Properties of fuzzy set spaces with Lp metrics

Abstract

This paper discusses the properties of the spaces of fuzzy sets in a metric space with $L_p$-type $d_p$ metrics, $p\ge 1$. The $d_p$ metrics are well-defined if and only if the corresponding Haudorff distance functions are measurable. In this paper, we give several fundamental conclusions on the measurability of these Haudorff distance functions. Then we give some characterizations of total boundedness, relative compactness and compactness in fuzzy set spaces with $d_p$ metrics. At last we present the completions of fuzzy set spaces with $d_p$ metrics.