On the existence of probability measures on fuzzy measurable spaces

Fuzzy Sets and Systems 43 (1991) 173-181 North-Holland

173

On the existence of probability measures on fuzzy measurable spaces Anatolij Dvure~.enskij* Department of Probability and Mathematical Statistics, Faculty of Mathematics and Physics, Comenius University, Mlynskd dolina, CS-842 15 Bratislava, Czechoslovakia Received April 1989 Revised August 1989

Abstract: We present some classes of fuzzy measurable spaces which possess at least one probability measure.

Keywords: Fuzzy measurable space; probability measure; o-ideal.

1. Introduction Let (g2, 5¢) be a measurable space. It is well known that it possesses at least one probability measure. Indeed, fix an to a g2. Then the Dirac measure 6,o: b°---~ {0, 1} defined via 6o,(A) = 1 if to c A and 6,o(A) = 0 if to ~A, A a If, is a probability measure on 9°. This transparent fact is not evident for fuzzy measurable spaces. In [2], it has been proved only that any fuzzy measurable space admits at least one finitely additive measure. We recall that according to [1], a fuzzy measurable space ( = F-quantum space) is a couple (X, M), where X is a nonvoid set and M ~ [0, 1]x such that (i) (ii) (iii) (iv)

1 a M; a a M implies a' := 1 - a a M; t._.Jn=la. := s u p a . a M whenever ½~ M.

(1.0)

{a.} ~M;

In [3], the set M is called a soft fuzzy o-algebra, and this structure has been studied together with motivation in [1,2] within the framework of axiomatic models of quantum mechanics using fuzzy set ideas. The set M may be regarded as a partially ordered set in which we define f <~ g, for f, g e M, if f(x) <~g(x) for any x e X. Using Zadeh's fuzzy complementation ': f ~ . f ' = 1 - f for any fuzzy set f e M, we see that ' satisfies two conditions: (i) ( f ' ) ' = f for any f ~ M; (ii) f ~
174

A. Dvure6enskij

(nn=l a,)' = U~=~ a'n, {an} c M, hold. We recall that M is a Boolean o-algebra (for definiton see [6]) iff M consists exclusively of crisp subsets. A simple nonLrivial example of fuzzy measurable space is a couple (X, M), when X = [0, 1] and M = {0, 1, i, 1 - i, max(i, 1 - i), rain(i, 1 - i)}, where i(x) = x for any x • X. A probability measure is a mapping m :M---~ [0, 1] such that

m(aUa')=l m

for a n y a • M ,

ai = ~ m(ai) i=1

(1.1) (1.2)

whenever ai <~as for i --kj, {ai} ~ M. In [3], this mapping is called a P-measure. We recall that if the set M satisfies only conditions (i)-(iii) of (l.0) and if M possesses at least one mapping m with (1.1)-(1.2), then in view of 1 = ½,, automatically ½~M. In other words, M satisfying (i)-(iii) of (1.0) and containing ½, fails probability measures. In the present note, we exhibit some classes of fuzzy measurable spaces which have nonempty systems of probability measures.

2. Probability measures Let (X, M) be a fuzzy measurable space. We denote by P ( M ) the set of all probability measures on M. For any m • P ( M ) we have [3] m(a)<~m(b) if a ~< b, a, b • M. A set Q ~ P ( M ) is order-determining if, for a, b • M, the statement "m(a) ~ m ( b ) for any rn • Q " implies a ~-a ') (here W stands for weak). It is evident that the following assertions are equivalent: (i) a is a W-empty set (W-universum); (ii) a~½); (iii) a n a ' = a ( a U a ' = a ) ; (iv) a' is a W-universum (W-empty set). We denote by Wo(M) and WI(M) the sets of all W-empty sets and W-universes, respectively, from M. Two fuzzy sets a and b from M are W-separated [3] and we write a _Lb if a ~
Probability measures on fuzzy measurable spaces

175

Generalizing the notion of a e-ideal known for e-algebras of subsets, we say that a subset I of M is a a-ideal of M if (i) a O a ' ~ < I f o r a n y a • M ; (ii) a • M , b • l , a ~ < b , i m p l y a • l ; (iii) U~=l an • I whenever {an} c I; (iv) if a n b • I for some b • WI(M), then a • 1. Denote by I0 = {a • M: there is a c e W I ( M ) such that a n c • W0(M)},

(2.1)

Then by [2], I0 c I for any a-ideal I of M. Suppose that I is a e-ideal, and put a - - l b iff a N b ' , a' n b • I . Then [2]: (i) a --i b implies a' - 1 b'; (ii) iff an - t b n , for n/> 1, then U~=I an --1 U~=l bn and (-~=1 an ~1 f"~n=l bn. Denote by [ a ] l = { b • M : b - l a } . Then M / l : = { [ a ] t : a e M } is a Boolean aalgebra (in the sense of Sikorski's definition [6]; for the proof see [2]) if we define the Boolean operations ' , V , A , as follows: [a]~:=[a']j, a a M , and V~=l [an]t := [U~=l an],, A~=, [an], := [("~=1 an],, {an} c M. Moreover [0], and [111 are the minimal and maximal elements in M / I . According to [4], we define K ( M ) as the set of all subsets A c X such that there is a fuzzy set a • M with {a >½} c A g {a >/½}.

(2.2)

(The set {a >½} denotes {x • X : a(x) ~>½}, similarly for {a ~>½}.) Theorem 2.1. Let (X, M ) be a f u z z y measurable space. Then K ( M ) is a e-algebra o f subsets o f a set X. I f m is a probability measure on M, then the function e = Pm : K(M)--> [0, 1] defined via

(2.3)

P ( A ) -- m ( a ) ,

where a • M and A • K ( M ) are connected through (2.2), is a probability measure on K ( M ) with P({a = 1}) = 0 f o r any a • M.

(2.4)

Moreover, if m, n • P ( M ) , m :/: n, then Pm =/:pn. Conversely, let P be any probability measure on K ( M ) with (2.4), then the mapping mp : M---> [0, 1] defined via mp(a) = P ( A ) ,

a • M,

(2.5)

if a • M and A • K ( M ) satisfy (2.2), is a probability measure on M, and if Q is another probability measure on K ( M ) , then mp :/: m o. In addition, m = rap,° and P=Pmp"

Proof. The first part follows from Theorem 9 of [4]. The last assertion may be proved as follows. If, for A , B • K ( M ) we have simultaneously {a >½} c A c {a/> ½} and {a > ½} ~_ B ~_ {a/> ½}, then (2.4) entails P ( A ) = P ( B ) , so that m e ( a ) is well defined by (2.5). Evidently, me(a U a') = 1 for any a • M. Moreover, we

A. Dvure6enski]

176

have rap(a) = P({a > 1}), for any a e M. Hence, if ai _1_aj for i 4:j, then

mp(~n an ) = p ( {~Jn an > 1 } ) = p ( u {an> I}) = ~ P({an > 1}) = ~' mp(an) n

n

when we use the disjointness of subsets {an > I}, n i> 1. Suppose now m :/: n, then there is a fuzzy set a e M with m(a) :/: n(a). Hence, Pm({a > ½}) = m(a) :/:n(a) = Pn({a > ½}). Analogously, we proceed with other properties. [] The problem of the description of probability measures on fuzzy soft o-algebras is also evident from the following assertion. T h e o r e m 2.2. If a soft fuzzy a-algebra M possesses at least one subset a e M with

{a = 1} :/: fk, then K(M) possesses at least one probability measure P which does not fulfill (2.4). Proof. Suppose the contrary. Then P(M) coincides, due to T h e o r e m 2.1, with the set of all m = me defined via (2.5). Let a be a noncrisp fuzzy set from M. Then the following equalities hold

1 = me(a U a') = e({a U a' > I}) = P(X), for any probability measure P on K(M). Consequently, X = {a U a ' > ~} when we use the order-determination of the set of all probability measures on K(M), and this contradicts our assumptions. [] A mapping m : M---> [0, ~) such that m(a U a') = r e ( l ) for any a ~ M is said to be a finitely additive measure (measure) if m ( L _ J i ~ T a i ) = ~ i e T m ( a i ) , a i 1 aj for i 4:j, holds for any finite (contable) set T. We say that a finitely additive measure m is purely finitely additive if every measure m ' such that 0 ~< m ' ~< m is identically equal to zero. The following analogue of the Yosida and Hewitt decomposition theorem holds.

Theorem 2.3. Every finitely additive measure m on M can be uniquely written as

the sum of a measure ml and purely finitely additive measure. Proof. Let m be a finitely additive measure on M. Then P = Pm defined via (2.3) is a finitely additive measure on K(M). Due to the classical result of Yosida and Hewitt [7], there is a unique measure P1 and a unique purely finitely additive measure P2 on K(M) such that P = P1 + P2. It is evident that Pi({a = ½})= 0 for any a e M, i = 1, 2. In view of T h e o r e m 2.1, the mappings ml and m2 induced by •°1 and P2, respectively, by (2.5), are a measure and purely finitely additive measure on M. Therefore,

m(a) = P({a > I}) = el({a > 1}) + P2({a > ½}) = ml(a) + m2(a) for any a e M.

Probability measures on fuzzy measurable spaces

177

W e assert that m2 is a purely finitely additive m e a s u r e on M. I n d e e d , if m ' is a m e a s u r e on M with 0 ~< m' <~m2, then for P ' induced by m ' we have 0 ~< P ' <~ P2so that P ' = 0. T h e uniqueness of the d e c o m p o s i t i o n for m follows f r o m the uniqueness of that for P o n K(M). []

3. Existence criteria In the p r e s e n t section, we give s o m e sufficient conditions for the existence of at least one probability m e a s u r e on fuzzy m e a s u r a b l e spaces. L e m m a 3.1. (i) Let Io be the minimal o-ideal of M, defined via (2.1). Then a ~ Io

iff there is a sequence {a, } ~ M such that {a>½}c_ 0

{an=½} •

(3.1)

n=l

(ii) Let {b,} be an arbitrary sequence of fuzzy sets from M. Then

0 {b,, = ½} :/:X.

(3.2)

n=l

P r o o f . (i) S u p p o s e that a e I0. T h e n t h e r e is a c ~ WI(M) such that a N c ~< ½, which gives {a > ½} c_ {c = ½}. If we put a , = c, for any n / > 1, then we have (3.1). S u p p o s e n o w (3.1) holds. T h e n

{a>½}~ 0 (a,,=½}=0 {a,,Ua',,=½}c{c=½}, n=l

where c = n ~ = l (ii) Calculate

(a,, U a').

©(bo= ½ } =

n=l

n=l

0 {b,,Ub'=½}c_{b=½}~X, n=l

where b = n ~ = , (b,, tO b',,).

[]

N o w we shall d e v o t e o u r attention to the B o o l e a n o - a l g e b r a M/lo, w h e r e 1o is the minimal o-ideal of M. H e r e [0], 0 4= [1],0 which is g u a r a n t e e d by (iv) of (1.0). It is easy to see that if M consists exclusively of crisp subsets of X, then 1o = {0} and M/Io = M. F o r o u r aims we shall write 5 := [a]t0, a ~ M. It is not hard to show that M admits at least o n e probability m e a s u r e . I n d e e d , let m be a probability m e a s u r e on M, t h e n / z m :5~-->m(a), a 6 M, is a probability m e a s u r e on M/lo. C o n v e r s e l y , any probability m e a s u r e /z on M/lo induces a probability m e a s u r e m , on M via mr(a ) = Iz(5), a ~ M. W e say that a B o o l e a n o - a l g e b r a A is o-distributive if

V

t~T

A at~ = ~/~sT t~r a,~(,)

seS

(3.3)

A. Dvureeenskij

178

holds for any two-indexed sequence {at,: t ~ T, s ~ T} _ A . Due to Sikorski [6, Theorem 19.2], a Boolean o-algebra A is o-distributive iff, for any nonzero a c A , and any sequence {an} c_A, there exists {e(n): n i> 1} c {0, 1} such that

a ^ /~ a e~") :/: 0,

(3.4)

n=l

t where an0 = an, a .1 = a,, for n/> 1.

T h e o r e m 3.2. Let (X, M ) be a f u z z y measurable space. I f X is at most countable, then M possesses at least one (two-valued) probability measure, and M/lo is o-distributive.

Proof. (i) Let X = {Xl, x 2 . . . . } . We assert that there is at least one point Xo e X such that a(xo) 4: ½, for any fuzzy set a e M. Indeed, suppose the converse. Then, for every xn ~ X, we find an an e M such that an(xn) = ½. Then X = I..)n {an = ½}, which contradicts (3.2) of Lemma 3.1. Therefore, we may define a mapping mxo:M--* {0, 1} via mx°(a) =

{01 if a(x0) < ½, if a(xo) > ½.

(3.5)

It is easy to verify that mx,, is a probability measure on M. (ii) Let ~i 4= 0 and {tin: n 1> 1 } ~ M/lo be given. Inasmuch as a "/'to0, for every W-universum c ~ M, there is a point x e X such that c(x) > ½ and a(x) > ½. We assert that there is a point x~ e {a > l} such that b(xO :/: ½ for any b ~ M. Indeed, in the opposite case, for any xk ~ {a > ~}, there is a bk ~ M such that bk(xj,) = ½. Then {a > ½} c_ 1._3~{bk = ½}, which, in view of (i) of this lemma, means that t i = 0 . Therefore, we can choose e ( n ) ~ {0, 1} such that a,~m(x0>½. Then a fq On aen(n) ¢li1o. [] If N c_ M, then there is a soft fuzzy a-algebra M ( N ) containing N, M ( N ) c M and such that if Mt c M is aft another soft fuzzy e-algebra containing N, then M ( N ) c M1. Lemma 3.3. Let N c M, and define Mo = N U {0, 1} and, f o r any ordinal o l > 0 , we put M~=

~,

~

: ~ e Mt~ [or ~ < ol .

(3.6)

Then

(i)

(ii)

N c Moc_M1 c M2 c . . • c M , c_Mt3c_M, c M , f o r any n >- 1 and all 0 < t8 < ol.

M(N) = U M~,, ~
where K2 is the first uncountable cardinal.

Proof. It is standard.

[]

(3.7)

Probability measures on fuzzy measurable spaces

179

T h e o r e m 3.4. Any countably generated soft f u z z y o-algebra possesses at least one

(two-valued) probability measure, and M / Io is o-distributive; moreover, M / lo is o-isomorphic to some o-algebra of subsets. Proof. (i) Suppose that {b,}~=l ~_ M is a generator of M and put c = n , lib, U b',). Then U , {b, = ½} ~_ {c = ½} 4= X, and there exists a point x0 • X such that c(xo) :~ ½ and b,(xo) 4=½ for any n/> 1. We may find a real number 6, 0 < 6 < ½, such that b,(xo) • U : = [0, ½ - 6] U [½ + 6, 1] for any n 1> 1. In fact, let us put 6 = C(Xo) - ½. We assert that a(xo) • U for any a • M. This fact follows from (3.6) and (3.7). Hence, the mapping m~ o defined via (3.5) is a two-valued probability measure on M. (ii) Suppose that rid=0 and let {an}n=l~M/Io be given. Then { a } U U , { a , , b , } is also a countable generator of M. Define d = n ~ _ l ( a , U a ' , ) n n ~ - i (bn U b',). Then d is a W-universum of M. Since ti =/=0, we can find a point x~ • X such that a(xO > ½ and c(xO > ½. Similarly as in (i), there exists a constant 6, 0 < 6 < ½ , such that a,(x~), b n ( X l ) • U : = [ O , ½ - 6 ] U [ ½ + 6 , 1]. L e m m a 3.3 entails b(xO • U for any b • M. Define e(n) • {0, 1} such that a~(n~(xl) > ½. Then a n n ~ = l a e~") ~ I0. It is easy to verify that the Boolean o-algebra M/lo is countably generated, and its generator is, for example, {t~,: n i> 1}, where {a,} is a generator of M. Due to [6, Theorem 24.5], any o-distributive Boolean o-algebra is o-isomorphic to some o-algebra of crisp subsets of some set. [] T h e o r e m 3.5. Suppose there is a W-universum c • M such that

c<~aUa ' foranya•M.

(3.8)

Then M possesses at least one (two-valued) probability measure, and M/lo is o-distributive. Proof. Choose a point x0 • X such that C(Xo) > 1. Then by (3.8), the mapping mx,, defined via (3.5) is a two-valued probability measure on M. Now let ti 4= 0 and {ti,} ~ M/lo be given. There exists a point xl • X such that a(xO > 1 and c(xl) > 1. Choosing e(n) • {0, 1} such that ae(")(Xl) > ½, we have, due to (3.8) and (2.1), a n n ~ = l a,~~n) ~ I0. [] Remark 3.6. Any M consisting only of crisp subsets fulfils (3.8). L e m m a 3.7. If M is countably generated soft f u z z y o-algebra, then there is a W-universum c with (3.8). Proof. Suppose that {b.}~=L is a generator of M and put c = n ~ - i (b~ 0 b'.). We claim that this c is that in question. Indeed, put K = {a c M: a U a ' ~>c}. It is evident that (i) 0,1 e K; (ii) if a e K, then a' e K; (iii) {b.} ~_ K; (iv) if {an} ~_ K, then U n a . • K. This follows from the following: let x • X. Then either there exists an integer no such that a.,~(x)>~c(x) which entails (Un a . ) ( x ) ~ an,~(x)

A. Dvure6enskij

180

c(x), or a,(x) < c(x) for any n. Therefore, c(x), so that, (Un a, U (Un a,)')(x) >>-c(x). The properties (i)-(iv) give us K = M.

(Un an)(x) <~c'(x)

and (An a')(x) >I

[]

Let a<~M, put Ma = {b e M: b U b ' = a U a'}. Then Ma and Mc are either identical or disjoint, and M = UaEMMa. We say that a soft fuzzy o-algebra M is countably decomposable if there is a sequence {cn}~=~ ~_ W~(M) such that

M = U Me.

(3.9)

n

It is evident that any M consisting only from crisp subsets is countably decomposable.

Theorem 3.8. Any countably decomposable soft fuzzy o-algebra possesses at least

one (two-valued) probability measure, and M /Io is o-distributive. Proof. Suppose (3.9) holds in M. If we put c = An c , , then, for any a ~ M , we have c ~< a U a'. Our result now follows from T h e o r e m 3.5. [] The following example gives a soft fuzzy o-algebra which does not fulfill the above theorems, and for which K(M), M/lo and P(M) are described.

Example 3.9. Let X be an uncountable set. For any A ~ X, such that either it or its complement has countably many points, we define

aA(x)

=

0 1

ifx ~A} if X e A if A is countable,

1

ifxeA}

aA(X) = 1 if X ~ A if

Ac

is countable.

Then U n a A = a U . A . A system M = { a A : A ~ X , A or X - A is countable} is a soft fuzzy o-algebra of fuzzy subsets of a set X. M possesses only one (two-valued) probability measure m such that m ( a a ) = 0 if A is countable, otherwise m(aa) = 1. It is evident that P(M) is not order-determining. Moreover, K(M) = {A ~ X : A or X - A is countable}, Io = {aa: A is countable}, M/lo=

{0,1}. Problem. Does any soft fuzzy o-algebra possess at least one probability measure?

Acknowledgement The author is very indebted to the referee for his valuable comments.

Probability measures on fuzzy measurable spaces

181

References [1] A. Dvure~enskij and B. Rie~an, On joint observable in F-quantum spaces, Busefal 35 (1988) 10-14. [2] A. Dvure~enskij and B. Rie~an, On joint distribution of observables for F-quantum spaces, Fuzzy Sets and Systems 39 (1991) 65-73. [3] K. Piasecki, Probability of fuzzy events defined as denumerable edditivity measure, Fuzzy Sets and Systems 17 (1985) 271-284. [4] K. Piasecki, On fuzzy P-measures, in: Proc. First Winter School on Meas. Theory, Lipt. J~in Jan. 10-15 (1988) 108-112. [5] J. Pykacz, Quantum logics and soft fuzzy probability spaces, Busefal 32 (1987) 150-157. [6] R. Sikorski, Boolean Algebras (Springer-Verlag, Berlin, 1964). [7] K. Yosida and E. Hewitt, Finitely additive measures, Trans. Amer. Math. Soc. 72 (1952) 46-66.