A note on measures in fuzzy topological spaces

Fuzzy Sets and Systems 54 (1993) 341-345 North-Holland

341

A note on measures in fuzzy topological spaces Beloslav Rie~an Mathematical Institute, Slovak Academy of Sciences, Bratislava, Czechoslovakia Received April 1992 Revised June 1992

Abstract: Regularity of a fuzzy measure on a fuzzy topological space is considered. First it is proved that regular fuzzy sets form a fuzzy o-algebra. Then the subadditive case is studied as well as a connection between additivity and subadditivity (the Alexandrov theorem). Keywords: Fuzzy measure; fuzzy topological space.

1. Introduction

In the classical measure theory topological considerations play an important role. Therefore a similar situation can be expected in the fuzzy case, too. Recall first some basic definitions [1, 2]. Definition 1.1. Let Xq=O, 0 : / : o ~ c ( 0 , 1 ) x. ~: is said to be a fuzzy a-algebra, if the following conditions are satisfied: (i) f e E ~ f ± : = l - f e ~ . (ii) L e ~ ( n = l , 2 . . . . ) ~ V f . : = s u p f , e o*. A mapping/, : ~---~ (0, oo) is called a fuzzy measure, if the following conditions are satisfied: (i) f, g e J ; , f < - g ~ #(f)<~g(g). (ii) /~(f) + ~u(g) = / ~ ( f v g) + / a ( f ^ g) for all f, g e ~. (iii) (f,)~ c ~, f,,,,~f ~ tt(f,,)--o l~(f ). Definition 1.2. Let X 4: 0, 3- c (0, 1) x. 3- is said to be a fuzzy topology and (X, 3 ) a fuzzy topological space, if the following conditions are satisfied: (i) 0x, lx e 3". (ii) f, g e 3 - ~ f A g e 3-. (iii) (f~)~ c 3- ~ V~f~ e 3-. Definition 1.3. Let 3 - c o ~ c (0, 1) x, ~ b e a fuzzy a-algebra, O-be a fuzzy topology. An element f e is said to be regular, if # ( f ) = inf{#(g); g open, g ~>f} = sup{#(h); h compact, h ~
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2. Regular fuzzy sets We shall assume only the following properties of compact or open fuzzy sets, respectively. They are satisfied, if the fuzzy topological space is compact.

Assumptions 2.1. There is given a fuzzy o-algebra ~ c (0, 1 )x and two subsets C, U c ~ satisfying the following conditions: (i) f • C ~ 1 - f • U . (ii) f • U ~ 1-f•C. (iii) f / • U (i = 1, 2 . . . . ) ~ ~/7=lf • U. (iv) f, g e C ~ f v g e C . Proposition 2.2. If i~ : ~---> (0, oo) is a fuzzy measure and (f,), is a sequence of regular elements of then Vn=lfn is a regular fuzzy set. Proof. Let e be an arbitrary positive number. Then there

are

hi E

C,

gl c U such that

E

hi <~f <~gi,

P(gi)

--

(2.1)

I~(hi) < 2i+-----~

By the second property o f / , and by (2.1), /~(gl v g2) - / ~ ( h l

V

h2) = p(g,) + ~(g2) - ~(ga A g2) -- (/~(h,) + It(hE) - I~(h~ A hz)) E

because of (2.1) and obtain

hi A

E

h2 ~ gl A g2. Hence #(h~ n

^ h2) -

g(g~ A g2) ~<0. Similarly by induction we

e

(2.2)

~(iglgi) - ~(igl hi) < i~=12i+l" By (2.2) and the third property of a measure it follows

"(i91 gi) -- "(igl hi) ~2" n h i/~/~=~ hi, there is n Since Vi=~ "(i91 h i ) - " ( i ~

(2.3) such that (2.4)

hi)<2"

Hence by (2.1), (2.3) and (2.4) we obtain

9 h i < - Vi = 1f i ~ g gi =l1 ,

i=1

Vhi•C,

i=1

9gl•U,

i=1

and therefore ~/7=lf~ • ~ is a regular element. Defmilion 2.3. A measure/~ : ~:---~ (0, 1) is called a fuzzy probability meaure, if /~(f) +/~(f±) = 1 for every f • ~.

Theorem 2.4. If I~ : ~--+ (0, 1) is a fuzzy probability measure, then the family of all regular elements of F is a fuzzy o-algebra.

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Proof. By Proposition 2.2 it suffices to prove that f x is a regular element, whenever f is regular.

Choose g E U, h ~ C such that h <~f ~ g , /z(g) - / t ( h ) < e. Then

1-g<-l-f<~l-g,

1-gEC,

1-hEU,

/~(1 - h) - / z ( 1 - g) = 1 - / z ( h ) - (1 - / , ( g ) )

< e.

T h e o r e m 2.5. Let C be the family of all G6 compact fuzzy subsets of a compact topological space, ~ be

the fuzzy o-algebra generated by C, Iz : ~---> (0, 1) a fuzzy probability measure. Then Iz is regular, i.e. all members of ~ are regular. P r o o L It suffices to prove that every f e C is regular. But to every f E C there are f, ~ U such that f, ",af.

Since f ~ / ~ f ± , we obtain ~(f) = 1 -/,(f±)

= 1 - lim ~ ( f ~ ) = lim u(f,,). n---~

n----~o~

Therefore there is n such that f ~
3. Fuzzy submeasures

Definition 3.1. A fuzzy o-algebra ~ c (0, 1} x is called DF-o-algebra, if f, g e ~:, g ~ (0, oo) defined on a DF-a-algebra is called fuzzy submeasure, if the following conditions are satisfied: (i) f, g e ~, f <<-g ~ It(f)<~l~(g). (ii) f, g, h E ~ , f ~ (0, ~) be a submeasure. Then the following properties are satisfied:

(iv) (v) (vi) (vii) (viii) (ix)

lt(f v g) <- # ( f ) + #(g) for every f, g E ~. P(VT=lf,) <~~ = , I~(f~) for every f E ~; (i = 1 , . . . , n). f, g • ~, f <~g ~ /~(g) - / ~ ( f ) ~t~(f). (f,,),, = ,~ ~ /-l(Vn=lfn) <~ E~=I/*(f,).

Proof. The properties (iv) and (vi) follow by (ii), (v) is a consequence of (iv), (vii) and (viii) are consequences of (iii) and (vi), and (ix) follows by (v) and (vii). Definition 3.3. Let X, ~, C, U satisfy the Assumptions 2.1, ~: be a DF-a-algebra,/z : ~:---> (0, oo) be a

fuzzy submeasure. We shall say that f e ~: is regular, if 0 = inf(#(g - h); h <~f ~
Proof. First let f be regular. Hence to every e > 0 /u(g - h) < e. Then

1-g<-l-f<~l-h,

1--gEC,

/~((1 - h) - 1 - g) = ~ ( g - h) < e,

and hence 1 - f is regular, too.

1-hEU,

there are such h E C, g ~ U

that h<~f<~g,

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B. Rie~an / Measures in fuzzy topological spaces

Let (fn), ~ ~: be a sequence of regular elements. Then to every e > 0 and every n • N there are h, • C, g, • U such that E

h~<~f,<~g~,

tt(gn - h.) <2~+1 •

Then /~(V g~ - V h,) ~
(3.1)

Further VT=I h~ - V~'=1 h,-~0, hence there is n such that

~.l(iV=lhl -- iV=lh i ) <

(3.2,

2.

By (3.1), (3.2) and the second property of fuzzy submeasure It (i_~l g~ -

i_~l h i )

< e,

~c¢/ fi <~ ~/ gi, ~/ hi <~ i=l i=l i=l

~/ h i • C , i=1

i=l

gi • U.

Hence V~=I fi is regular.

4. Alexandrov's theorem In the preceding sections we deduced regularity from continuity. The Alexandrov theorem works in the opposite direction. We shall present here a fuzzy variant of the theorem. Definition 4.1. A subset C of a fuzzy a-algebra ~: will be called a compact family, if the following implication holds:

¢¢ If (fn), c C, /~ f / ~ 0

for every n, then /~ f~ :/= 0.

i=1

i=1

A function # : ~---~ (0, ~) is called compact, if there is a compact family C c ~ such that 0 = inf{~(f - g); g • C, g <~f} for every f • :~. Theorem 4.2. Let ~; be a DF-o-algebra. Let ~t : ~--~ (0, ~) be a compact, subadditive function (i.e. f ~ /~(f) ~< ~(g) + g(h). Then the following implication hold:

(fo)o

(fn)

Proof. By subadditivity it follows that l i m ~ # ( f n ) = c exists. We want to prove that c = 0. Assume c > 0 . Then there is compact C c ~ such that to every n there exists gn ~
i=1

gi ~ 0

for every n.

In the opposite case there is n w i t h / ~ =n~ gi - 0. But then

(/,-gi

i=1

(4.1)

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which is impossible. But by (4.1) we obtain that 0 ~ Ai~l gi ~< Ai~l fi, which is a contradiction with the assumption that/~i~1 f~ = 0.

References [1] C.L. Chang, Fuzzy topological spaces, J. Math. Anal. Appl. 24 (1968) 182-190. [2] E.P. Klement, Fuzzy a-algebras and fuzzy measurable functions, Fuzzy Sets and Systems 4 (1980) 83-93. [3] A.P. Sostak, Twenty years of fuzzy topology: basic ideas, notions and results, Usp. Mat. Nauk 44 (1989) 99-147 (in Russian).