)=I, i = l , 2 . . . . .
s(x) = ti~ if G,(x) = l, s = l, 2 . . . . .
n, m,
foranyx•X, for any x • X,
(where El, G~, oli, tis satisfy the condition given in Definition 3.6), then {aq . . . . . a,} = {ti~ . . . . . tim}, and if ai = ti,, then Ei = t~. Proof. For any a~i • {oq . . . . . a~,}, since /~i # ~, there exists ~ • X such that /~i(g) = L and hence s(£) = a~i. Observe that ((-JT=l Gs)(£) = X(£) = I. There exists G, such that (~,(~) = L and therefore =
= ti, • { t i , . . . . .
tim}.
That is to say, {aq . . . . . o~,} c {til . . . . , tim}" Similarly, we may prove {ill . . . . . tim} ~- {aq . . . . .
a~,}.
Fuzzy integralson L-fuzzy sets
67
Moreover, if c~i = fls, observe that cei # crj, i 4: j, fls 4= ~,, s # t, and then for any
x eX, Ei(x) = I ¢~ s(x)=
o{i ~
s(x)=
~s ~
G s ( x ) = 1.
Consequently/~; = Gs-
Theorem 3.8. Let fi~ e ~, f e )VI+. For any s e H,
s(x)=o~i
i f P ~ ( x ) = l , i = l , 2 . . . . . n, f o r a n y x e X ,
where Ei, o~isatisfy the condition given in Definition 3.6. If we define Q,i(s) = VT=, [ ~ A 0(A n/~i)1, then f f dO = s esup QA(s), ll(f) where H(f) = (s; s ~
sEH(f)
Ee~ L \E(x)=I
/
In fact, for any/~ e ~ , whenever/~ # t~ and inf~(~)=tf(x) > 0, if we take OCo= ~(x)=/ inf f(x),
So(X')={O°
if/~(x') = I, i f / ~ ( x ' ) = I,
for any x' e X,
then So e H ( f ) and f ( x ) ] ^ 0(.4 n/~). sup Qi(s) >I QA(so) = C~o^ 0(/l n ~) = (_inf \E(x)=l /
s~H(f)
It follows that
t~e~ L \/~(x)=l
sEH(I)
J
On the other hand, for any s e H(f),
s(x)=o6
if/~i(x)=I,i=l,
2,...,n,
for a n y x e X .
So there exists iv (1 ~< iv ~
Qa(s) = Q [o~, ^ ~(A n E,)I = ~,o ^ t~(A n Lo).
i=l
Since aio <<-inf~,o(x)=tf(x), we have
Q.~(s)<~( inf f ( x ) ) A . ( A f ~ E , o , ~ S u p [(inx)f=lf(X)) A0(A('~/~)]"
\t~,o(x)=l
/~
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Qiao Zhong
Consequently, sup QA(s)<-sup [(_inf f ( x ) ) A O ( f l n F . ) ] . s~H(:)
~6j# L YE(x)=I
/
The proof of this theorem is complete.
4. Properties of fuzzy integrals on L-fuzzy sets We shall discuss several properties for fuzzy integrals on L-fuzzy sets in this section.
Proposition 4.1. Let o~, fl • [0, ~), { c~n} = [0, oo). (1) F, ~ F~, where F~(x)={Io
iff(x)~o<, F~(x)={I i f f ( x ) < o~, 0
iff(x)>a, i f f ( x ) <- re,
xeX.
(2) If o~<--fl, then Fa~F~, F~=Fa. (3) If o~,,~o: and o<, < o<, then A*~=I F~ =A~=~ F~ = F~. (4) If a, Xao: and tr,> te, then U~=I F~, =U~=~ Fa = F~. Proof. (1) Obvious. (2) For any x e X, if f (x)/> fl, then f ( x ) >i ol, and hence F~(x) ~< Fo,(x). That is, F~ c Fo, Similarly, we have F~ c Fa(3) Let a~n/~o<, o<, < a~. We have F~. = F , for any n, so it follows that n~=l F~. = Fo<. Moreover, if (f-)~=l F j ( x ) = I, x ~ X, then F~.(x) = I for any n, namely f ( x ) > o~. It follows from a~, ~ a~ that f ( x ) >i o~, and hence F~(x) = I. Thus we obtain n ~ = l F~o ~ F~. The proof of n~=~ Fo,. = F~ is similar. (4) If tr,",ate, a~,>a~, then Fo, cF~, and therefore U~=IF,~.~F~. Furthermore, if F~(x) = I, x ~ X, namely f ( x ) > re, by using o<,',,are, there exists no such that f ( x ) >I O:,o. Thus F,.o(X) = L and hence (U~=l F J ( x ) = I. That is, F,~= U~=l F~. Similarly, we can prove U~=l F~. = F~.
Theorem 4.2. The fuzzy integrals on L-fuzzy sets have the following properties (wherefl, B e 4, f, fl,)~ e AI+): (1) If O(fI) = O, then f,~f dO = O. (2) If f , i f dO = O, then O(,~ n F6) = O. (3) If fl <-f2, then fAfl dO <~f~if2 dO. (4) ,4 c/~, then ~,if dO <- I a f dO. (5) For any a e [0, ~), Ia a dO = a ^ O(fi0. (6) J'A (A v f2) dO/> I~i fl dO v J'A fz dO. (7) ~A (fl A f2) dO ~< J'~ifl dO A J'A f2 dO. (8) ~Auaf dO/> ~,if dO v ~ f dO. (9) J'Anfif dO ~ ~ i f dO A ~ f d0. (10) J'~i (f + a) dO ~< J'~if dO + j'~i a dO, a e [0, oo). (11) For any a e [0, oo), if Ill -f21 <- a, then IJ',ifl dO - J'~if2 dO[ ~< a.
Fuzzy integralson L-fuzzy sets
69
Proof. (1) Obvious.
(2) Suppose that ~Af dO = 0 and 0(,/i n Fo)= c > 0. By using Proposition 4.1, we have U~=~ (A O Fl/~) = A n F~. It follows from the continuity from below of 0 that l i m ~ 0(A n F1/~)= fi(.,~ n Fo)= c, and therefore there exists no such that O(A n FI/~o)>t½c. Thus we have fAf d 0 = sup [oc^o(AnF~)l>~--^o(AnFl,.o)>~--^½c>O. 1 1 ~lO.~l no no This is a contradiction. (3) Let F~ (k = 1, 2) be the L-fuzzy set such that
F~(x)={Io If f~
iffk(x)~>°¢' f o r a n y x ~ X . irA(x) < o:,
FIcF],
and hence 0(fi n F 1) ~<0(.A n F~). Thus we have
I,~ $1 dO ~
F~(x)={Io ifa>-°~'
if a < o~,
foranyxEX.
Then
~,={~
ifa>~°:" ifa < o~.
We have
~ia dO = =
sup [c~ ^ 0(fi, n F'~,)] v sup [a~ ^ 0(A n / ~ ) ] sup
[o~ ^ o ( A ) ]
v 0 = a ^ 0(,4).
(6) Observe that f l v f2 ~>fk, k = 1, 2. It follows from (3) that .~A( f l v rE) d0/> i'd fk d0, k = 1, 2. Consequently,
SA(fl v A) dO >"S,ifl dO V f,4A dO" (7) The proof of this conclusion is similar to (6). (8) By using (4), we have
fAu f dO >~f,if dO,
f~iu f dO >~f f dO,
it follows that SAu~f dO >!fAf dO v S~f dO. (9) The proof of this conclusion is similar to (8).
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Qiao Zhong
(10) By Theorem 3.5, we have f~i ( f + a ) d 0 = s u p [ ( - i n f
~ . ~ L \E(x)>O
~< sup [ ( _ i n f
~_~. L \E(x)>O
~ < s u p [ ( inf
~:¢~ L \/~(x)>O
(f(x)+a))^O(An/7-g)]
f(x))^o(AnPO+(a^O(4nPo)] /
/
= yff dO+ f adO. (11) It follows from If1 -AI ~ a that fl <~f2 + a, and by (3), (10), we have
f AdO<~ fA (f2 + a)dO ~
Definition 4.3. f • AI + is called fuzzy integrable on A, if f~if d0 < oo. Theorem 4.4. f • lf'l+ is fuzzy integrable on 4, if and only if there exists aCo• [0, oo), such that 0(4 n F~o) < oo. Proof. If there exists fro e [0, ~), such that 0(A n Fo,o) = a < ~, then 0(A n F~) ~< 0 ( 4 n F~,o) = a for any tr > O¢o. Hence
f.ifdO= oce[o, sup~o] [~ ^ 0(4 nF~)] v ~e(~o,~) sup [oc ^ o(4 n F~)]<<-OCoV a < ~. Conyersely, if for any ~ • [0, ~), 0(A n F~) = ~, then
f f d0 =
sup ~e[0,~)
[a~ ^ 12(4 n F~)] =
sup o~e[o,~)
o¢ = ~.
Theorem 4.5. Let 4 e :~, a e [0, ~). Then: (1) SAfd0~>o¢ ¢:~ Vile[0, o0, 0(4AF~)~>a~. Therefore 0(4AF~)~>o~ SAf d0/> o¢. (2) SAf dO <~oc <:# O(A n F~) <<-oc. Hence O(A n Fo~)<~oc ~ f~if dO <~oc. (3) f,~fdO=tr<=~ Vfle[O,a), o(AnF,)>~>~o(AnF~). Particularly, if 0 ( 4 ) < ~ , then fAfdO =tr <=>0 ( 4 AF,~)~> o ~ > 0 ( 4 AFt). Proof. (1) It is sufficient to consider the case a~ e (0, ~). If 0 ( 4 n F~)/> oc for any fl < a¢, then
f . f dO I> t~[o, sup [fl ^/i(,4 n F,)] I> sup [fl A a~] = sup fl = a~. ,~) tie[o, o0 p~[o, ~)
Fuzzy integrals on L-fuzzy sets
71
On the other hand, if there exists f l < oc, such that /~(~i N F a ) < o~, then /~(A N Fr) <~/~(-4 N Ft~) whenever r I> fl, and thus we have
~df d/i =
sup r ~[0,fl)
[r
A
~ ( d fq Fr)]
V
supoo) rE[[J,
[r A
/~(d f3 fr)]
~ tr, then there exists O~o> o~, such that J'~if d/~ I> fro. It follows by (1) that /~(ANFt3)>~ O¢o whenever f l < O~o. Particularly, if we take /3 = ½(O¢o+ o0, then o~/~(A n F~)/> O~o> o~. On the other hand, suppose that /~(A fq F&) > o~. I f / 3 ~ a ; , /3n > o~, it follows from Proposition 4.1 that O~= ~ (A N Ft~n)= A ~ F~. By using the continuity from below of/~, we have lim~__,oo/~(Afq Fan) =/~(A N F~) > a~, and hence there exists O¢o> oc, such that/~(A ~ F~o) > or. Thus f ;tf d~ >>-O~o^ (l(A f3 F~o) > o~. The equivalence relation is proved. The rest is evident. (3) First of all, by using (1) and (2), the first equivalence follows immediately. Next, if/~(A) < ~ , we take a sequence {G} ~[0, o~), such that r~,~o6 r~< o~. Using Proposition 4.1, we have (-~=1 (AfqF, n ) = - 4 ~ F ~ - It follows from the continuity from above of /~ that lim~_,~/~(A fq F~o)=/~(/i ~ F~). By the above results, we obtain the second equivalence. 5. Convergence for sequence of measurable functions
In this section, some convergences for a sequence of measurable functions will be defined, and relations between these concepts will be discussed. Definition 5.1. Let {f~, f } c / ~ , A • ~. Write
f)(x) =
0 iff~(x)-~f(x)
i.e. ]imf~(x) # f ( x )
, for any x • X.
(1) If A c / 9 , then we say {f~} converges to f everywhere on A, and denote it by fn~+f on A. (2) If there exists/~ • ~ w i t h ¢(/~) = 0, such that f , - ~ f on A Cl/~c, then we say {f, } converges to f almost everywhere on A, and denote it by f~ .... , f on .A. (3) If there exists /~ e ~ with /~(fil fq/~¢) =/~(fi0, such that f,,~:-~f on .2, f7/~¢, then we say {f~ } converges to f pseudo-almost everywhere on .4, and denote it by fn p ..... , f o n d . Definition 5.2. Let {fn, f } c h4, A • :~. Write
T~(x) = { lo where a~ • [0, ~].
if If~(x) - f ( x ) l / > o~, for any x • X, if Ifn(x) - f ( x ) l < o~,
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Qiao Zhong
(1) If for any given e > 0 , #(/i n TT)---, 0 as n---, o~, then we say {f~} converges in fuzzy measure # to f on ,4, and denote it by f~ ~ f on .4. (2) If for any given e > 0, when n ---, o0 we have/~(.4 n (TT)C)---~/~(A), then we say {f,} converges pseudo-in fuzzy measure/~ to f on .,i, and denote it by f~ p'o ~f on/~, (3) {f~ } is said to F-mean converge to f on A, if lim (_ If~ - f l d/~ = O. n--.~oo J A
The following conclusions give some relations between these concepts. Proposition 5.3. f f f~ a.e., f on A, and # is null-subtractive, then f~ p.a.e, f on .4.
Proposition 5.4. f f f, r, , f on X, and ~ is autocontinuous from below, then for any A ~ ~, we have f,, ~ f on A. It is clear that the above conclusions are true. Theorem 5.5. F-mean convergence is equivalent to convergence in fuzzy measure. Proof. If f~ ~---~-f on .4, then for any given e > 0, there exists no such that /~(,4nTT/2)~no. It follows by Theorem 4.5(1) that SA ] f ~ - f l d # < e as n 1> no, namely {f~ } F-mean converges to f on A. Conversely, if {f~} does not converge in fuzzy measure/~ to f on ,zi, then there exist e > 0, 6 > 0 and a sequence {ni}, such that #(A O T~') > 6 for every ni. It follows that Sd Ifn, - f l d# >i e ^ #(.An TT,)/> e ^ 6 > 0 for every hi. That is to say, {fn } does not F-mean converge to f on A.
6. Monotone convergence theorems for a sequence of fuzzy integrals In this section, we shall prove two monotone convergence theorems for a sequence of fuzzy integrals on L-fuzzy sets. Denote
F~(x) =
I
F ~ ( x ) = { I0 where ~ e [0, oo].
iff~(x) >/a, if f,(x) < ~,
for any x ~ X,
iff.(x)>o:, foranyxeX, if f,(x) ~< a~,
Fuzzy integralson L-fuzzy sets
73
Proposition 6.1. Let {f~, f } c ~4 +, A ~ ~;. (1) I f f , ~ f o n A, then n ~ = l (A" O F ] ) = , 4 OF~. (2) Iff~,,~fon ft, then U~=x (A N F ] ) = A n F~. Proof. We only prove (2). The proof of (1) is similar to (2). For (2), it is sufficient to verify
.4(x) ^
F~,(x) = .4(x) A F&(x) for any x e X.
In fact, if/5(x) = 0, where
I /)(x)=
0
iffn(x)/~f(x), iff,(x),~f(x),
x~X,
then ,'~(x) = 0 since .4 =/5, The equality is obviously true w h e n / ) ( x ) = 0. We now assume t h a t / 5 ( x ) = L Observe that f.(x) <~f(x) when /)(x) = I. If V~=l F~,(x) = I (otherwise, V~=a F](x) must be 0), there exists n such that F~,(x)= I, namely f . ( x ) > a. Hence f(x) ~>f,(x) > a, that is, V&(x) = I. Thus we have .4(x) ^ (V~=l F](x)) <~
,4(x) ^ F~x). On the other hand, if F~,(x) = 1, namely f ( x ) > a, sincef,(x),~f(x), there exists no such that f . 0 ( x ) > a. That is, F]°(x)= 1, and therefore V~=l F~,(x)= I. Thus .4(x) ^ F~x)<-A(x) ^ (V~=lV~(x)).
Theorem 6.2 (Monotone Convergence Theorem). Let ( f , , f } = At +, .4 ~ #. If
f. ,,~f on A, then I~ f. dO ,~ I,i f dO. Proof. Let fn/~f on A . First of all, we have
f, fndO<~f, ifn+ldO<~f4fdO,
n=l,
2
In fact, observed that .zi c D, where
ff)(x)= {Io if f"(x)/~f(x)' iff,(x)pf(x),
x ¢X,
and fi~ n / ) n F ] c fi, n / ) n F ] +1 c A n / ) N F~. It follows immediately that
f
f n/°dO fn/.+ldO--'o+ldO, f
Write c = fA f d0. If c = 0, then 0 <~ j'A fn dO ~< S~if d0 = 0, n = 1, 2 . . . . . for this case, the conclusion of the theorem is obviously true.
so that
Qiao Zhong
74
Now, suppose that c > 0. It is clear that limn__,=J'z fn dO ~
Theorem 6.3 (Monotone Convergence Theorem). Let { f , , f } ~ M+, .4 e ~. If f ~ f on .4, and there exist no and a constant c <~~ a f d~ (0 <~c), such that 0(.4 n F7 °) < ~, then SAfn d0",aS~if dO. Proof. Suppose that fn ",af o n / i . We easily prove
f, f d~ <~fAL+l dO<-f,ifn dO,
n=l,
2. . . .
(see the proof of Theorem 6.2). Write c = S,if dO. When c = oo, from ~ = S,i f d/~ ~< S,i f~ dO ~<0% n = 1, 2 , . . . , we easily see that the conclusion of the theorem is evidently true. In the following, we assume that c < oo. Clearly, limn__,=J'afn dO >~c. Suppose that lim,__.=S,if, d/~ > c, there exists co > c, such that limn__,=S a f , dO > Co, and hence j'Af~ dO > Co for every n. Using Theorem 4.5(2), we have #- (A- n Fco) " > Co for every n. Furthermore, by using Proposition 6.1, it follows that n ~ = l (A- n Fco)n_- A- n nO Fc0. Observe the continuity from above of 0 and the condition 0(A n F~0) ~(.4 O F7-°) < ~. We have limn-~= 0(A n FTo) = 0(/i N Fco) ~ Co. Thus it follows from Theorem 4.5(1) that J'~if dO i> Co > c. This is a contradiction.
7. Everywhere and (pseudo) almost everywhere convergence theorems for a sequence of fuzzy integrals In this section, we shall establish an everywhere convergence theorem and a (pseudo) almost everywhere convergence theorem for a sequence of fuzzy integrals on L-fuzzy sets. Write gn = i n f f , i~n
H~x)={Io
hn = supra, i~n
if hn(x) > a, if hn(x) <~o6
for any x e X,
where a~ e [0, ~]. Some other signs used in this section coincide with those given in the preceding sections.
75
Fuzzy integralson L-fuzzy sets
Theorem 7.1 (Everywhere Convergence Theorem). Let {f,, f } c M+, A ~ ~. If f,~:>f on A, and there exist no and a constant c < ~ A f d(t (c>~O), such that # ( A n H7 °) < ~, then lim,__,~S/if. d0 exists, and is equal to f.,~f dO. Proof. By using Theorem 4.2, and observing that gn <~f~ <~h., it follows that
~Ag. d0~
h,",af on ft,
and it follows from Theorems 6.2 and 6.3 and the condition O(An H7 o) < ~ that lim.__.~f,i g. dO = lim._.~ SLih. d o = ~.if dO. Hence we know that lim.__,= J'~if. dO exists and equals ~i f dO. We easily prove the following conclusion: Theorem 7.2. Let O be null-subtractive (resp. O be pseudo-null-subtractive with
respect to fi, where A ~ ~). Then for any B ~ ,~, we have whenever O(B) = 0 (resp. O(A n B ~) = O(A) < ~).
.~.~n~°f dO = ~,~f dO
Theorem 7.3 (Almost Everywhere Convergence Theorem). Let .71 ~ ~, {f~, f }
~'1+, and O be null-subtractive. If f~ ~-:-~ f on A, and there exist no and a constant c ~ S,if dO (c >! 0), such that O(ft n H~ °) < ~, then lim~__,~S~f~ d0 exists and is equal to f ,i f dO. Proof. Suppose that f~ _z=~f on A, namely there exists /~ e # with 0(/~) = 0,
such that fn~-~f on .4 O/~c. Observing that 0 is null-subtractive and by the condition 0(A n / ~ n HT-0 ~<0(.4 n H7-°) < ~, it follows from Theorems 7.1 and 7.2 that ~im SAfn d0 =~im f,ingofn d 0 = fAng f d0 = SAf d0. Theorem 7.4 (Pseudo-Almost Everywhere Convergence Theorem). Let {fn, f } c
A'I+, fl ~ ~, O(fl)< ~, and O be pseudo-null-subtractive with respect to ft. If f, ~ f on ,4, then lim,_,~ J'~if~ d0 exists, and is equal to S,i f dO. Proof. Let f~ p. . . . .
) f on A. Then there exists/~ e o~, such that 0(A n / ~ ) = 0(A) and fn ~-~f on A O/~c. Observe that 0(,4 n/~c o HT) ~<0(A) < ~ for any c e [0, ~]. It follows from Theorems 7.1 and 7.2 and the pseudo-null-subtraction of 0 that
.-~ ~
.-~
~of.
~ o f d~
f d~.
Qiao Zhong
76
8. Convergence (pseudo) in fuzzy measure theorem for a sequence of fuzzy integrals In this section, we shall prove a convergence (pseudo) in fuzzy measure theorem for a sequence of fuzzy integrals on L-fuzzy sets. The notation used in this section is identical with those given earlier. Moreover we denote
{ I0 T~(x) =
iflf"(x)-f(x)l~°~' if If,(x) - f ( x ) l < tr,
foranyx~X,
where tr~ [0, ~].
Proposition 8.1. Let {f,, f } c M+, b, c ~ [0, oo). Then
(1) Fc"+~= F~+~U T~, (2) F~-b ~ F~ n (T~,) ~ (where b < c).
Proof. Suppose that x ~ X. When Fc+o(X) = I. F"c+zo(X)<<-F~+o(X) v TT,(x) is evident. We now assume that Fc+o(x)# L To prove FT+2o(X)~< F~+b(x)v T~(x), it is sufficient to verify that T'~(x) = I when F~+zo(X) = L In fact, if F~+zo(X) = L then f~(x) I> c + 2b. Observe that Fc÷b(X) ~- I, namely f ( x ) < c + b. We have If, - f l I> f, - f / > c + 2b - (c + b) = b, that is, T'~(x) = I. Consequently, F~"÷2oc Fc+o tO T~,. (2) For any x ~ X, if (F~ n (T~')~)(x) = I, then F~(x) = I and (T~)¢(x) = I, and hence f ( x ) >- c and Jr,(x) - f ( x ) l < b. Thus we have f , ( x ) > f ( x ) - b I> c - b. That is to say, F~_b(x) = I. Consequently, Fc"-o = F~ n (T~) ~. Theorem 8.2 (Convergence pseudo in fuzzy measure theorem). Let {f~,f} c ~I +, ,4 ~ ~, O ('4 ) < ~, and fz be pseudo-autocontinuous with respect to A. l f f, p ¢' , f on A, then lira,__.= J',ifn dO exists, and is equal to SYtf dO. Proof. Denote c = j'.if dO. Evidently, c ~ c t>/~(.4 n F~÷~). On one hand, since f , ~ f on ,4, then for any given e > 0,
~(An(TT)C)--,O(A)
as n--,~.
Observe that/~ is pseudo-autocontinuous from above with respect t o / i , we have
~((T7 n A ) u (A n Fc+9)-o a(A n re+9. Moreover, it follows from Proposition 8.1(1) that
A n F~+2~ ~ (T7 n A) u ( A n F~+D, there exists no such that /~(A f3 F~+2~) ~< 0((T7 h A ) U (A n Fc+~)) < O(fl n F¢+~) + e <~c + 2e as n t> no. Thus from Theorem 4.5(2), we obtain
Af~dft<<-c+2e
as n ~>no.
On the other hand, in order to prove that there exists ho such that
Fuzzy integrals on L-fuzzy sets
77
c - 2e ~< J',if, dO as n I> ho, it is sutficient to consider the case c > 0. For any given e e (0, c), observe that 0 is pseudo-autocontinuous from below with respect to fi~. Then 0 ( A n F~ n (TT)C) --, 0 ( A n F~).
Furthermore, by using Proposition 8.1(2), we have
AAF~_, - = a-
n F c n (TT)~.
Hence there exists ho, such that
O(AAF'~_,)>~O(AAFcA(TT)C)>O(fiLNF~)-e>.--c-e
a s n ~ > h o.
It follows from Theorem 4.5(1) that
,J~dO>~c-e>~c-2e
as n ~>ho.
Consequently, lim._,= ~i f. dO = ~i f dO. Theorem 8.3 (Convergence in Fuzzy Measure Theorem). Let {fn, f } ~ ~,I+, and
O be autocontinuous. If f~ f' ~f on X, lim,__.~ S~if~ dO = .~i f dO.
then for any A ~ ~,
we have
Proof. Suppose that fn ~---~-~fon X. Observe that 0~<0(,4ATT)<~0(TT)--*0
foranyfi~e~, e>0.
Then 0(fii n TT)--* 0 as n--, oo. Write: c = .f~if d0. (i) If c < 0% then by using Theorem 4.5(3) the following holds:
O(,'i A Fc_~) >~c >>-O(A A Fc+,), and by Proposition 8.1(1), we h a v ~ A n Fnc+ze = (t~ n Fc+E) O (t~ n Tn), for e v e r y given e > 0. It follows from the autocontinuity from above of 0 that 0((A n F~+~) U ( A n TT))--, 0(A n Fc+~). Hence there exists no, such that
O(A O V'~+z~)<<-O((A n Fc+~) U (A O TT))
as n ~>n0.
Using Theorem 4.5(2), we obtain J'~if, dO ~
78
Qiao Zhong
#(A o Fc-, n (TT)c) ~ #(A n F~_~) as n--->00, and hence there exists t]0, such that
#(/i n
F7-2~) I>/~(A n F¢_, n (TT) c)
>[,(,4nFc_~)-e>>-c-2e
as n ~>~o.
So, by using Theorem 4.5(1), we obtain c-2e<<-f,if~d# as n~>t~o. Consequently, lim~__.~J':i f~ d/~ = c. (ii) Suppose that c = ~ . For any a:e[0, ~), by Theorem 4.3, we have /](Ai n F~)= oo. Observe that :i O FTv~/i O FN+I O (TT) c is true for any N > 0, and by using the autocontinuity from below of/~ there exists no, such that
O(AnF~v)I>/~(A n FN+I n
(TT) ¢) t> N
as n I> no.
Thus it follows from Theorem 4.5(1) that .f~if, dO t> N as n ~>no. Consequently, lim~__,~I,iL d/~ = o0 = c. Corollary 8.4 (F-Mean Convergence Theorem). Let {f~, f } c K¢+, and [t be autocontinuous. If {f, } F-mean converges to f on X, then for any ,4 ~ 4, we have lim~__.®S~if~ d# = J'/i f d/~.
Remark. If we take L = [0, 1], the conclusions given in this paper are identical with those proved in [6, 7, 12]. Acknowledgement The author is very much indebted to Prof. H.-J. Zimmermann and the referees for their critical reading of the manuscript and many valuable suggestions for improvements.
References [1] G. Birkhoff, Lattice Theory, Amer. Math. Soc. Coll. Pubi., 3rd ed. (AMS, Providence, RI, 1979). [2] D. Dubois and H. Prade, Fuzzy Sets and Systems: Theory and Applications (Academic Press, New York, 1980). [3] J.A. Goguen, L-fuzzy sets, J. Math. Anal. Appl. 18 (1967) 145-174. [4] E.P. Klement, Fuzzy o-algebras and fuzzy measurable functions, Fuzzy Sets and Systems 4 (1980) 83-93. [5] Qiao Zhong, Riesz's theorem and Lebesgue's theorem on the fuzzy measure space, BUSEFAL 29 (1987) 33-41. [6] Qiao Zhong, The fuzzy integral and the convergence theorems, BUSEFAL 31 (1987) 73-83. [7] Qiao Zhong, On fuzzy measure and fuzzy integral on fuzzy set, Fuzzy Sets and Systems 37 (1990) 77-92. [8] Qiao Zhong, On the extension of possibility measures, Fuzzy Sets and Systems 32 (1989) 315-320. [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. Thesis, Tokyo Institute of Technology (1974).
Fuzzy integrals on L-fuzzy sets
79
[11] Wang Zhenyuan, Asymptotic structural characteristics of fuzzy measure and their applications, Fuzzy Sets and Systems 16 (1985) 277-290. [12] Wang Zhenyuan and Qiao Zhong, Transformation theorems for fuzzy integrals on fuzzy sets, Fuzzy Sets and Systems 34 (1989) 355-364. [13] L.A. Zadeh, Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems 1 (1978) 3-28.
FUZZY
INTEGRALS
61
ON L-FUZZY
SETS
QIAO Zhong Department of Basic Sciences, Hebei Institute of Architectural Engineering, Zhangjiakou, Hebei, China Received September 1988 Revised December 1988
Abstract: The fuzzy measures and fuzzy integrals for Zadeh's fuzzy sets were studied by Qiao and Wang [5, 6, 7, 12]. In this paper we consider the real-valued fuzzy measures and fuzzy integrals for L-fuzzy sets (where L is a pseudocomplemented infinitely distributive complete lattice). On a fuzzy o-algebra of L-fuzzy sets, the concepts of fuzzy measure and fuzzy integral are introduced, some equivalent forms of fuzzy integrals on L-fuzzy sets are given. Moreover we discuss several properties of fuzzy integrals on L-fuzzy sets, and prove some convergence theorems for a sequence of fuzzy integrals on L-fuzzy sets.
Keywords: Lattice; L-fuzzy set; fuzzy measure; fuzzy integral on L-fuzzy set.
1. Introduction
The fuzzy measure and fuzzy integral, defined on a classical o-algebra, were studied by Sugeno [10], Ralescu and Adams [9] and Wang [11] by dropping the o-additivity property of classical measures and replacing it by monotonicity and continuity. Qiao and Wang [5, 6, 7, 12] generalized the concepts of the fuzzy measure and fuzzy integral to a fuzzy o-algebra of Zadeh's fuzzy sets (where a fuzzy set is a mapping from a nonempty set X to the interval [0, 1]). In this paper we shall establish a theory of fuzzy measure and fuzzy integral on a fuzzy o-algebra of L-fuzzy sets (where a L-fuzzy set is a mapping from X to the pseudocomplemented infinitely distributive complete lattice L). In Section 2 of this paper, we shall introduce the concepts of fuzzy o-algebra and fuzzy measure for L-fuzzy sets, and some structural characteristics of fuzzy measures will be given. In Section 3, fuzzy integrals on L-fuzzy sets will be defined, and some equivalent forms of such a type of fuzzy integrals will be proposed. In Section 4, we shall discuss several properties of fuzzy integrals on L-fuzzy sets. In Section 5, some convergences for a sequence of measurable functions will be defined, and the relations between these concepts will be discussed. In Section 6, we shall prove two monotone convergence theorems for a sequence of fuzzy integrals on L-fuzzy sets. In Section 7, an everywhere convergence theorem and a (pseudo-) almost everywhere convergence theorem for a sequence of fuzzy integrals on L-fuzzy sets will be studied. Finally, in Section 8, we shall give the convergence (pseudo) in fuzzy measure theorem for a sequence of fuzzy integrals. Throughout this paper, L denotes a pseudocomplemented infinitely distributive 0165-0114/90/$03.50 © 1990---Elsevier Science Publishers B.V. (North-Holland)
Qiao Zhong
62
complete lattice, namely the lattice L satisfies the following conditions: (1) For any H c L, /jheH h and VhEn h exist in L. (2) ForanyHcL, MEL,
(3) There exists a mapping N : L + L, such that N(N(a)) = a for any a E L, and if a S b, then N(b) s N(a) for any a, b E L. That is, N is a pseudocomplementation on L. Obviously, N(Z) = 8, N(0) = Z, where Z and 8 are respectively the greatest and least element of L. (By condition (l), Z and 8 exist.) In this paper, X denotes a nonempty set, 5$(X) = {A; A : X- L} is the class of all L-fuzzy sets on X, SPL(X) = {E’; Z? : X-+ {I, e}}; evidently, CPL(X)c &(X). We shall use the following definitions and notation: Let A, Z?E .9$(X), {A,; t E T} c 9=(X), where T is any index set. AcB
e
A(x)SB(x)
foranyxEX;
(A U B)(x) = A(x) v B(x)
for any x E X;
(A fl B)(x) = A(x) A B(x)
for any x E X;
A’(x) = A@(x))
for any x E X,
where N is a pseudo-complementation Observe that
on L.
(u A,>== I?,“‘,
(lP)‘=A,
IET
((FTAJC =,VT”‘.
Thus, 9”(X) has a lattice structure induced pointwise by L, namely 9=(X) is a pseudocomplemented infinitely distributive complete lattice. The greatest element of S=(X) is the L-fuzzy set X: X(x) = Z for any x E X. The least element of S=(X) is the L-fuzzy set 4%g(x) = 8 for any x E X. For a class SPt(X), we must observe the fact X, fl E SPJX),
and
Xc = I, 8’ =X.
ForanyEES)L(X), Z?flZ?=#, i?UE=X. We make the following conventions: ,‘I,‘*‘=& ini {a,; a,
l-$)=X, l
[O, 031)= w,
where 0 is the classical empty set.
o*,==o,
Fuzzy integrals on L-fuzzy sets
63
2. Fuzzy measure on fuzzy ~r-algebra of L-fuzzy sets In this section, we shall introduce the concepts of fuzzy o-algebra and fuzzy measure for L-fuzzy sets. Moreover, we shall give some structural characteristics of fuzzy measures. Definition 2.1. A nonempty subclass # of ,~L(X) is called a fuzzy o-algebra, if it satisfies the following conditions: (1) t~, X e ~ ; (2) if .4 • ~, then `4c • ,~; (3) if {/i.} m o~, then U~=~/i~ • ~. Evidently, ffL(X) and ~L(X) are fuzzy o-algebras. If ~ is a fuzzy o-algebra, then ~ = ~L(X) n ~ is a fuzzy o-algebra. Observe that the fuzzy o-algebra satisfies: (4) For any {/i~} ~ ~, n n = l / i n • ~" In this paper, .~ will always denote a fuzzy o-algebra. Definition 2.2. A mapping/~ : ~---> [0, oo] is said to be a fuzzy measure on o~, if and only if: (1) /i(0) = 0. (2) For any A, B • ~, if `4 ~/~, then/~(/i) ~
Definition 2.3. /i is called autocontinuous from above (resp. from below), if /~(/~)---~ 0 implies/i(/i U/3~)--~/~(/i) (resp./~(/i n/~)-->/fi(/i)), w h e n e v e r / i • ~, {/~} c ~. /fi is called autocontinuous, if it is both autocontinuous from above and from below. Definition 2.4. Let `4 • #, /](`4)< ~. /i is said to be pseudoautocontinuous from above with respect to /i (resp. from below with respect to /i), if for any {/~,} c ~, when /~(/~,, n/i)--, 9(`4), we have /~((/)~, n:i) u/~)~ ~(/~) (resp. /~(/~. n/~)--~/~(/~)) whenever E e A n ~. /i is called pseudo-autocontinuous with respect to `4, if it is both pseudo-autocontinuous from above with respect to `4 and from below with respect t o / i ( w h e r e / i n ~ = {.4 n / ) ; / ) • ~}). Definition 2.5. /~ is called null-subtractive, if we have /i(`4 n/~c) =/fi(`4) whene v e r / i , B • ~, /~(B) = O.
64
Qiao Zhong
Definition 2.6. Let A e 4 , t~(.A) < oo. t~ is said to be pseudo-null-subtractive with respect t o i{, if for any E ~ A fq 4, we have #(/~ fl/~) =/~(/~) whenever/~ ~ 4,
O(B A A) =/~(A). It is clear that the following conclusions are true.
Proposition 2.7. If fz is autocontinuous from below, then it is null-subtractive. Proposition 2.8. If fz is pseudo-autocontinuous from below with respect to ,4, then it is pseudo-nuU-subtractive with respect to A.
3. Fuzzy integral on L-fuzzy set In this section, we shall introduce the concept of fuzzy integral on an L-fuzzy set, and some equivalent forms of such a type of fuzzy integral will be given. Definition 3.1. A mapping f :X--} ( - ~ , ~) is called a measurable function on 4, if F~ • 4 for every cr • [ - ~ , oo], where F~ is the L-fuzzy set such that
Fo,(x) =
I
i f f ( x ) I> or, for any x ~ X. i f f ( x ) < c~,
Denote h4 = {f; f is a measurable function on 4},
gft + = { f ; f e ift, f >~O} . Definition 3.2. Let ,,1 e 4, f ~ M+. The fuzzy integral of f on .4 with respect to/~ is defined by
f f d/~ = #e[0,SUp~o][c~ ^/~(A n F~)], where a ~ [0, oo], and F~ is as in Definition 3.1. Remark. When L = [0, 1], J'~f d~ is the fuzzy integral defined in [6, 7, 12]. If we take L = {0, 1}, then f ~ f dg is the fuzzy integral in [9, 11]. In the following, we shall give equivalent forms of the fuzzy integral on an L-fuzzy set.
Proposition 3.3.
f
fd#=
sup [ c ~ ^ / ~ ( A N F , ) ] =
ote[o,~)
sup [ c ~ ^ # ( A N F ~ ) ] .
oce(o,~)
Fuzzy integrals on L-fuzzy sets
65
Theorem 3.4.
fA
f d/2 = sup [o~ ^ / 2 ( 4 n F~)] = sup [o~ ^ ta(A n F~)], ae[O,~]
0~ (0, ~ )
where Fc~is the L-fuzzy set such that I
/ff(x)>o¢,
F~(x) = 0 iff(x) ~ o~,
foranyxeX.
Proof. We only prove SAf dO = sup~to,® 1 [a ^ / 2 ( 4 n F,)]. First of all, we observe that F , • ~ for every a~ • [0, ~]. In fact, for any natural number n, we have F~+l/n c F,, where {I
F~+v~(x) = . ~
iff(x)>~ o¢+ 1/n, if f (x) < a¢ + 1/n,
for any x • X,
and hence U : = l F~+l/n c FaOn the other hand, when f(x) > ~, x • X, there exists a natural number no such that f(x) >i oc + 1~no, x • X. Hence F , ~ F~+l/,o, and therefore F a ~ U:=~ F~+v~. Thus Fa = U : = l F~,+v, • J;. Furthermore, by using the monotonicity of/~, we h a v e / ~ ( 4 n F~,) I>/2(4 n Fa) for any o¢ • [0, ~], and therefore S.if d/~ I> sup~lo, o~1 [o¢ ^ / 2 ( 4 ¢q Fa)]. If we assume that
f f d/2 >
sup ,~1o,~1
[~ ^ / ~ ( 4 n
F~)I =
b,
then there exists ~ > 0, such that supo,~to ~oI [c~ ^ # ( 4 n F~)] > b + e, and thus there exists O¢o, such that O¢o^/2(A n F~_0)> b + E, namely Oro> b + e and /~(4nF~0)>b+e. It follows that / ~ ( A n F ~ ) t > / ~ ( 4 n F ~ o ) > b + e , and therefore sup [o~ ^f~(4nF~)]>~(b+ e ) ^ f ~ ( 4 n
~[o,~1
bF~+~)=b+e > b .
This is a contradiction. Theorem 3.5.
fAf d/2 = sup [ ( inf f ( x ) ) A /~(4CI/~)] E e ~ t. \t~(x)=l
/
where ~ = ~L(X) n ~. Proof. First, for any given cr e [0, ~], We have infF.tx)=~f(x) >! 0:. Since F~ • ~ , then
Qiao Zhong
and hence
f, i f d/~ ~
L \E(x)=I
¢q/~)].
/
Furthermore, observe that ~ c ~. It follows that sup[( i~f__f(x))^~(flNE)]<~sup[(
~'~
E
inf f ( x ) ) ^ / f i ( A N / ~ ) ] .
Ee,~ L \~'(x)> 0
Finally, for any given /~ • ~-, if we take o~'= therefore
I
inf~(x)>of(x),
then /~ c F~., and
It follows that sup[(
inf f ( x ) ) ^ f t ( , A f q l E , ) J < ~ f f d f t .
/~e~: L \~'(x)>0
/
The proof of this theorem is complete. Definition 3.6. s • M+ is called a nonnegative simple function on ~, if there exist E1 . . . . . En•~ (where ~ = 3 a L ( X ) f q ~ :, /~i#~, i = 1 , 2 . . . . . n, l~,i[")JEj~'~ , i # j , I,_,J~=l/~i= X) and real numbers o:1. . . . . or, • [0, oo) (where tri # o:j, i # j ) such that for any x • X, s(x)=~
ifL(x)=I,i=l,
2. . . . .
n.
Denote the set of all nonnegative simple functions on ~ by H. The following conclusion shows that the nonnegative function given in Definition 3.6 has unique representation. Proposition 3.7. I f s • H
S(X)=Offi
has two representations:
ifE, i(X)=I, i = l , 2 . . . . .
s(x) = ti~ if G,(x) = l, s = l, 2 . . . . .
n, m,
foranyx•X, for any x • X,
(where El, G~, oli, tis satisfy the condition given in Definition 3.6), then {aq . . . . . a,} = {ti~ . . . . . tim}, and if ai = ti,, then Ei = t~. Proof. For any a~i • {oq . . . . . a~,}, since /~i # ~, there exists ~ • X such that /~i(g) = L and hence s(£) = a~i. Observe that ((-JT=l Gs)(£) = X(£) = I. There exists G, such that (~,(~) = L and therefore =
= ti, • { t i , . . . . .
tim}.
That is to say, {aq . . . . . o~,} c {til . . . . , tim}" Similarly, we may prove {ill . . . . . tim} ~- {aq . . . . .
a~,}.
Fuzzy integralson L-fuzzy sets
67
Moreover, if c~i = fls, observe that cei # crj, i 4: j, fls 4= ~,, s # t, and then for any
x eX, Ei(x) = I ¢~ s(x)=
o{i ~
s(x)=
~s ~
G s ( x ) = 1.
Consequently/~; = Gs-
Theorem 3.8. Let fi~ e ~, f e )VI+. For any s e H,
s(x)=o~i
i f P ~ ( x ) = l , i = l , 2 . . . . . n, f o r a n y x e X ,
where Ei, o~isatisfy the condition given in Definition 3.6. If we define Q,i(s) = VT=, [ ~ A 0(A n/~i)1, then f f dO = s esup QA(s), ll(f) where H(f) = (s; s ~
sEH(f)
Ee~ L \E(x)=I
/
In fact, for any/~ e ~ , whenever/~ # t~ and inf~(~)=tf(x) > 0, if we take OCo= ~(x)=/ inf f(x),
So(X')={O°
if/~(x') = I, i f / ~ ( x ' ) = I,
for any x' e X,
then So e H ( f ) and f ( x ) ] ^ 0(.4 n/~). sup Qi(s) >I QA(so) = C~o^ 0(/l n ~) = (_inf \E(x)=l /
s~H(f)
It follows that
t~e~ L \/~(x)=l
sEH(I)
J
On the other hand, for any s e H(f),
s(x)=o6
if/~i(x)=I,i=l,
2,...,n,
for a n y x e X .
So there exists iv (1 ~< iv ~
Qa(s) = Q [o~, ^ ~(A n E,)I = ~,o ^ t~(A n Lo).
i=l
Since aio <<-inf~,o(x)=tf(x), we have
Q.~(s)<~( inf f ( x ) ) A . ( A f ~ E , o , ~ S u p [(inx)f=lf(X)) A0(A('~/~)]"
\t~,o(x)=l
/~
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Qiao Zhong
Consequently, sup QA(s)<-sup [(_inf f ( x ) ) A O ( f l n F . ) ] . s~H(:)
~6j# L YE(x)=I
/
The proof of this theorem is complete.
4. Properties of fuzzy integrals on L-fuzzy sets We shall discuss several properties for fuzzy integrals on L-fuzzy sets in this section.
Proposition 4.1. Let o~, fl • [0, ~), { c~n} = [0, oo). (1) F, ~ F~, where F~(x)={Io
iff(x)~o<, F~(x)={I i f f ( x ) < o~, 0
iff(x)>a, i f f ( x ) <- re,
xeX.
(2) If o~<--fl, then Fa~F~, F~=Fa. (3) If o~,,~o: and o<, < o<, then A*~=I F~ =A~=~ F~ = F~. (4) If a, Xao: and tr,> te, then U~=I F~, =U~=~ Fa = F~. Proof. (1) Obvious. (2) For any x e X, if f (x)/> fl, then f ( x ) >i ol, and hence F~(x) ~< Fo,(x). That is, F~ c Fo, Similarly, we have F~ c Fa(3) Let a~n/~o<, o<, < a~. We have F~. = F , for any n, so it follows that n~=l F~. = Fo<. Moreover, if (f-)~=l F j ( x ) = I, x ~ X, then F~.(x) = I for any n, namely f ( x ) > o~. It follows from a~, ~ a~ that f ( x ) >i o~, and hence F~(x) = I. Thus we obtain n ~ = l F~o ~ F~. The proof of n~=~ Fo,. = F~ is similar. (4) If tr,",ate, a~,>a~, then Fo, cF~, and therefore U~=IF,~.~F~. Furthermore, if F~(x) = I, x ~ X, namely f ( x ) > re, by using o<,',,are, there exists no such that f ( x ) >I O:,o. Thus F,.o(X) = L and hence (U~=l F J ( x ) = I. That is, F,~= U~=l F~. Similarly, we can prove U~=l F~. = F~.
Theorem 4.2. The fuzzy integrals on L-fuzzy sets have the following properties (wherefl, B e 4, f, fl,)~ e AI+): (1) If O(fI) = O, then f,~f dO = O. (2) If f , i f dO = O, then O(,~ n F6) = O. (3) If fl <-f2, then fAfl dO <~f~if2 dO. (4) ,4 c/~, then ~,if dO <- I a f dO. (5) For any a e [0, ~), Ia a dO = a ^ O(fi0. (6) J'A (A v f2) dO/> I~i fl dO v J'A fz dO. (7) ~A (fl A f2) dO ~< J'~ifl dO A J'A f2 dO. (8) ~Auaf dO/> ~,if dO v ~ f dO. (9) J'Anfif dO ~ ~ i f dO A ~ f d0. (10) J'~i (f + a) dO ~< J'~if dO + j'~i a dO, a e [0, oo). (11) For any a e [0, oo), if Ill -f21 <- a, then IJ',ifl dO - J'~if2 dO[ ~< a.
Fuzzy integralson L-fuzzy sets
69
Proof. (1) Obvious.
(2) Suppose that ~Af dO = 0 and 0(,/i n Fo)= c > 0. By using Proposition 4.1, we have U~=~ (A O Fl/~) = A n F~. It follows from the continuity from below of 0 that l i m ~ 0(A n F1/~)= fi(.,~ n Fo)= c, and therefore there exists no such that O(A n FI/~o)>t½c. Thus we have fAf d 0 = sup [oc^o(AnF~)l>~--^o(AnFl,.o)>~--^½c>O. 1 1 ~lO.~l no no This is a contradiction. (3) Let F~ (k = 1, 2) be the L-fuzzy set such that
F~(x)={Io If f~
iffk(x)~>°¢' f o r a n y x ~ X . irA(x) < o:,
FIcF],
and hence 0(fi n F 1) ~<0(.A n F~). Thus we have
I,~ $1 dO ~
F~(x)={Io ifa>-°~'
if a < o~,
foranyxEX.
Then
~,={~
ifa>~°:" ifa < o~.
We have
~ia dO = =
sup [c~ ^ 0(fi, n F'~,)] v sup [a~ ^ 0(A n / ~ ) ] sup
[o~ ^ o ( A ) ]
v 0 = a ^ 0(,4).
(6) Observe that f l v f2 ~>fk, k = 1, 2. It follows from (3) that .~A( f l v rE) d0/> i'd fk d0, k = 1, 2. Consequently,
SA(fl v A) dO >"S,ifl dO V f,4A dO" (7) The proof of this conclusion is similar to (6). (8) By using (4), we have
fAu f dO >~f,if dO,
f~iu f dO >~f f dO,
it follows that SAu~f dO >!fAf dO v S~f dO. (9) The proof of this conclusion is similar to (8).
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Qiao Zhong
(10) By Theorem 3.5, we have f~i ( f + a ) d 0 = s u p [ ( - i n f
~ . ~ L \E(x)>O
~< sup [ ( _ i n f
~_~. L \E(x)>O
~ < s u p [ ( inf
~:¢~ L \/~(x)>O
(f(x)+a))^O(An/7-g)]
f(x))^o(AnPO+(a^O(4nPo)] /
/
= yff dO+ f adO. (11) It follows from If1 -AI ~ a that fl <~f2 + a, and by (3), (10), we have
f AdO<~ fA (f2 + a)dO ~
Definition 4.3. f • AI + is called fuzzy integrable on A, if f~if d0 < oo. Theorem 4.4. f • lf'l+ is fuzzy integrable on 4, if and only if there exists aCo• [0, oo), such that 0(4 n F~o) < oo. Proof. If there exists fro e [0, ~), such that 0(A n Fo,o) = a < ~, then 0(A n F~) ~< 0 ( 4 n F~,o) = a for any tr > O¢o. Hence
f.ifdO= oce[o, sup~o] [~ ^ 0(4 nF~)] v ~e(~o,~) sup [oc ^ o(4 n F~)]<<-OCoV a < ~. Conyersely, if for any ~ • [0, ~), 0(A n F~) = ~, then
f f d0 =
sup ~e[0,~)
[a~ ^ 12(4 n F~)] =
sup o~e[o,~)
o¢ = ~.
Theorem 4.5. Let 4 e :~, a e [0, ~). Then: (1) SAfd0~>o¢ ¢:~ Vile[0, o0, 0(4AF~)~>a~. Therefore 0(4AF~)~>o~ SAf d0/> o¢. (2) SAf dO <~oc <:# O(A n F~) <<-oc. Hence O(A n Fo~)<~oc ~ f~if dO <~oc. (3) f,~fdO=tr<=~ Vfle[O,a), o(AnF,)>~>~o(AnF~). Particularly, if 0 ( 4 ) < ~ , then fAfdO =tr <=>0 ( 4 AF,~)~> o ~ > 0 ( 4 AFt). Proof. (1) It is sufficient to consider the case a~ e (0, ~). If 0 ( 4 n F~)/> oc for any fl < a¢, then
f . f dO I> t~[o, sup [fl ^/i(,4 n F,)] I> sup [fl A a~] = sup fl = a~. ,~) tie[o, o0 p~[o, ~)
Fuzzy integrals on L-fuzzy sets
71
On the other hand, if there exists f l < oc, such that /~(~i N F a ) < o~, then /~(A N Fr) <~/~(-4 N Ft~) whenever r I> fl, and thus we have
~df d/i =
sup r ~[0,fl)
[r
A
~ ( d fq Fr)]
V
supoo) rE[[J,
[r A
/~(d f3 fr)]
~
In this section, some convergences for a sequence of measurable functions will be defined, and relations between these concepts will be discussed. Definition 5.1. Let {f~, f } c / ~ , A • ~. Write
f)(x) =
0 iff~(x)-~f(x)
i.e. ]imf~(x) # f ( x )
, for any x • X.
(1) If A c / 9 , then we say {f~} converges to f everywhere on A, and denote it by fn~+f on A. (2) If there exists/~ • ~ w i t h ¢(/~) = 0, such that f , - ~ f on A Cl/~c, then we say {f, } converges to f almost everywhere on A, and denote it by f~ .... , f on .A. (3) If there exists /~ e ~ with /~(fil fq/~¢) =/~(fi0, such that f,,~:-~f on .2, f7/~¢, then we say {f~ } converges to f pseudo-almost everywhere on .4, and denote it by fn p ..... , f o n d . Definition 5.2. Let {fn, f } c h4, A • :~. Write
T~(x) = { lo where a~ • [0, ~].
if If~(x) - f ( x ) l / > o~, for any x • X, if Ifn(x) - f ( x ) l < o~,
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Qiao Zhong
(1) If for any given e > 0 , #(/i n TT)---, 0 as n---, o~, then we say {f~} converges in fuzzy measure # to f on ,4, and denote it by f~ ~ f on .4. (2) If for any given e > 0, when n ---, o0 we have/~(.4 n (TT)C)---~/~(A), then we say {f,} converges pseudo-in fuzzy measure/~ to f on .,i, and denote it by f~ p'o ~f on/~, (3) {f~ } is said to F-mean converge to f on A, if lim (_ If~ - f l d/~ = O. n--.~oo J A
The following conclusions give some relations between these concepts. Proposition 5.3. f f f~ a.e., f on A, and # is null-subtractive, then f~ p.a.e, f on .4.
Proposition 5.4. f f f, r, , f on X, and ~ is autocontinuous from below, then for any A ~ ~, we have f,, ~ f on A. It is clear that the above conclusions are true. Theorem 5.5. F-mean convergence is equivalent to convergence in fuzzy measure. Proof. If f~ ~---~-f on .4, then for any given e > 0, there exists no such that /~(,4nTT/2)
6. Monotone convergence theorems for a sequence of fuzzy integrals In this section, we shall prove two monotone convergence theorems for a sequence of fuzzy integrals on L-fuzzy sets. Denote
F~(x) =
I
F ~ ( x ) = { I0 where ~ e [0, oo].
iff~(x) >/a, if f,(x) < ~,
for any x ~ X,
iff.(x)>o:, foranyxeX, if f,(x) ~< a~,
Fuzzy integralson L-fuzzy sets
73
Proposition 6.1. Let {f~, f } c ~4 +, A ~ ~;. (1) I f f , ~ f o n A, then n ~ = l (A" O F ] ) = , 4 OF~. (2) Iff~,,~fon ft, then U~=x (A N F ] ) = A n F~. Proof. We only prove (2). The proof of (1) is similar to (2). For (2), it is sufficient to verify
.4(x) ^
F~,(x) = .4(x) A F&(x) for any x e X.
In fact, if/5(x) = 0, where
I /)(x)=
0
iffn(x)/~f(x), iff,(x),~f(x),
x~X,
then ,'~(x) = 0 since .4 =/5, The equality is obviously true w h e n / ) ( x ) = 0. We now assume t h a t / 5 ( x ) = L Observe that f.(x) <~f(x) when /)(x) = I. If V~=l F~,(x) = I (otherwise, V~=a F](x) must be 0), there exists n such that F~,(x)= I, namely f . ( x ) > a. Hence f(x) ~>f,(x) > a, that is, V&(x) = I. Thus we have .4(x) ^ (V~=l F](x)) <~
,4(x) ^ F~x). On the other hand, if F~,(x) = 1, namely f ( x ) > a, sincef,(x),~f(x), there exists no such that f . 0 ( x ) > a. That is, F]°(x)= 1, and therefore V~=l F~,(x)= I. Thus .4(x) ^ F~x)<-A(x) ^ (V~=lV~(x)).
Theorem 6.2 (Monotone Convergence Theorem). Let ( f , , f } = At +, .4 ~ #. If
f. ,,~f on A, then I~ f. dO ,~ I,i f dO. Proof. Let fn/~f on A . First of all, we have
f, fndO<~f, ifn+ldO<~f4fdO,
n=l,
2
In fact, observed that .zi c D, where
ff)(x)= {Io if f"(x)/~f(x)' iff,(x)pf(x),
x ¢X,
and fi~ n / ) n F ] c fi, n / ) n F ] +1 c A n / ) N F~. It follows immediately that
f
f n/°dO fn/.+ldO--'o+ldO, f
Write c = fA f d0. If c = 0, then 0 <~ j'A fn dO ~< S~if d0 = 0, n = 1, 2 . . . . . for this case, the conclusion of the theorem is obviously true.
so that
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74
Now, suppose that c > 0. It is clear that limn__,=J'z fn dO ~
Theorem 6.3 (Monotone Convergence Theorem). Let { f , , f } ~ M+, .4 e ~. If f ~ f on .4, and there exist no and a constant c <~~ a f d~ (0 <~c), such that 0(.4 n F7 °) < ~, then SAfn d0",aS~if dO. Proof. Suppose that fn ",af o n / i . We easily prove
f, f d~ <~fAL+l dO<-f,ifn dO,
n=l,
2. . . .
(see the proof of Theorem 6.2). Write c = S,if dO. When c = oo, from ~ = S,i f d/~ ~< S,i f~ dO ~<0% n = 1, 2 , . . . , we easily see that the conclusion of the theorem is evidently true. In the following, we assume that c < oo. Clearly, limn__,=J'afn dO >~c. Suppose that lim,__.=S,if, d/~ > c, there exists co > c, such that limn__,=S a f , dO > Co, and hence j'Af~ dO > Co for every n. Using Theorem 4.5(2), we have #- (A- n Fco) " > Co for every n. Furthermore, by using Proposition 6.1, it follows that n ~ = l (A- n Fco)n_- A- n nO Fc0. Observe the continuity from above of 0 and the condition 0(A n F~0) ~(.4 O F7-°) < ~. We have limn-~= 0(A n FTo) = 0(/i N Fco) ~ Co. Thus it follows from Theorem 4.5(1) that J'~if dO i> Co > c. This is a contradiction.
7. Everywhere and (pseudo) almost everywhere convergence theorems for a sequence of fuzzy integrals In this section, we shall establish an everywhere convergence theorem and a (pseudo) almost everywhere convergence theorem for a sequence of fuzzy integrals on L-fuzzy sets. Write gn = i n f f , i~n
H~x)={Io
hn = supra, i~n
if hn(x) > a, if hn(x) <~o6
for any x e X,
where a~ e [0, ~]. Some other signs used in this section coincide with those given in the preceding sections.
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Fuzzy integralson L-fuzzy sets
Theorem 7.1 (Everywhere Convergence Theorem). Let {f,, f } c M+, A ~ ~. If f,~:>f on A, and there exist no and a constant c < ~ A f d(t (c>~O), such that # ( A n H7 °) < ~, then lim,__,~S/if. d0 exists, and is equal to f.,~f dO. Proof. By using Theorem 4.2, and observing that gn <~f~ <~h., it follows that
~Ag. d0~
h,",af on ft,
and it follows from Theorems 6.2 and 6.3 and the condition O(An H7 o) < ~ that lim.__.~f,i g. dO = lim._.~ SLih. d o = ~.if dO. Hence we know that lim.__,= J'~if. dO exists and equals ~i f dO. We easily prove the following conclusion: Theorem 7.2. Let O be null-subtractive (resp. O be pseudo-null-subtractive with
respect to fi, where A ~ ~). Then for any B ~ ,~, we have whenever O(B) = 0 (resp. O(A n B ~) = O(A) < ~).
.~.~n~°f dO = ~,~f dO
Theorem 7.3 (Almost Everywhere Convergence Theorem). Let .71 ~ ~, {f~, f }
~'1+, and O be null-subtractive. If f~ ~-:-~ f on A, and there exist no and a constant c ~ S,if dO (c >! 0), such that O(ft n H~ °) < ~, then lim~__,~S~f~ d0 exists and is equal to f ,i f dO. Proof. Suppose that f~ _z=~f on A, namely there exists /~ e # with 0(/~) = 0,
such that fn~-~f on .4 O/~c. Observing that 0 is null-subtractive and by the condition 0(A n / ~ n HT-0 ~<0(.4 n H7-°) < ~, it follows from Theorems 7.1 and 7.2 that ~im SAfn d0 =~im f,ingofn d 0 = fAng f d0 = SAf d0. Theorem 7.4 (Pseudo-Almost Everywhere Convergence Theorem). Let {fn, f } c
A'I+, fl ~ ~, O(fl)< ~, and O be pseudo-null-subtractive with respect to ft. If f, ~ f on ,4, then lim,_,~ J'~if~ d0 exists, and is equal to S,i f dO. Proof. Let f~ p. . . . .
) f on A. Then there exists/~ e o~, such that 0(A n / ~ ) = 0(A) and fn ~-~f on A O/~c. Observe that 0(,4 n/~c o HT) ~<0(A) < ~ for any c e [0, ~]. It follows from Theorems 7.1 and 7.2 and the pseudo-null-subtraction of 0 that
.-~ ~
.-~
~of.
~ o f d~
f d~.
Qiao Zhong
76
8. Convergence (pseudo) in fuzzy measure theorem for a sequence of fuzzy integrals In this section, we shall prove a convergence (pseudo) in fuzzy measure theorem for a sequence of fuzzy integrals on L-fuzzy sets. The notation used in this section is identical with those given earlier. Moreover we denote
{ I0 T~(x) =
iflf"(x)-f(x)l~°~' if If,(x) - f ( x ) l < tr,
foranyx~X,
where tr~ [0, ~].
Proposition 8.1. Let {f,, f } c M+, b, c ~ [0, oo). Then
(1) Fc"+~= F~+~U T~, (2) F~-b ~ F~ n (T~,) ~ (where b < c).
Proof. Suppose that x ~ X. When Fc+o(X) = I. F"c+zo(X)<<-F~+o(X) v TT,(x) is evident. We now assume that Fc+o(x)# L To prove FT+2o(X)~< F~+b(x)v T~(x), it is sufficient to verify that T'~(x) = I when F~+zo(X) = L In fact, if F~+zo(X) = L then f~(x) I> c + 2b. Observe that Fc÷b(X) ~- I, namely f ( x ) < c + b. We have If, - f l I> f, - f / > c + 2b - (c + b) = b, that is, T'~(x) = I. Consequently, F~"÷2oc Fc+o tO T~,. (2) For any x ~ X, if (F~ n (T~')~)(x) = I, then F~(x) = I and (T~)¢(x) = I, and hence f ( x ) >- c and Jr,(x) - f ( x ) l < b. Thus we have f , ( x ) > f ( x ) - b I> c - b. That is to say, F~_b(x) = I. Consequently, Fc"-o = F~ n (T~) ~. Theorem 8.2 (Convergence pseudo in fuzzy measure theorem). Let {f~,f} c ~I +, ,4 ~ ~, O ('4 ) < ~, and fz be pseudo-autocontinuous with respect to A. l f f, p ¢' , f on A, then lira,__.= J',ifn dO exists, and is equal to SYtf dO. Proof. Denote c = j'.if dO. Evidently, c ~ c t>/~(.4 n F~÷~). On one hand, since f , ~ f on ,4, then for any given e > 0,
~(An(TT)C)--,O(A)
as n--,~.
Observe that/~ is pseudo-autocontinuous from above with respect t o / i , we have
~((T7 n A ) u (A n Fc+9)-o a(A n re+9. Moreover, it follows from Proposition 8.1(1) that
A n F~+2~ ~ (T7 n A) u ( A n F~+D, there exists no such that /~(A f3 F~+2~) ~< 0((T7 h A ) U (A n Fc+~)) < O(fl n F¢+~) + e <~c + 2e as n t> no. Thus from Theorem 4.5(2), we obtain
Af~dft<<-c+2e
as n ~>no.
On the other hand, in order to prove that there exists ho such that
Fuzzy integrals on L-fuzzy sets
77
c - 2e ~< J',if, dO as n I> ho, it is sutficient to consider the case c > 0. For any given e e (0, c), observe that 0 is pseudo-autocontinuous from below with respect to fi~. Then 0 ( A n F~ n (TT)C) --, 0 ( A n F~).
Furthermore, by using Proposition 8.1(2), we have
AAF~_, - = a-
n F c n (TT)~.
Hence there exists ho, such that
O(AAF'~_,)>~O(AAFcA(TT)C)>O(fiLNF~)-e>.--c-e
a s n ~ > h o.
It follows from Theorem 4.5(1) that
,J~dO>~c-e>~c-2e
as n ~>ho.
Consequently, lim._,= ~i f. dO = ~i f dO. Theorem 8.3 (Convergence in Fuzzy Measure Theorem). Let {fn, f } ~ ~,I+, and
O be autocontinuous. If f~ f' ~f on X, lim,__.~ S~if~ dO = .~i f dO.
then for any A ~ ~,
we have
Proof. Suppose that fn ~---~-~fon X. Observe that 0~<0(,4ATT)<~0(TT)--*0
foranyfi~e~, e>0.
Then 0(fii n TT)--* 0 as n--, oo. Write: c = .f~if d0. (i) If c < 0% then by using Theorem 4.5(3) the following holds:
O(,'i A Fc_~) >~c >>-O(A A Fc+,), and by Proposition 8.1(1), we h a v ~ A n Fnc+ze = (t~ n Fc+E) O (t~ n Tn), for e v e r y given e > 0. It follows from the autocontinuity from above of 0 that 0((A n F~+~) U ( A n TT))--, 0(A n Fc+~). Hence there exists no, such that
O(A O V'~+z~)<<-O((A n Fc+~) U (A O TT))
as n ~>n0.
Using Theorem 4.5(2), we obtain J'~if, dO ~
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Qiao Zhong
#(A o Fc-, n (TT)c) ~ #(A n F~_~) as n--->00, and hence there exists t]0, such that
#(/i n
F7-2~) I>/~(A n F¢_, n (TT) c)
>[,(,4nFc_~)-e>>-c-2e
as n ~>~o.
So, by using Theorem 4.5(1), we obtain c-2e<<-f,if~d# as n~>t~o. Consequently, lim~__.~J':i f~ d/~ = c. (ii) Suppose that c = ~ . For any a:e[0, ~), by Theorem 4.3, we have /](Ai n F~)= oo. Observe that :i O FTv~/i O FN+I O (TT) c is true for any N > 0, and by using the autocontinuity from below of/~ there exists no, such that
O(AnF~v)I>/~(A n FN+I n
(TT) ¢) t> N
as n I> no.
Thus it follows from Theorem 4.5(1) that .f~if, dO t> N as n ~>no. Consequently, lim~__,~I,iL d/~ = o0 = c. Corollary 8.4 (F-Mean Convergence Theorem). Let {f~, f } c K¢+, and [t be autocontinuous. If {f, } F-mean converges to f on X, then for any ,4 ~ 4, we have lim~__.®S~if~ d# = J'/i f d/~.
Remark. If we take L = [0, 1], the conclusions given in this paper are identical with those proved in [6, 7, 12]. Acknowledgement The author is very much indebted to Prof. H.-J. Zimmermann and the referees for their critical reading of the manuscript and many valuable suggestions for improvements.
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