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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][0,) be continuous and strictly decreasing with f(1)=0. The probabilistic semi-metric space (S,) is called a probabilistic f-metric structure provided the mapping :S×SD + (where (p,q) is denoted Fpq) satisfies the following triangle inequality: Given ϵ>0 there is a δ>0 such that f(Fpr(ϵ))<ϵ, whenever Max(f(Fqr(δ)),f(Fqr(δ)))<δ·

It is shown that every Menger (θ-Menger) space (S,,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 ϵ-λ uniformity. Two recipes for constructing a metric for the ϵ- λ uniformity are given which use the function f.

Reviewer: R.Tardiff

MSC:
60A99Foundations of probability theory
54A05Topological spaces and generalizations
54E99Topological spaces with richer structures
20M15Mappings of semigroups