Measures of fuzzy sets

219

Fuzzy Sets and Systems 9 (1983) 219-227 North-Holland Publishing Company

MEASURES

OF FUZZY

SETS

QU Yinsheng Department

of Biochemistry,

Tianjin

Medical

College,

Tianjin,

People’s

Republic

of China

Received December 1980 Revised December 1981 In this paper, a fuzzy set is considered as a point of a segment in Riesz space. Using the Daniel1 representation theorem of positive linear functionals on Riesz space, it is demonstrated that v(A), the u-finite measure of fuzzy set A, equals JoA(u)v(du). Some fundamental properties of measures and signed measures of fuzzy sets are discussed. The law of large numbers in fuzzy occasions and a few useful properties of the conditional probability of fuzzy events are obtained. Keywords: Soft u-algebra, Soft u-field, Fuzzy measurable Measure of a fuzz-y set, Almost everywhere crisp.

space, Fuzzy measure space,

Introduction In 1968, L.A. Zadeh defined the probability of fuzzy events. In the so called soft sciences, such as, biology, medicine, psychology, economics, etc., this idea is especially valuable. Nevertheless, since Zadeh’s original work, there has been limited research in this area. In this paper, we build a mathematically sound foundation for the measure of fuzzy sets. It is needless to say that the concept of the measure of fuzzy sets is distinct from that of the fuzzy measure of sets developed by Sugeno [l] 1. De Morgan

algebra, Riesz space and fuzzy sets

Suppose in a distributive lattice M, there exist a maximum minimum element 0, and for each a E M, there is a complement the following conditions: (1) 0==1, I’= 0, (2) ( ac)c= a, (3) (ai v at)‘=

element i and a a’ which satisfies

a; A a& (a, A a2)c = at v a; (De Morgan law). Then M constitutes a De Morgan algebra. When M is a a-complete lattice, it is called a De Morgan u-algebra or soft o-algebra. If a v uc = I and a A a’ = 0, then we call it the Boolean element in M. When all of its elements are Boolean, M is called a Boolean algebra. In any De Morgan algebra, there are at least two Boolean elements: 0 and I. All Boolean elements in M form a Boolean algebra called the kernel of M. 01650114/83/0000-0000/$03.00

0 1983 North-Holland

Y. Qu

220

Let Q, b be two elements of lattice L, and as b. The set {XEL:

xsb

and a~x}

is called a segment in L, denoted by [a, b]. Suppose that 9 is a Riesz space (i.e. a semiordered linear space), 0 is the zero element, and I is an arbitrary element such that I> 0. For every Q E [0, I], we define uC= I- a. Then the segment [0, I] is a De Morgan algebra. Let % be the set of all real-valued functions defined in the set U. Then % is a complete Riesz space. Let IU be the constant mapping such that I,(u) = 1, for all is a complete De Morgan algebra u E U. Then [0, I”] (usually denoted 9 t,& whose elements are called the fuzzy sets in U. For A E [0, I,,], and u E U, A(u) is the membership of u with respect to A. If {Ai} is a countable (or finite) class of fuzzy sets, and xi Ai < I,,, then A = xi Ai is the sum of {Ai}, and {Ai} is a partition of A. If (Y is a non-negative real number, A E~o.lu~, and aA s L then aA is the scalar product of A by CY. For A B E~eCo,r,l,AB is the algebraic product of A and B. The union and intersection of fuzzy sets A and I3 are denoted by A VII and A A B, respectively.

2. The soft u-field A family of fuzzy sets 9- is called the soft a-field. when the following conditions are satisfied: (1) I”E9-; (2) if A, I3 ES-, and AsB, then B-AE~-; (3) if {Ai}e9-, then VT=i AiE~;_; (4) if A, B EL+;_, and A+B
Proposition 2.1. Any soft a-field 9-

is generated by its kernel Sk.

of fuzzy sets

Measures

221

Proof. As indicated above, (Sk)- c 9-. Now we will demonstrate 9-c (Sk>-. Suppose A ~9~; we must prove that A is 9,-measurable, i.e. A, = {u: A(u) > CY}belongs to Sk, or & E 9-. In fact, (YES-, so CYVAES-, avA-a ~9~. It follows that B = 1” - ((Yv A - a) E 9-. Therefore, lim B” = ji B”EK. nII=1 But

so m

A,=I,-/h\

It=1

B”E%.

Cl

We call (U, 9-) the fuzzy measurable space, where 5% is a soft o-field. Suppose W, (Sx)-) and W, (SyJ-1 are two fuzzy measurable spaces. For A E (5Q)-. and B E (SQL, we define [AxB](X,

Y)=A(X).B(Y)

as the direct product of A and B. We call A X B the fuzzy measurable rectangle in the product space XX Y. The smallest soft o-field containing all fuzzy measurable rectangles in XX Y is called the product soft o-field in XX Y, denoted by ($x)-x (%I-. It can be proved that the product soft a-field is generated by the product o-field 9X x.9,, that is, (sx)- X (Py)- = (9X X sy)-.

3. The measures of fuzzy sets

Suppose (U, 9J is a fuzzy measurable space. An extended real-valued function v defined on 8- is called a signed measure, if it satisfies the following conditions (u can only take one infinite value of +m and --co, and there is at least one fuzzy set A ES-, such that v(A) is finite): (1) if A, B E 9-, and A + B < &,, then v(A + B) = v(A) + v(B)

(additivity);

(2) if A,, ES;-, and A, t A, then 4%)

t u(A)

(continuity

from below).

When v is non-negative, it is called a measure. If u is a (signed) measure on 8-, and 9 is the kernel of 9-, then v is a (signed) measure on 9. When v is a measure on the soft a-field 9-, we call (U, 9-, V) the fuzzy measure space. When I/ is a probability measure, the elements in 8- are called fuzzy events.

Y. Qu

222

Proposition

3.1. Suppose v is a signed measure on 9-, {A)=%-,

and E=I 4 s

1”. Then

v f A, = f v(Ai) ( i=l ) i=l IfB,JO,BnES-, =1,2 ,..., v(h) Proposition

non-negative

4 0

(denumerable

additiuity).

and there exists B, such that v(Bk)
(continuity

then

from above).

3.2. Suppose v is a signed measure on 9-, real number, such that CYA
A ES-

and 01 is a

(positive homogeneity).

v(aA) = W(A)

(3.1)

In fact, (3.1) can be deduced from additivity and continuity from below. We might regard the signed measure on 9- as a restriction of a certain linear functional on Riesz space R’ which is formed by all g-measurable functions to the segment [0, IU] (=$;-). For measure v, it might be viewed as a restriction of some positive linear functional on 3’. Proposition 3.3. If v is a signed measure on 9v+(A) = sup{v(B): B E 9-, B G A}.

v-(A) = -inf{v(B):

B ES-,

Then v+ and v- are measures on 9v= 1/+-v-.

for each A ES-,

B G A}. and

Proof. Since v(O) = 0, vf ZO and v’(O) = 0. Assume A,+A,=GI,. For B,sA,, B,
v(BJ

= v(B, +B,)s

let

that

Al, A2~9-,

and

v+(A, +A*).

It follows that v’(A,) + v+(A2) s V+ (A, +A*). Now we take B GA, + AZ, and define B,=BAA,, B,=Ov(B-A,)sA,. So that B,
It

G v+(A,)

+ v+(A,).

follows

that v+(A1 + A*) 6 v’(A,) + v+(AJ. That is, v+(A, + v+(A2). Let B, t A, then v+(A,)< v’(A), so that lim, v+(A,) G v’(A). Let B
+ AZ) =

v+(A,)

v-(A) = -inf(v(B):

B ELF-, B
= (-v)‘(A),

v- is also a measure on S-. Furthermore, v-(A) = -inf{v(A) - v(A -B): B E 9-, B
El

B E .9-, B, t B.

v+(A,),

Measures

of fuzzy sets

Proposition 3.4. Let v be a a-finite (U, %). Then, for any A E 9-, v(A) =

measure

A(u)v*(du)

JlJ

where v* is the restriction

on the fuzzy

223

measurable

space

(3.2)

of v to the kernel 9.

Proof. Suppose that 9’ represents the Riesz space formed by a class of realvalued functions on U, 1” E 9’, and E is a positive linear functional on 92’ which satisfies the following conditions: (1) J-3,) = 1; (2) if f,, ES’, and f,, J 0, then E(f,,> k 0. Then, from the Daniel1 representation theorem [3], on the u-field generated by 9%’ (i.e. the smallest u-field which makes all functions in 3 Bore1 measurable) there exists a unique probability measure P, such that every f E 9’ is P-integrable and for the functional E E(f) = j

u

(3.3)

f(u)R(du).

Let 3’ be the set of all 9 measurable functions, then v might be regarded as the restriction of certain positive linear functional E’ to 9[,.rU1. Suppose v is a finite measure, E’(I,) = v(U) = a
In particular, v(A)

= v(A)/a! = I, AP(du). for B Es, =

v*(B)/a

= P(B),

so

A(u)v*(du).

Iu

If v is a a-finite measure, then there exist {A,} c 9, such that U = x=i A,, (for ordinary sets, addition is the union of disjoint sets) and for each O,, v(A,)
Iu

Since v(A) =x=, v(A) =

A(u)v:(du)

=

5A.

A(u)v*(du).

v,,(A), I

“A(u)v*(du).

The proof is completed.

0

The above proposition is an extension theorem. It says that if v is a u-finite measure on the u-field 5, and 9- is the soft u-field generated by 9, then there exists a unique extension of v on 9- which is given by (3.4). This theorem is parallel to the Caratheodory extension theorem, which says that if v is a u-finite measure on the field go, and 9 is the u-field generated by

Y. Qu

224

so, then there exists a unique extension of v on 9. The two theorems can be deduced from the Daniel1 theorem, so the latter is the more profound extension theorem. (3.4) coincides with the definition of probability of fuzzy events given by Zadeh

[41.

4. Examples

(1) Suppose (X, 9-) is a fuzzy measurable space. For a fixed x0 E X, let 6,,(A) = A(x,), A E 9. Then 6, is a probability measure on 9-. It is called the Dirac measure at x0. In particular, for an ordinary measurable set A,

h,,W) = 10

when x0 E A, otherwise.

(2) Suppose (X, Y) is an ordinary measurable C(A)

=

cardinality M

of the set A

space. For A E .Y, we define

when A is finite, otherwise.

C is called the counting measure. Let X = {xi, i E I} be a finite or a countable set. C is a o-finite measure on (X, 9’), where 9’ is the set consisting of all subsets in X. For any A E 9, C(A) = j- A(x)C(dx) X

C(A) is called the cardinality

= c A(xi). is1

of the fuzzy set A.

(3) (X, 9,, V) is a fuzzy measure space, DE.%-, each A ES-, v,(A) = v(AD)/v(D) is a probability

5. Probability

and O< v(D)
of fuzzy events

Suppose 5 is a measurable mapping from the probability space (0,8, P) to (X, gx, P,), i.e. if A E Sx then &-‘(A) E 9, or S(E) = 9 where 9(E) is the u-field generated by 5. PE is the image of P under the mapping 5, i.e. for every A E Sx, P,(A) = P(t-‘(A)). -Suppose (sx)-. is the soft o-field generated by sx. For A E (&)-, we define &-‘(A)-A Q 5, i.e. 5-‘(A)(o) = [A 0 5](o) = A(e(a)> = A(x), where x = t(w) for all 0 E 0. The smallest soft c-field containing a{,$} is called the soft o-field generated by 5.

Measures

Proposition

5.1. If A E (9&)-,

of fuzzy sets

then e-‘(A)

225

E (%{E})-, and

P,(A) = P(5-‘(A)). 2 a} = E-‘{x: A(x) 3 (Y}, and A is 9x measurable, i.e. Proof. Since {o: t-‘(A)(o) we have {w: ~-‘(A)(~)~cx}E~(,$}, that is, [-‘(A) is S(t){x: A(x)aa!}~S~, measurable, therefore t-‘(A) E (9(t))-. F rom the transformation theorem of integration we have

P,(A) =I

A(x)P,(dx)

=

[A 0 [](w)P(dw)

= P@-‘(A)).

X

5.2. The law of large numbers. Let 5 be a random variable as above. Suppose A E 9-,, the n observations of E (or the outcomes of n random experiments performed independently) are x1, x2,. . . , x,. The grade of occurrence of the fuzzy event A in the kth experiment is ck = A(x,). We define ~n=~!~lk=

i

Ah,)

and

3

n

k=l

as the number and the frequency of occurrence of the fuzzy event A in the n experiments, respectively. Since {&} is a sequence of IIIUtUally independent random variables with the same distribution (their distribution function is F(y) = P,({x: A(x)c Y})), and E& = P,(A) < ~4, we have from the law of large numbers

lim ’)zk=, 2 (lk -Eck) =o (a.s.), ,Ii.e. lim %= P,(A)

n-

n

(a.s.).

This is the law of large numbers for fuzzy events. Furthermore, since O
<=

[A(x)-P,(A)FP,(dx)s

1,

we conclude that the sequence {&} observes the central limit

holds with respect to y E R uniformly, Bi = 2 Var & = nj k=l

so

where

[A(x)-P,(A)]‘P,(dx). X

theorem,

i.e..

Y. Qu

226

5.3. Conditional probability. Assume g(o) is a P-integrable function. For any D E 5~, Jn D(w)g(o)P(do) is the integral of g on the fuzzy set D, denoted as jD g(w)P(dw). When P(D) > 0,

1 j g(w)dP P(D) D i’s the average of g on D. For A ~9-, the average of the membership function of A on D is the probability of the fuzzy event A conditioned by fuzzy event D, denoted as P,(A) or P(A/D): PD(A)

P(AD) =P(D) JA(o)P(dw)

1

=p(D)

D

For fuzzy set A, A, = { o: A(o) > 0) is the support of A, Ak = {w : A(w) = 1) is the kernel of A, and Ar = As-AI, is the fuzzy region of A. If P(AJ = 0, we say that the fuzzy event A is almost everywhere crisp with respect to the measure P. When one of the two fuzzy events A and D is almost everywhere crisp, we have P(AD)

= P(A AD).

then the necessary and sufficient condiProposition 5.4. Let A, D E 9~, P(D)>O, tion for P,(A) = 1 is thut D, is contained in A,, except for a null meusure set. Proof.

By definition, p (A) JD.

A(o)Db)dP

D ID,

D(w)

dp

(D is the support of 0). Therefore,

J

PD(A) = 1 is equivalent

to

D(w)(l-A(w))dP=O.

D.

Since P(D,) 5 P(D) > 0, this condition A), except for a null measure set. Corollary.

everywhere

Suppose P(D)>O.

Then crisp with respect To P.

is reduced to D, c Ak (Ak is the kernel of

P,(D)

= 1 if and

only

if D

is almost

If {Ai, i E -r) is a fuzzy measurable partition of 0, i.e. 0 = Cicl Ai, Ai E 9- (i E I) and P(Ai) > 0, i E I, then the class {Ai, i E I) is said to form a complete system of fuzzy events. It is easily seen that if D E 9-, P(D) > 0, and {Hi, i = 1,2, . . . , n} is a complete system of fuzzy events, then P,(H,)=P,(D)P(fl)/

t

PH~(D)P(HJ-

k=l

This is the Bayes formula

for the fuzzy events.

Measures

of fuzzy

sets

227

Suppose 5: (C&9, P) + (X, sx, PE) and 7: (Q.9, P) + (Y, gv, P,,) are two random variables. Let p(w) = (t(o), q(w)) = (x, y); the image of P under the measurable mapping p is denoted by Ph. If D E (.Fv)-, P,,(D) > 0, and A E (.Fx)-, then the probability of fuzzy event A conditioned by fuzzy event D is

P,(A) =

P([-‘(A)

. v-l(D)

P(rl-‘(D))

= Pti(A P,(D)

Two fuzzy events A and D are independent P(AD)

XD) ’

if

= P(A)P(D).

If Y- and 9- are two classes of fuzzy sets, and every A E .9’- and B E 9- are independent, then the classes Y- and 9- are said to be independent. 5.5. If 5 and q are two independent random variables ity space (f&5, P), then (9(t))and (S{q})-. are independent.

Proposition

on the probabil-

Proof. Suppose A E (%{5})-, B E (%{q})-, P(M) = E[AB]. Since A and B are NtJ- and Nd- measurable, respectively, they might be represented as the Bore1 measurable functons of 5 and q, i.e. A = (p(t), B = r,+(q). Because 5 and q are independent, so are q(r) and 4(q). Therefore E[AB]= E[(~(,$)I,!J(~)]=

ECdS)I . E[d~,(rl)l=EA . EB = P(AP@Q. References [l] [2] [3] [4]

M. Sugeno, Fuzzy measures and fuzzy integrals, Trans. SICE (Japan, 1972) 218-226. P.R. Halmos, Measure Theory (Van Nostrand, New York, 1951). R.B. Ash, Measure, Integration and Functional Analysis (Academic Press, New York, 1972). L.A. Zadeh, Probability measures on fuzzy events, J. Math. Anal. Appl. 23 (1968) 421-427.