## 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

### MSC:

 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