On fuzzy number-valued fuzzy measures defined by fuzzy number-valued fuzzy integrals II

Fuzzy Sets and Systems 48 (1992) 257-265 North-Holland

257

On fuzzy number-valued fuzzy measures defined by fuzzy number-valued fuzzy integrals II* ZhangGuang-Quan Department of Mathematics, Hebei University, Baoding, Hebei, 071002, P.R. China Received April 1990 Revised December 1990

Abstract: We will discuss some properties inherited by fuzzy number-valued fuzzy set functions and fuzzy number-valued fuzzy measures defined by fuzzy number-valued fuzzy integrals on the fuzzy set, for example null-additivity, autocontinuity and uniform autocontinuity, etc.

Keywords: Fuzzy sets; fuzzy number; fuzzy measure; fuzzy integral.

1. Introduction Zhang [1, 2] introduced the fuzzy distance and limit of fuzzy numbers, and proved some elementary properties of the fuzzy limit of fuzzy numbers. Zhang [3] introduced also fuzzy number-valued fuzzy measure and fuzzy number-valued fuzzy integral on the fuzzy set, and discussed some properties of them and obtained a series of results similar to results in [4,5]. Suzuki [6] discussed some properties inherited by set functions defined by fuzzy integrals on the classical set. Zhang [7] extended Suzuki's results, and gave a sufficient condition for fuzzy number-valued fuzzy set functions defined by fuzzy number-valued fuzzy integrals on the fuzzy set to be a fuzzy number-valued fuzzy measure. This paper will discuss some properties inherited by fuzzy number-valued fuzzy measures defined by fuzzy number-valued fuzzy integrals on the fuzzy set, In Section 2 of this paper, we will recall the

fuzzy number-valued fuzzy measure and fuzzy number-valued fuzzy integral on the fuzzy set introduced by Zhang. In Section 3, we will discuss the properties inherited by fuzzy number-valued fuzzy measures defined by fuzzy number-valued fuzzy integrals on the fuzzy set. Throughout this paper, let X be a nonempty set, F*(X) be the fuzzy o-algebra of fuzzy subsets of X, R be the set of all real numbers, F* be the set of all fuzzy numbers. This paper is continuation and development of [3] and [7]. All concepts and signs not defined in this paper may be found in [1,2, 3, 7].

2. Elementary concepts and theorems We have introduced the following concepts and given the following results on fuzzy numbers and fuzzy number-valued fuzzy measures and fuzzy number-valued fuzzy integrals on the fuzzy set. By the decomposition theorem of fuzzy sets,

a= U

Z~lO. I I

Z[a~-,a~-],

for every ti • F*.

Definition 2.1. For every d, b • F*, we say that = d +/~, if for every ~. • (0, 1], c~=a~-+b~

and

c~=a~-+b~.

We say that ? = t i - / ~ , if for every ~.• (0, 1], c~- = a~- - b~- and c~- = a~- - b~-. We say that ? = tiv/~, if for every ). • (0, 1],

c~=a~vb~-

and

c~-=a~vb~-.

We say that g = ~ ^ b, if for every A • (0, 1], * This research is supported by the National Natural Fund of China.

c~=a~Ab~-

0165-0114/92/$05.00 (~) 1992--Elsevier Science Publishers B.V. All rights reserved

and

c~-=a~-Ab~-.

Zhang Guang-Quan / Fuzzy number-valued fuzzy measures H

258

Definition 2.2. For every & 6 • F*, we say that a ~<6, if for every ~. • (0, 1], a; <~b-; a n d a-~ <~b~;.

We say that a < 6 , ko • (0, 1], such that +<

a~o < b ~ o or

axo

if ~i~<6 and there exists

is said to converge to a in fuzzy distance /5, denoted by (/5) l i m a . = a

or

a. --->a (n---> ~),

n--+oo

if for arbitrary e > 0, there exists an integer N > 0, such that

+

bxo.

We say that d = 6, if a~-<6 and

Definition 2.5. Let {a,} c F*, ti e F*. The {a,}

/5(a,, a) < e as n > N .

6~
Theorem 2.2. Let {G} c F*, a ~ F*. Then {a,} converges to d in fuzzy distance/5 if and only if {a~}, {a~+~} converge respectively to a-;, a~ uniformly for every A e(0, 1] in the usual distance of real numbers.

Let F* = {& d > 0 , a • F*}. Definition 2.3. For any positive real number M, + if there exists ~.o•(0, 1] such that M < a~,, or af~,<-M, then the fuzzy number a is called fuzzy infinity, written ~.

Theorem 2.3. Let {&,}, {/~}, {6,} = F*, a e F*. If for every n, a, <~6, <~G, and (/5) lira a. = a,

(/5) lim 6, = &

n----}oo

Definition 2.4. /5(a, 6) defined by the following equality (*) is called a fuzzy distance of both fuzzy numbers a and 6,

n--..~ oe

then (/5) lira 6. = a. n--.~o¢

/5(d, 6 ) =

(_J ~ , [ l a l - b ? l ,

k~lO, H

Theorem

sup la~ - b~l v la~" - b~-I],

(,) for every a, 6 e F*. It is easy to see that, if a, 6 are real numbers, then /5(a, 6) = la - 61. Theorem 2.1. Whenever a, 6 ~ F*, (1) /5(ti, 6)/> 0, and a = 6 if and only g

t~(a, 6) = 0; (2) p(a, 6) =/5(6, d); (3) whenever g • F*, we have p(a, 6) <~ p(a, e) + p(e, 6); (4) /5(a + 6, a + e) =/5(6, g); (5) #(a - 6, a - e) = p(& e);

(6) / 5 ( b - a , e - a ) = / 5 ( 6 , e); (7) if a <~6 <<-e, we have

[~(a, 6) <~p(a, e),

and /5(6, e) <~p(a, e);

(8) if d <~~ <~6 and d <~d <~b, we have p(e, d) <~p(a, 6).

2.4. Let

{a,} c F*,

a, 6 ~ F * ,

and

(/5) lim,_,= & = a. If for every n, &,<---b,, (resp. b<--d,), then a<<.6 (resp6<~d).

2.6. A fuzzy number-valued fuzzy measure ((z)fuzzy measure) on F*(X) is a fuzzy number-valued fuzzy set function #:F*(X)--> F*, with the properties: (ZFM1) #(0) = 0; (ZFM2) if ,4 c / t then #(.4) ~<#(/t); (ZFM3) if dlCd2c.., and { A n } c F * ( X ) , then Definition

# ( (~-J~ = e{n) --" (P) .~=¢ lim/~ (A-); (ZFM4) if A, ~ d 2 ~ " " and {e{.} c F * ( X ) , and there exists no such that i,(Z{.o) 4=o~, then #

.4, =(/5) lira#

n).

(X, F*(X)) is called a fuzzy measurable space, and (X, F*(X), g) is called a (z)fuzzy measure space.

Zhang Guang-Quan / Fuzzy number-valued fuzzy measures II Definition 2.7. A fuzzy number-valued fuzzy set function /t is called null-additive (resp. nullsubtractive), if we have #(,4 U/~) =/t(Tt) (resp. /t(A n / ~ ) = #(A)), whenever/i,/~ e F*(X) and /t(t~) = o. Definition 2.8. A fuzzy number-valued fuzzy set

function /t is called autocontinuous from above (resp. from below), if we have (~) lim/t(A

U B.) =/t(A)

(resp. (/5)lim # ( A n n ~

B~.)=/t(A)),

whenever A • F*(X), {/)n} C F*(X) with

259

6, then /t(/~) - e < / t ( ~ u P) <~/t(~) + e (resp./t(E) - e ~
Fo,={x;f(x)>~oc},

o~•R,

and

(~) lim/t(/~.) = 0.

XF~(X) = /t is called autocontinuous, if /t is both autocontinuous from above and autocontinuous from below. Definition 2.9. A fuzzy number-valued fuzzy set

function/t is called F-additive, if

/t(A u/~) =/t(A) v/t(t~), whenever A, B • F*(X). Definition 2.10. A fuzzy number-valued fuzzy

set function /t is called sub-additive (resp. sub-subtractive), if we have

/t(A

u

iff X q F~.

The set of all fuzzy measurable functions is denoted by M*, the set of all non-negative fuzzy measurable functions is denoted by M*. Definition 2.13. Let (X, F*(X),/t) be a (z)fuzzy measure space, z~ •F*(X), f • M * . The fuzzy number-valued fuzzy integral ((z)fuzzy integral) of f on A with respect t o / t is defined by

f

fd/t=

U

xe[O, ll

Z[ sup a ^ (/t(A A XF~));, Loce[O,~)

(2.1)

B)
(resp./t(fii

n/3¢)/>/t(.,{) -/t(/~)),

where F~ = { x ; f ( x ) / >

Theorem 2.5. If a fuzzy number-valued fuzzy set function /t is sub-additive (resp. subsubtractive), then it is autocontinuous from above (resp. from below). Theorem 2.6. If a fuzzy number-valued fuzzy set function /t is autocontinuous from above (resp. from below), then it is null-additive (resp. null-subtractive ).

~r}, o~ • [0, o~).

Theorem 2.7. SAfd/t defined by (2.1) has the following elementary properties: (1) if /t(fl)=O, then SAf d/t=O, for any feM*; (2) if fl ~fz, then fall d/t ~ SAfz d/t; (3) if f a r d/t = O, then/t(ft m ;~F,,)= 0; (4) for any constant a e [0, oo),

f a d/t Definition 2.11. A fuzzy number-valued fuzzy set function/t is called uniformly autocontinuous from above (resp. from below), if for every e > 0, there exists 6 > 0 such that, whenever /~, P • F*(X), /~ n P = ~ (resp. P ~/~) a n d / t ( P ) <

q

sup ~ ^ (/t(A n x~o)h

o~e[o,~)

a ^ /t(A);

(5) if fl c B, then f A f d/t <~~Bf d/t; (6) for any constant a • [0, oo),

fa (f + a) d/t <- fAf d# + a A /t(A).

260

Zhang Guang-Quan / Fuzzy number-valued fuzzy measures 11

Theorem 2.8. S/if d/z :# ~ if and only if there exists o: • [0, oo) such that/Z(7~ n ZFD¢ ~.

Then the fuzzy number-valued fuzzy set function induced by f is a fuzzy number-valued fuzzy measure if and only if f • FM(/Z).

Definition 2.14. For a given and fixed function f • M * , and 77"F*(X)--~F*, we define

q:(/~) = L f d/z,

for every/~ • F*(X).

(2.2)

r/f is called the fuzzy number-valued fuzzy function induced by f (and /Z). Throughout paper, r/ is used instead of r/f if there is confusion, and all the integrals should understood in the sense of (2.1).

set the no be

Proposition 2.1. Let r1 be the fuzzy number-

valued fuzzy set function induced by f. Then: (1) for any o: >i 0 and E • F*(X), ,r

^/z(~

3. Properties inherited by

Let (X, F*(X),/Z) be a o-finite fuzzy numbervalued fuzzy measure space and r/ be the fuzzy number-valued fuzzy set function induced by some non-negative function f. In this section, the function f is not supposed to be in FM(/Z), and in general r/does not satisfy (ZFM4). The fuzzy number-valued fuzzy set function r/ has some properties in common with/Z. If/Z is F-additive, then rI is also F-additive, because

sup

= (sup

n x,o) ~ rl(~) ~ ~r v / z ( ~ n x,o);

(2) r/(/~) ~
o(P). Let

AM={f;f•M*,ff

f d / z } • Z * (see[I, 7]) n

for any {/~,} c F * ( X ) with/~,/* or/~,",a }.

for real-valued functions f and g. In this section, we will present some theorems on the inheritance of the properties of/Z. Theorem 3.1. If /Z is null-additive (resp. subiadditive, null-subtractive, sub-subtractive ), then 71 is null-additive (resp. sub-additive, nuU-subtractive, sub-subtractive ). Proof. (1) Let E , P • F * ( X ) , E n F = O , and r/(/~) =0. Then, by using Theorem 2.7(3), we have

For a given/Z, we define

/z(P n

FM(/Z) = { f ; f • A M and/Z(X{~-
Therefore, for every a~> 0, /Z(F n X,D = 0, and hence

Definition 2.15. A fuzzy number-valued fuzzy set function/Z is called o-finite, if there exists an increasing sequence of fuzzy subsets {/~,) such that

/z(/~,):#o~, n = l , 2 . . . . ,

and

X = 0 /~n. n=l

Theorem 2.9. Let (X, F*(X),/Z) be a o-finite

fuzzy number-valued fuzzy satisfying the condition Iz(l~) ¢ ~ and/Z(P) ¢ /~(/~ U F) #: ~.

measure space

(2.3)

x,O = o.

~((~ u P) n x,o) = ~((E n x,O u (P n x,o)) =/z(k n

x~o),

respectively,

/z(~ n P° n zr~) =/z((~ n P~ n XFo) u (~e n XFo n x,~D) =/z(~ n x,o) n (P n ZFo)c)

=/z(~ n xO), by using that /Z is null-additive, respectively null-subtractive. That is,

,J(E U F) = ,7(E) (resp. r/(/~ N/~¢) = r/(/~)).

Zhang Guang-Quan / Fuzzy number-valued fuzzy measures I1 (2) L e t / ~ , / ~ e F*(X). Since

sup

-

a^(b+c)<--.a^b+a^c,

sup

respectively,

ae[0,~)

a ^ (b - c ) > ~ a ^ b - a ^ c ,

-

261

o~ ^ (/u(P Cl )¢vo))], a ^ (U(/~ n ZFo))~+

sup cv ^ (U(P n )(F.))~-]

ae[0,~)

for every a, b, c e R, it follows that

o(E u P) =U

X~lO,H

~.[ sup LaelO, ~)

sup a'e[O,~¢)

<~ U

Z~[0, II

o~^ (~((~ u P) n z.o))~, o~ ^ O,((E u P) n x,,o))~-] J

Z[ sup tr ^ (/~(E n XFo) L a~[O,~)

+ ~(P n x~))~, sup c~ ^ (g(/? n Xeo)

~.e[o.~)

+]

+ u(P n x~o))~ =

U

Ze[O,1]

In the following, if /~(/~)>0,

we suppose

(~(t~))o > 0.

x[ sup (~ ^ (~(~? n x~o))ik a'e[O,~)

3.2. If ~ is autocontinuous from above (resp. from below), then rI is autocontinuous from above (resp. from below). Theorem

+ c~ ^ (~(P n z.O);), sup (o~ ^ (~(/~ n Z'O)~#e[0,

Corollary 3.1. Let (X, F*(X), t~) be a Oo-finite (z)fuzzy measure space. If f e FM(I~), then (z)fuzzy measure ~1 is null-additive (resp. sub-additive, null-subtractive, sub-subtractive), whenever I~ is null-additive (resp. sub-additive, nuU-subtractive, sub-subtractive), where a Oofinite (z)fuzzy measure space is a o-finite (z)fuzzy measure space satisfying the condition (2.3).

)

3.1. Let 0 < Tl(E ) ~ ~, then there exists a, b e R such that Lemma

+ ~ ^ (~(P n x,~o))~

0 < 2 a ~< r/(/~) ~< b < ~ respectively that

and

n(E n P ) =

U X~IO, I]

~.[ sup o~ ^ (t~(/~ n Pc n z~D);,

r/(/~) =

U

;~[ sup

o~ A (/~(/~ n x~o));,

sup

c~ A (/~(/~ n x . 9 ) ; / . J

)tel(I, 11 k a~el2a,b ]

L a~[O,~)

sup ~ ^ ( ~ ( ~ n Pc n x~o))~

c, el2a,bl

oce[o,~)

Since r/(/~) 4= ~, there trl > 0 with oq < t r o such that

Proof.

Z~lO, 11 k ~ l O ,

)

- . ( P n x,O)i-, sup a ^ ( ~ ( ~ n ze~)

/t(/~ n )~F~,,)4: ~

and

exists

]~(/~ N )~F~,) > 0.

Put

o~lO,~)

- ~ ( P n x,0)~

0 < 2a = oq ^ (/~(/~ N )¢F.,))ff

a, ^ (~(/~ n x.o,))Z XelO,ll

k o~e[0,p )

A - - O~ ^ ( U ( P

and n

x,O)~+ ),

sup ( a A (g(/~ n x,0)~-

Applying Proposition 2.1(3), we obtain

o~e[0,~)

q

- ~ ^ (u(t n x,0);)|

i> ~

;t[ sup a ^ (t~(/~ n x , 0 ) ~

Z~[O, 11 l_ a~el0,~ )

~b = tx(, v (/a(/~ n x...))~- < ~.

_1

0 < 2 a ~< r/(/~) ~< lb < b
cro > O,

Zhang Guang-Quan / Fuzzy number-valued fuzzy measures I1

262

that is

L e m m a 3.2. Let 0 < rl(E ) ~ ~o, and a, b satisfy

the preceding lemma. For integers m >1 1, put

2a = 2a ^ (/~(/~ D Xr~))x-,

o~k=k(b-a)/m+a,

xm));.

2a = 2a A (B(/~ n

....

,m,

,7~(~)

Therefore sup

k=0,1

=

o~ A (U(/~ n XF~))~ -(+)

~el0,Za)

~< 2a ~< sup

a~e[2a,b]

~ A (/~(/~ n XF~)); (+).

/~[ sup

U (Or'k+ 1 A ( ~ ( / ~ n ZF~,,~))~-, )~e[O,1] LO~k~m-I

sup a~+, ^ (,(/~ n x~o)h o~,~,~-1

and

On the other hand, b = 2Cro v 2(/~(E n

X~o,,))~-

=

U

~.e[O,1]

Z[

sup

sup

O~k ~m--1

and ~< -o v (t~(E n XFo,,)); = ½b ~ b. Therefore o~e(b,=)

cr A

(/~(/~ n

~7(E) = (~5) lim tl~(/~ ) = (~5) lira ~Lm(/~). Proof. For a~ with •..,m-l, wehave

XFa))~-(+)

o,,e(b,o~)

a~ A (/~(E n x,O); (+),

<~~,+, ^

fork=0,1,...,m-1, (~m(E))~ -(+) =

and hence r/(/~) =

U

;t[ sup

).~[0,11 L ae[O,~)

U

cr A (/~(/~ n XF~))z-,

v

ol A (/~(/~ n

v

;~e[0,1l

sup

ae[2a,:~)

sup

2~[ s u p

Lo~e[2a, )

x[ sup

~.~[0.11 L o:el2,a,b I

sup

ae[2.a,b]

)(,Fa))-~,

x,~,)); (+),

and hence

c~, A (~(~ n x,~o,~)); (+)

~<

sup

ol A (/~(/~ n XF.))~-(+)

O~k ~m--1

a',+,A(~(EnXF.))~-(+)

---- ( ~ m ( E ) ) ~ "(+),

It follows that

(,7~(t~)); (+) ~ (,7(t?)); {÷} ~< (,7~(~));( +}

a~ A (/~(/~ n ,~F.))~-

o~e[2a,~)

sup = U

sup

n

O~k~m-- I

a, A (~(/~ n z~))i-

ofel0,2a)

U

;t•[0,1],

~< sup

~.[ sup

Ae[0, 11 La'e[0,2a)

sup

=

(U(E

O~k ~m - 1

sup a. ^ ( ~ ( ~ n z~D), =

k=0,1,

~
<~(u(E n z,O); (+) = b A (u(E n x J ) ; (+) sup

o~,~a~o~,+~,

a, ^ (~(/~ n Xv~,+,)); (+}

~< sup (/~(/~ n z~.))~-(+)

a'e[2a,b]

+]

e~ ^ (~(~? n X~o**,))~ •

Then it follows that

( u ( E n X~))~ ~ (U( ~ n Z~o,,))~

sup

~x, A ( ~ ( E n J~Fa,+,))~-,

kO~k<.m-1

,r A (~(/~ n x~D)I-]

a~ v ( p ( E n XFa))~-,

+]

for any ;t e (0, 1]. That is, (r/L(/~) ~< r/(/~) ~< r/~(/~),

m = 1, 2 . . . . .

Now choose m so large that oq < 2a. Then we get

a~ A (~(~ n xF°)h

(~(~ n X~o,));(+)~ > ,rl.

a~ ^ (~(~ n z~o));,

Indeed if (/~(/~ (7 X s , ) ) ; (+) < ch holds, then the contradition

q

~ ^ (u(/~ n z~-o))x .

(n(~")); (+) ~< ,r, v (~(~ n x < ) ) ; (+} = a , < 2a ~ (r/(E))~ -(+)

Zhang Guang-Quan / Fuzzy number-valuedfuzzy measures II

for any )~ e (0, 1] follows. Thus for some k ~ 1 must hold

(~(~));~+~ = o~+~ ^ (~(~ n x~o));~+L

263

That is,

for any n/> no. Thus the lemma is proved.

Indeed if k = 0 holds, then the contradiction (r~(~'))~ -~+~ = m ^ ( u ( E n Z,~o,,)); ~+~ = c,.,

^ (I~(E n

< oq

A

x ~ ) ) ; ~+~

(#(~, n z , % ) ) ; ~+~

< ~ A 2(t~(~ n XF~,,)); (+) = oq < 2a ~< (y(E));~+~ follows. Thus ( ~ ( ~ ) ) ; ~ + ~ - (n~(~));~+~ = ~ + ~ A (u(~' n z~;,,)); ~+~ - a.,,_,

A

Proof of Theorem 3.2. Let /~,/~, e F*(X) and r/(/~,)--->0. The theorem is trivial in case r](/~) = ~. Let r/(/~) = 0; then #(/~ n ZF~) = 0 for any ~r>0 by Theorem 2.7(3). Thus by the autocontinuity from above (and hence nulladditivity) of/~, we have

~(E U E.) = U z[ sup, ~ ^ (~(~ u ~.) n x,o))L ~.elO,]l I-aelO. ) sup

( # ( ~ n x~o,)); <+~

= (o~+, - o~_,)

^

( ~ ( ~ n x , o ) ) ; (+~

a ^b-a

AC<~a ^ ( b - c ) .

(~(~ u ~.) n x~;));1 J

=

= 2(b - a)/m ^ (l~(E n £F~JIZ .. ~-(+~ = 2(b - a)/m is deduced by the help of the inequality

A

U

ZelO, ll

z[ sup . ^ (/~(/~. n Z~o));, t~10,

)

sup o~ ^ ( u ( ~ . n z~o))~ = r/(/~,)---~0= r/(/~)

as n--->~.

Now the conclusion is easily obtained. For any e > 0, take M/> 2(b - a)/e. Then

Now we assume that 0 < r/(/~) ¢ ~, and let a and b be determined as in Lemma 3.1. By Lemma 3.3 and the assumption for/~, we have

( n ~ ( ~ ) ) 2 +) - ( n ~ ( # ) ) ; ~+~< e

(/5) lim/z(/~ U/~.) n ZF~) = It(E n X~$)

as m/> M, for every ~l e (0, 1]. Therefore lim (r/2(~:));~+~ = lira (t/~(E))~"(+) trl~

(3.1)

n~C

for a fixed fl > 0. Thus we may choose no such that

ttl~c~

= (~(~));~+), uniformly for every ~l e (0, 1]. That is, (,6) lim q ~ ( E ) = (¢5) lim q~(/~)= ~1(/~).

Lemma 3.3. Let rt(E,)---~O as n--->~. Then

l~(E n ZF~)--->0 for any fixed/3 > O.

< ½b + ½b=b, for any ~ > oq~ and n ~>no (put /3= oq~ in (3.1), and note that /~(~en ~v,) is decreasing in o0. Hence it follows that

n(E U E.) =

U

X~[O, 1 ]

Z[ sup

k2a~o:~b

Proof. Fix fl > 0, and take any e with 0 < e
^ (~(/~. n x~))~(+) < ( n ( ~ . ) ) ; ~+~ < ~, for every /l e(0, 1] and for any n>~no. This means that

( u ( e n x,O);~+)<~ (~(~. n x~)); ~*' < e, for any n ~ n o uniformly for every Z e (0, 1].

sup

~ ^ (U(/~ U E,) n

x,o))-;,

+]

o~ ^ ( u ( ~ u ~ . ) n x~o)h

,

for any n ~ n o . We define ~?u(/~UE.) and r/~(~ U E.) for a and b as in Lemma 3.2. Then

~ ( ~ u ~.) < ~(~ u ~.) < n~(E u ~.), m = 1, 2 . . . . .

n 1> no.

Applying (3.1) to c~g, k = 0 , 1 . . . . .

(3.2) rn-l,

we

Zhang Guang-Quan / Fuzzy number-valued fuzzy measures H

264

as /~(je)< 6. We may assume that 6 ~< e. The proof will be completed if we show that

obtain lim (r/~(/~ U/~.))Z (÷)

n~oc

=

sup

O~k~
.~-(+)

o~+~^t~tt~ n

uniformly for every ). • (0, 1]. That is, n---~oc

(3.3)

r/(/~ N [ ' 0 ~> r/(/~) - e,

(3.4)

for every /~ • F*(X), r/(_P) < 6. By Proposition 2.1(1), we have

= (,g(/~));(+),

(15) lim r/~(/~

q(E U F) <~rl(E ) + e,

U/~.) =

r/~(/~).

^ ~ ( P n z,o) -< n ( P ) < 6,

for any a~ > O, and hence

t,(P n z , 9 < 6,

By the same way,

(3.5)

for any c~/>6. In case rl(/~UF)~
(15) lira r/~(E U E,) = r~Lm(/~). n---~e,~

Therefore, by using Lemma 3.2,

(r/(/~ U F));(+) ~< e q(/~) = (15)lira r/~(/~) = (p) lim (p) lira r/Lm(L" U/~.) m~

m~oc

n~:¢

(r/(E U F));(+) > e.

Put A = {(r/(/~ U ¢));(+); (r/(/~ U F))i -(+) > e, ~, • (0, 1]};

n ~ "~

~< (15) lim (15) lira r/(/~

or

U/~.)

8 = { ( ~ ( ~ u P));(+~; (~(t? u P ) ) i -~+~ -< e,

z • (o, 11).

= (15) lim q(/~ U/~.)

For any (r/(/~" U F));(+) • A, we have

n~OC

(t5) lim (15) lira r/mV(/~ U/~.)

(n(E U #));(+) =

sup

o~[o,~)

~ ^ ( u ( E u P) n x , o ) ) ; <+)

~< e v sup c~ ^ (/~(/~ U F) n XF~)); (+)

= (/5) lim r/~(/~) = r/(/~),

o~elE,~)

m---~.oc

which means that

=

rl(/~) = (~) lim r/(E U/~.).

~< (n(t~ u P));(+),

n----}oc

sup

0~' A ( l ~ ( / ~

g l~') n XF~,))X - ( + )

that is, The proof of the alternative statement of the theorem is similarly done. Corollary 3.2. Let (X, F*(X), t~) be a Oo-finite (z)fuzzy measure space. If f • FM(li), then (z)fuzzy measure ~l is autocontinuous from above (resp. from below), whenever # is autocontinuous from above (resp. from below).

(n(t~ u P));(+~ = sup a~ ^ (~(/~ U/~) n XFa))~-(+). Therefore

~< sup u ^ (/~(/? n Z e ) ) i -(+) + e

Theorem 3.3. If I~ is uniformly autocontinuous,

ae[e,~)

then r1 is also uniformly autocontinuous.

( n ( E ) ) ; (÷~ + e,

Proof. For arbitrary given e > 0, since ~ is uniformly autocontinuous, then there exists 6 = 6(e) > 0 such that

and for any (r/(/~ U/7))i -(÷) • B, we have

~(~)

It follows that

-

e

~<~(~' u

#) ~< ~(E) + e,

~ ( ~ ) - e -< t~(E n pc) ~< t~(t?)

+

~,

(n(P~ u P ) ) ; ( * ) <~ ~ <~ (r/(E)); (+~ + e.

(~(P~ u P));(+~ <~(r~(E))£ (+~ + e,

Zhang Guang-Quan / Fuzzy number-valued fuzzy measures II for every ~, • (0, 1], that is

,7(E u P) ,7(E) +

E.

T h e p r o o f of (3.4) is similarly done.

Corollary 3.3. Let (X, F * ( X ) , l~) be a oo-finite ( z ) f u z z y measure space. I f f • F M ( # ) , then ( z ) f u z z y measure r1 is uniformly autocontinuous, whenever ~ is uniformly autocontinuous.

4. Conclusion W e have p r o v e d that a fuzzy n u m b e r - v a l u e d fuzzy m e a s u r e r/ defined by a fuzzy n u m b e r valued fuzzy integral on the fuzzy set with respect to a fuzzy n u m b e r - v a l u e d fuzzy m e a s u r e

265

/t inherits F-additivety, null-additivety, a u t o c o n tinuity and uniformly autocontinuity, etc.

References [1] Zhang Guangquan, Fuzzy distance and limit of fuzzy numbers, Basefal 33 (1987) 19-30. [2] Zhang Guangquan, Fuzzy continuous function and its properties, Fuzzy Sets and Systems 43 (1991) 159-171. [3] Zhang Guangquan, Fuzzy number-valued fuzzy measure and fuzzy number-valued fuzzy integral on the fuzzy set, Fuzzy Sets and Systems, to appear. [4] Zhang Guangquan, FSC-fuzzy measure and fuzzy integral on the fuzzy set, J. Hebei University 4 (1988) 11-23. [5] Wang Zhenyuan, The autocontinuity of set function and fuzzy integral, J. Math. Anal. Appl. 99 (1984) 195-218. [6] H. Suzuki, On fuzzy measures defined by fuzzy integrals, J. Math. Anal. Appl. 132 (1988) 87-101. [7] Zhang Guangquan, On fuzzy number-valued fuzzy measures defined by fuzzy number-valued fuzzy integrals I, Fuzzy Sets and Systems 45 (1992) 227-237.