On a statistical generalization of metric spaces.(English)Zbl 0063.08119

From the text: In a very interesting paper Karl Menger [Proc. Natl. Acad. Sci. USA 28, 535–537 (1942; Zbl 0063.03886)] has recently introduced a statistical generalization of semi-metric and metric spaces. According to Menger a set $$R$$ of elements (points) is called a statistical semi-metric space if with each pair of points $$p$$ and $$q$$ of the space $$R$$ a real function $$F(x; p, q)$$ is associated satisfying the following conditions:
1. $$F(x;p,q)=0$$ for $$x<0$$ and $$\lim_{x\to\infty} F(x;p,q)=1$$.
2. $$F(x;p,q)$$ is a non-decreasing function of $$x$$ and continuous to the left.
3. $$F(x;p,q)=F(x; q, p)$$ for any pair of points $$p$$ and $$q$$. 4. $$F(x;p,p)=1$$ for any $$x > 0$$.
The function $$F(x;p,q)$$ can be interpreted as the probability distribution function of the distance of $$p$$ and $$q$$; i.e., for any value $$x$$, $$F(x;p,q)$$ denotes the probability that the distance of $$p$$ and $$q$$ is less than $$x$$. In all that follows a distribution function will mean a function of a real variable $$x$$ which satisfies conditions 1 and 2.
As a statistical generalization of the triangular inequality in metric spaces the following inequality has been proposed by Menger: For any three points $$p$$, $$q$$ and $$r$$ we have
5. $$T[F(x;p,q), F(y;q,r)] < F(x+y;p,r)$$ where $$T(a,b)$$ is a function of two variables satisfying certain conditions.
A statistical semi-metric space is called a statistical metric space if inequality 5 is satisfied for all triples, $$p, q$$ and $$r$$.
Menger’s generalization of the triangular inequality has the drawback that it involves an unspecified function $$T(a,b)$$ and one can hardly find sufficient justification for a particular choice of this function. Furthermore the notion of “between” introduced by Menger on the basis of inequality 5 has the properties of the between relationship in metric spaces only under restrictive conditions on the distribution functions $$F(x;p,q)$$.
Here we propose another statistical generalization of the triangular inequality which is free from the above mentioned difficulties.

MSC:

 54E70 Probabilistic metric spaces 60B99 Probability theory on algebraic and topological structures

Zbl 0063.03886
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