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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,${\cal F})$ is called a probabilistic f-metric structure provided the mapping ${\cal F}: S\times S\to D\sp+$ (where ${\cal F}(p,q)$ 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,${\cal F},T)$ in which $\sup\sb{a<1}T(a,a)=1$ 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 54E99 Topological spaces with richer structures 20M15 Mappings of semigroups