Completeness with respect to the probabilistic Pompeiu-Hausdorff metric

TL;DR

This paper proves the completeness of various families of closed, bounded, compact, and convex subsets within complete probabilistic metric and normed spaces using the probabilistic Pompeiu-Hausdorff metric, extending classical results.

Contribution

It establishes the completeness of these families in probabilistic metric and normed spaces, including convex subsets in the sense of ;erstnev, which was previously unaddressed.

Findings

Families of closed nonempty subsets are complete under the probabilistic Pompeiu-Hausdorff metric.

Completeness extends to bounded, compact, and convex subsets in probabilistic metric and normed spaces.

Results generalize classical metric space completeness to probabilistic settings.

Abstract

The aim of the present paper is to prove that the family of all closed nonempty subsets of a complete probabilistic metric space $L$ is complete with respect to the probabilistic Pompeiu-Hausdorff metric $H$. The same is true for the families of all closed bounded, respectively compact, nonempty subsets of $L$. If $L$ is a complete random normed space in the sense of \v{S}erstnev, then the family of all nonempty closed convex subsets of $L$ is also complete with respect to $H$.