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**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.

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 |