Completeness with respect to the probabilistic Pompeiu-Hausdorff metric

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 $H$.