On the continuity of probabilistic distance

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".