Metrization of probabilistic metric spaces. Applications to fixed point theory and Arzela-Ascoli type theorem

Abstract

This work is devoted to the metrization of probabilistic spaces. More precisely, given such a space (G,D,⋆) and provided that the triangle function ⋆ is continuous, we exhibit an explicit and canonical metric σD on G such that the associated topology is homeomorphic to the so-called strong topology. As applications, we make advantage of this explicit metric to present some fixed point theorems on such probabilistic metric structures and we prove a probabilistic version of the Arzela-Ascoli theorem.