On the continuity of probabilistic distance

TL;DR

This paper extends a classical result on the continuity of probabilistic distances in probabilistic metric spaces, removing the restriction that the t-norm must be stronger than Łukasiewicz's, thus broadening the applicability of the theorem.

Contribution

The paper generalizes Schweizer and Sklar's continuity theorem to all continuous triangular norms, eliminating the previous restriction on the t-norm's strength.

Findings

The continuity of probabilistic distances holds under all continuous t-norms.

The extension broadens the class of probabilistic metric spaces where the theorem applies.

The result simplifies assumptions needed for convergence in probabilistic metric spaces.

Abstract

The famous result of B.~Schweizer and A.~Sklar [Pacific J Math 10(1960) 313--334 - Theorem 8.2] asserts that, given a probabilistic metric space $(X,\mathcal F,t)$, $\mathcal F=\{F_{p,q}:p,q\in X\}$, we have $F_{p_n,q_n}(x)\to F_{p,q}(x)$ provided that $F_{p,q}$ is continuous at $x$ and $t$ is continuous and stronger then {\L}ukasiwicz's $t$-norm. We extend this result to arbitrary continuous triangular norms, i.e.\ we omit the condition "$t$ is stronger then {\L}ukasiewicz's".