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Properties of fuzzy set spaces with $L_p$ metrics

Huan Huang

TL;DR

This paper gives some fundamental conclusions on the measurability of this function and gives the characterizations of compactness in fuzzy set space with d_p$ metrics.

Abstract

This paper discusses the properties of the spaces of fuzzy sets in a metric space with $L_p$-type $d_p$ metrics, $p\geq 1$. 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.

Properties of fuzzy set spaces with $L_p$ metrics

TL;DR

This paper gives some fundamental conclusions on the measurability of this function and gives the characterizations of compactness in fuzzy set space with d_p$ metrics.

Abstract

This paper discusses the properties of the spaces of fuzzy sets in a metric space with -type metrics, . The 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 metrics. At last we present the completions of fuzzy set spaces with metrics.
Paper Structure (8 sections, 61 theorems, 70 equations)

This paper contains 8 sections, 61 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Fuzzy sets and metrics on them
  3. Measurability of function $H([u]_\alpha, [v]_\alpha)$
  4. Properties of $d_p^*$ metric and $H_{\rm end}$ metric
  5. Characterizations of compactness in $(F^1_{USCG} (X)^p, d_p)$
  6. Characterizations of compactness in $(F^1_{USCG} (\mathbb{R}^m)^p, d_p)$
  7. Completion of $(F^1_{USCG} (X)^p, d_p)$
  8. Conclusions

Key Result

Proposition 3.1

Let $u \in F^1_{USC}(X)$ and $x_0 \in X$. Then $H([u]_\alpha, \{x_0\})$ is a decreasing function of $\alpha$ on $[0,1]$. So $H([u]_\alpha, \{x_0\})$ is a measurable function of $\alpha$ on $[0,1]$, which is equivalent to that $d_p(u, \widehat{x_0})$ is well-defined.

Theorems & Definitions (133)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Example 2.5
  • Remark 2.6
  • Proposition 3.1
  • proof
  • Remark 3.2
  • Remark 3.3
  • ...and 123 more