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A possibility measure is not a fuzzy measure
Fuzzy Sets and Systems 7 (1982) 311-313 North-Holland Publishing Company
311
SHORT COMMUNICATION A POSSIBILITY MEASURE IS NOT A F U Z Z Y MEASURE*
Madan L. PURI Department of Mathematics, Indiana University, Bloomington, IN 47405, U.S.A.
Dan RALESCU Department of Mathematics, University of Cincinnati, Cincinnati, OH 45221, U.S.A. Received December 1980 Revised J.omuary 1981 In this note we show that a possibility measure is not a particular type of fuzzy measure, except in trivial cases.
Keywords: Possibility measure, Fuzzy measure, Capacity.
L In~oduc~a
Let X be a set and let ~ be a (r-algebra of subsets of X. A fuzzy measure (Sugeno [6]) is a set function ~:,~ ~ [0, 1] with the properties:
(FM1) (FM2)
A ,'- B=> I.t(A)<~ I.t(B),
(FM3)
A 1 c A2 c . . "::) P~(?=l A,)=,~__~lim~(A,),
(FM4)
A l m A 2 ~ ..-=~p,(f~A , , ) = l i m.--~ /~(A.).=
Let ~(X) denote the set of all subsets of X, A possibility measure (Zadeh [7]) is a set functien ~ - : ~ ( X ) ~ [0, 1] with the properties:
(P1) (P2)
A = B::~r(A)<~ ~r(B),
(P3)
1r( A [.J, )
= sup "a'(A).
i~l
i~l
Clem'ly, any possibility measure is uniquely determined by a function f : X - - ~ [0, 1], via the formula w(A)=supf(x),
A=X.
x~A
* "['his work was supported by the National Science Foundation grant number IST-7918468.
0165-0114/82/0000-0000/$02.75 © 1982 North-HoUand
(*)
312
M.L. Purl, D. Ralescu
Indeed, it suffices to take f(x)=~r({x}), x ~ X . The next question arises, then, naturally: is any possibility measure a fnzzy measure? It is easy to see that any possibility measure satisfies (FM3). It was pointed out in [1,p. 1629; 3, p. 665; 6, p. 12] that the set function ~ defined by (*) is a fa~r_~ymeasure. We prove in the next section that this happens only in trivial cases. 2. M M remit We first give two counterexamples which show that, even in 'nice' cases, a possibility measure is not a fuzzy measure. Example 1. Take X = R , f (x)= 1, and ,r as defined by (.). Consider An = (n, oo); observe that ,r(N~=t A~) = 0 , while lim._~ ,r(A~)= 1, thus (FM4) is violated. Example 2. Take X = [0, 1], f(x)= 1 for x ~ [0, 1), and f ( 1 ) = 0. Consider A,, = [1 - l/n, 1]; thus 1r(NT,=1 An) = 0, while lim,__.,. 1r(A~) = 1. In the rest of this section, we suppose that X = R k (the k-dimensional euclidean space). The reader who feels that this asgumption is too restrictive may take X to be any metric space without isolated points. Theorem. Let f:Rk--*[0, 1] and , r ( A ) = s u p ~ A f ( x ) , A c R k be the associated possibility measure. If ar is a fuzzy measure, then f(x)= 0 at every point of continuity of f. Proof. Take a point Xo~ @k such that f is continuous at Xo. Define A , = {x ~ Rkl I[x - Xoll< l/n, x :/: Xo} (here [I Udenotes the euclidean norm in Rk). Obviously A , # 0; also A 1 D A 2 D . . . and N ~ = I A ~ = 0 . Since ,r satisfies (FM4), it follows that limn_
= 0.
Let (xj)j be a sequence such that xo = limi__~ x~, xj # Xo. If n ~> 1 is fixed, then x~.A~ for all ]>>-j,~. Thus f(Xj)<~SU~,:~A~f(x) for j>~j,. It follows that 0 ~< l i m s u p j _ _ ~ f ( x ~ ) < ~ s u p , ~ ( x ) = 1r(A~). Since this is true for any n>~l, we conclude that lim supi._~ f(xi)= 0, thus l i % , ~ f(x~)= 0. Finally, since f is continuous at Xo, it follows that f(xo)= limy-.~o f ( y ) = 0, which ends the proof. Let ~ be a possibility measure with a continuous 'dem~ity' f. I f ~ is a f u z z y measure, then ,r = O.
CorollmT.
Proof. Obvious. The next question is: when is a possibility measure a f , zzy measure? The implication is always true if the set X is finite. In general, as it c~m be seen from the above, ~- fails to satisfy (FM4). However, if f is upper semicontinuous, any possibility measure ar (given by (*)) is a capacity (i.e. satisfies (FM4), where the A , ' s are compact subsets of the topological space X; for more details, see [4, 5]).
A poss/bi//~ measure/s not a/uzzy measure
313
Rderenees [1] A. Kandel, Fuzzy sets, fuzzy algebra, and fuzzy statistics, Proc. of the IEEE 66 (1978) 1619-1639. [2] C. V. Negoita and D. A. Ralescu, Applications of Fuzzy Sets to Systems Analysis (Wiley, New York, 1975). [3] H. T. Nguyen, On fiuziness and linguistic probabilities, J. Math. Anal. Appl. 61 (1977) 658-671. [4] H. T. Nguyen, Some mathematical tools for linguistic probabilities, Fuzzy Sets and Systems 2 (1979) 53--65. [5] D. Ralescu, Measures, capacities, and optimization with inexact constraints, to appear. [6] M. Sugeno, Theory of fuzzy integrals and its applications, Ph.D. Thesis, Tokyo Inst. of Technology (1974). [7] L. A. Zadeh, Fuzzy se~,s as a basis for a theory of possibility, Fuzzy Sets and Systems 1 (1978) 3-28.
311
SHORT COMMUNICATION A POSSIBILITY MEASURE IS NOT A F U Z Z Y MEASURE*
Madan L. PURI Department of Mathematics, Indiana University, Bloomington, IN 47405, U.S.A.
Dan RALESCU Department of Mathematics, University of Cincinnati, Cincinnati, OH 45221, U.S.A. Received December 1980 Revised J.omuary 1981 In this note we show that a possibility measure is not a particular type of fuzzy measure, except in trivial cases.
Keywords: Possibility measure, Fuzzy measure, Capacity.
L In~oduc~a
Let X be a set and let ~ be a (r-algebra of subsets of X. A fuzzy measure (Sugeno [6]) is a set function ~:,~ ~ [0, 1] with the properties:
(FM1) (FM2)
A ,'- B=> I.t(A)<~ I.t(B),
(FM3)
A 1 c A2 c . . "::) P~(?=l A,)=,~__~lim~(A,),
(FM4)
A l m A 2 ~ ..-=~p,(f~A , , ) = l i m.--~ /~(A.).=
Let ~(X) denote the set of all subsets of X, A possibility measure (Zadeh [7]) is a set functien ~ - : ~ ( X ) ~ [0, 1] with the properties:
(P1) (P2)
A = B::~r(A)<~ ~r(B),
(P3)
1r( A [.J, )
= sup "a'(A).
i~l
i~l
Clem'ly, any possibility measure is uniquely determined by a function f : X - - ~ [0, 1], via the formula w(A)=supf(x),
A=X.
x~A
* "['his work was supported by the National Science Foundation grant number IST-7918468.
0165-0114/82/0000-0000/$02.75 © 1982 North-HoUand
(*)
312
M.L. Purl, D. Ralescu
Indeed, it suffices to take f(x)=~r({x}), x ~ X . The next question arises, then, naturally: is any possibility measure a fnzzy measure? It is easy to see that any possibility measure satisfies (FM3). It was pointed out in [1,p. 1629; 3, p. 665; 6, p. 12] that the set function ~ defined by (*) is a fa~r_~ymeasure. We prove in the next section that this happens only in trivial cases. 2. M M remit We first give two counterexamples which show that, even in 'nice' cases, a possibility measure is not a fuzzy measure. Example 1. Take X = R , f (x)= 1, and ,r as defined by (.). Consider An = (n, oo); observe that ,r(N~=t A~) = 0 , while lim._~ ,r(A~)= 1, thus (FM4) is violated. Example 2. Take X = [0, 1], f(x)= 1 for x ~ [0, 1), and f ( 1 ) = 0. Consider A,, = [1 - l/n, 1]; thus 1r(NT,=1 An) = 0, while lim,__.,. 1r(A~) = 1. In the rest of this section, we suppose that X = R k (the k-dimensional euclidean space). The reader who feels that this asgumption is too restrictive may take X to be any metric space without isolated points. Theorem. Let f:Rk--*[0, 1] and , r ( A ) = s u p ~ A f ( x ) , A c R k be the associated possibility measure. If ar is a fuzzy measure, then f(x)= 0 at every point of continuity of f. Proof. Take a point Xo~ @k such that f is continuous at Xo. Define A , = {x ~ Rkl I[x - Xoll< l/n, x :/: Xo} (here [I Udenotes the euclidean norm in Rk). Obviously A , # 0; also A 1 D A 2 D . . . and N ~ = I A ~ = 0 . Since ,r satisfies (FM4), it follows that limn_
= 0.
Let (xj)j be a sequence such that xo = limi__~ x~, xj # Xo. If n ~> 1 is fixed, then x~.A~ for all ]>>-j,~. Thus f(Xj)<~SU~,:~A~f(x) for j>~j,. It follows that 0 ~< l i m s u p j _ _ ~ f ( x ~ ) < ~ s u p , ~ ( x ) = 1r(A~). Since this is true for any n>~l, we conclude that lim supi._~ f(xi)= 0, thus l i % , ~ f(x~)= 0. Finally, since f is continuous at Xo, it follows that f(xo)= limy-.~o f ( y ) = 0, which ends the proof. Let ~ be a possibility measure with a continuous 'dem~ity' f. I f ~ is a f u z z y measure, then ,r = O.
CorollmT.
Proof. Obvious. The next question is: when is a possibility measure a f , zzy measure? The implication is always true if the set X is finite. In general, as it c~m be seen from the above, ~- fails to satisfy (FM4). However, if f is upper semicontinuous, any possibility measure ar (given by (*)) is a capacity (i.e. satisfies (FM4), where the A , ' s are compact subsets of the topological space X; for more details, see [4, 5]).
A poss/bi//~ measure/s not a/uzzy measure
313
Rderenees [1] A. Kandel, Fuzzy sets, fuzzy algebra, and fuzzy statistics, Proc. of the IEEE 66 (1978) 1619-1639. [2] C. V. Negoita and D. A. Ralescu, Applications of Fuzzy Sets to Systems Analysis (Wiley, New York, 1975). [3] H. T. Nguyen, On fiuziness and linguistic probabilities, J. Math. Anal. Appl. 61 (1977) 658-671. [4] H. T. Nguyen, Some mathematical tools for linguistic probabilities, Fuzzy Sets and Systems 2 (1979) 53--65. [5] D. Ralescu, Measures, capacities, and optimization with inexact constraints, to appear. [6] M. Sugeno, Theory of fuzzy integrals and its applications, Ph.D. Thesis, Tokyo Inst. of Technology (1974). [7] L. A. Zadeh, Fuzzy se~,s as a basis for a theory of possibility, Fuzzy Sets and Systems 1 (1978) 3-28.