Metric spaces of fuzzy sets. (English) Zbl 0704.54006

Diamond, Phil; Kloeden, Peter

Fuzzy Sets Syst. 35, No. 2, 241-249 (1990).

This article studies two classes of metrics on the space of normal fuzzy convex sets over \({\mathbb{R}}^ n\). For \(1\leq p<\infty\), \(d_ p(u,v)\) is the Lp-norm of the Hausdorff distance between the level sets of u and v. The metric \(\rho_ p(u,v)\) is the Lp-norm of the Lp-distance between the support sets of the level sets of u and v. The authors show that, for each p, \(d_ p\) and \(\rho_ p\) induce the same topology which is complete, separable and locally compact. As a consequence a strong law of large numbers for fuzzy random variables [E. P. Klement, M. L. Puri and D. A. Ralescu, Proc. R. Soc. Lond., Ser. A 407, 171-182 (1986; Zbl 0605.60038)] holds in all these spaces. A characterization of compact and locally compact subsets in terms of boundedness and p-mean equi-left continuity is also given.

Reviewer: A.J.Klein

MSC:

54B20	Hyperspaces in general topology
03E72	Theory of fuzzy sets, etc.
54E35	Metric spaces, metrizability
54A40	Fuzzy topology

Keywords:

space of normal fuzzy convex sets; Lp-norm of the Hausdorff distance; compact and locally compact subsets; boundedness; p-mean equi-left continuity

Citations:

Zbl 0605.60038

References:

[1]	Diamond, P., Fuzzy chaos, (1987), submitted for publication
[2]	Diamond, P., Fuzzy least squares, Inform. sci., 46, 141-149, (1988) · Zbl 0663.65150
[3]	Diamond, P.; Kloeden, P., Characterization of compact subsets of fuzzy sets, Fuzzy sets and systems, 29, 341-348, (1989) · Zbl 0661.54011
[4]	Graves, L.M., The theory of functions of real variables, (1946), McGraw-Hill New York · Zbl 0063.01720
[5]	Kaleva, O., Fuzzy differential equations, Fuzzy sets and systems, 24, 301-317, (1987) · Zbl 0646.34019
[6]	Klement, E.P.; Puri, M.L.; Ralescu, D.A., Limit theorems for fuzzy random variables, (), 171-182 · Zbl 0605.60038
[7]	Kloeden, P.E., Fuzzy dynamical systems, Fuzzy sets and systems, 7, 275-296, (1982) · Zbl 0509.54040
[8]	Kloeden, P.E., Chaotic mappings on fuzzy sets, (), 368-371, Tokyo
[9]	Lay, S.R., Convex sets and their applications, (1982), John Wiley New York · Zbl 0492.52001
[10]	Lyashenko, N.N., Statistics of random compacts in Euclidean space, J. soviet math., 21, 76-92, (1983) · Zbl 0506.60007
[11]	Puri, M.L.; Ralescu, D.A., The concept of normality for fuzzy random variables, Ann. probab., 13, 1373-1379, (1985) · Zbl 0583.60011
[12]	Vitale, R.A., Lp metrics for compact, convex sets, J. approx. theory, 45, 280-287, (1985) · Zbl 0595.52005