Characterization of compact subsets of fuzzy sets (Q1113122)

Many applications of fuzzy sets restrict attention to the convenient metric space (\({\mathcal E}^ n,D)\) of normal, fuzzy convex sets on the base space \({\mathbb{R}}^ n\), with D the supremum over the Hausdorff distances between corresponding level sets. We mention in particular the fuzzy random variables of \textit{M. L. Puri} and \textit{D. A. Ralescu} [Ann. Probab. 13, 1373-1379 (1985; Zbl 0583.60011)], the fuzzy differential equations of \textit{O. Kaleva} [Fuzzy Sets Syst. 24, 301-317 (1987; Zbl 0646.34019)], the fuzzy dynamical systems of the second author [Fuzzy Sets Syst. 7, 275-296 (1982; Zbl 0509.54040)] and the chaotic iterations of fuzzy sets of Diamond and Kloeden. In these papers specific results are often obtained for compact subsets of \({\mathcal E}^ n\), which raises the question of how to characterize such compact subsets. The purpose of this is to present a convenient characterization of compact subsets of the metric space (\({\mathcal E}^ n,D)\). Our main result is that a closed subset of \({\mathcal E}^ n\) is compact if and only if the support sets are uniformly bounded in \({\mathbb{R}}^ n\) and the support functions of Puri and Ralescu are equileftcontinuous in the membership grade variable \(\alpha\) uniformly on the unit sphere \(S^{n-1}\) of \({\mathbb{R}}^ n\). To this end we note that the support functions provides a means of embedding all of the space \({\mathcal E}^ n\) in a Banach space, which we exhibit explicitly, not just the subspace \({\mathcal E}^ n_{Lip}\) of `Lipschitzian' fuzzy sets considered by Puri and Ralescu.