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

Mohammed Bachir (1) , Bruno Nazaret (2)

1 Dynamique et contrôl Optimal
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

Schweizer, Sklar and Thorp proved in 1960 that a Menger space $(G,D,T)$ under a continuous $t$-norm $T$, induce a natural topology $\tau$ wich is metrizable. We extend this result to any probabilistic metric space $(G,D,\star)$ provided that the triangle function $\star$ is continuous. We prove in this case, that the topological space $(G,\tau)$ is uniformly homeomorphic to a (deterministic) metric space $(G,\sigma_D)$ for some canonical metric $\sigma_D$ on $G$. As applications, we extend the fixed point theorem of Hicks to probabilistic metric spaces which are not necessarily Menger spaces and we prove a probabilistic Arzela-Ascoli type theorem.

Keywords: Metrization of probabilistic metric space; Probabilistic 1-Lipschitz map; Probabilistic Arzela-Ascoli type Theorem; msc: 54E70; 46S50

Domains: Functional Analysis [math.FA]

Mohammed Bachir, Bruno Nazaret. Metrization of probabilistic metric spaces. Applications to fixed point theory and Arzela-Ascoli type theorem. 2019. ⟨hal-02112322v2⟩