The equivalence of convergences of sequences of fuzzy numbers and its applications to the characterization of compact sets

In this paper, with the help of the convergence of sequences of fuzzy numbers with respect to the Lebesgue measure, we study the relationship between convergences of sequences of fuzzy numbers with respect to the endograph metric, the sendograph metric and the image metric. We prove that these convergences are equivalent under proper conditions. In addition, by applying our result, we give a new characterization of compact sets in fuzzy number space with the sendograph metric.