Properties of fuzzy set spaces with $L_p$ metrics

TL;DR

This paper investigates the fundamental properties of fuzzy set spaces equipped with $L_p$ metrics, focusing on measurability, compactness, and completion, thereby advancing the mathematical understanding of these spaces.

Contribution

It provides new characterizations of measurability, compactness, and the completion process for fuzzy set spaces with $L_p$ metrics, filling gaps in the theoretical framework.

Findings

Measurability of Haudorff distance functions is characterized.

Conditions for compactness in fuzzy set spaces are established.

Descriptions of the completions of fuzzy set spaces with $L_p$ metrics are provided.

Abstract

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.