# Probabilistic Arzela-Ascoli theorem

## TL;DR

This paper establishes a probabilistic version of the Arzela-Ascoli theorem, characterizing the compactness of 1-Lipschitz functions on probabilistic metric spaces based on the compactness of the underlying space.

## Contribution

It introduces a probabilistic Arzela-Ascoli theorem, linking the compactness of 1-Lipschitz functions to the compactness of the probabilistic metric space.

## Key findings

- The space of 1-Lipschitz functions is compact if and only if the underlying space is compact.
- Provides a new characterization of compactness in probabilistic metric spaces.
- Extends classical Arzela-Ascoli theorem to probabilistic setting.

## Abstract

We prove that, in the space of all probabilistic continuous functions from a probabilistic metric space G to the set $\Delta$ + of all cumulative distribution functions vanishing at 0, the space of all 1-Lipschitz functions is compact if and only if the space G is compact. This gives a probabilistic Arzela-Ascoli type Theorem.