Properties of fuzzy set spaces with $L_p$ metrics

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