Some remarks on the triangle inequality in probabilistic metric spaces. (English) Zbl 0622.60007

Semin. Teor. Probab. Apl. 80, 8 p. (1986).
An alternative version of the triangle inequality for a subclass of probabilistic semi-metric spaces is proposed and studied. Let the function f: [0,1]\(\to [0,\infty)\) be continuous and strictly decreasing with \(f(1)=0\). The probabilistic semi-metric space (S,\({\mathcal F})\) is called a probabilistic f-metric structure provided the mapping \({\mathcal F}: S\times S\to D^+\) (where \({\mathcal F}(p,q)\) is denoted Fpq) satisfies the following triangle inequality: Given \(\epsilon >0\) there is a \(\delta >0\) such that \(f(Fpr(\epsilon))<\epsilon\), whenever \(Max(f(Fqr(\delta)),f(Fqr(\delta)))<\delta.\)
It is shown that every Menger (\(\theta\)-Menger) space (S,\({\mathcal F},T)\) in which \(\sup_{a<1}T(a,a)=1\) is also a probabilistic f-metric structure for any f and that all such probabilistic f-metric structures generate a uniformity for S which is equivalent to the \(\epsilon\)-\(\lambda\) uniformity. Two recipes for constructing a metric for the \(\epsilon\)- \(\lambda\) uniformity are given which use the function f.
Reviewer: R.Tardiff


60A99 Foundations of probability theory
54A05 Topological spaces and generalizations (closure spaces, etc.)
54E99 Topological spaces with richer structures
20M15 Mappings of semigroups


Zbl 0622.60008