On the equivalence of convergences of fuzzy sets (Q1357153)

Let \(\{u_n\}\) be a sequence of fuzzy sets with compact nonempty level sets and \(u\) be a fuzzy set, which in addition has continuous level sets i.e. \(\alpha \to L_\alpha (u)\) is continuous in the Hausdorff metric. Here \(L_\alpha(u)\) denotes the \(\alpha\)-level set of \(u.\) In the paper the authors characterize the convergence of \(\{u_n\}\) to \(u\) in the metric \(H(u,v)= \sup_{0\leq\alpha\leq 1}h(L_\alpha (u),L_\alpha (v)).\) For related results see also \textit{P. Diamond} and \textit{P. Kloeden} [Metric spaces of fuzzy sets (1994; Zbl 0873.54019)].