Probabilistic metric space

A probabilistic metric space is a generalization of metric spaces where the distance is no longer valued in non-negative real numbers, but instead is valued in distribution functions.

Let D+ be the set of all probability distribution functions F such that F(0) = 0: F is a nondecreasing, right continuous mapping from the real numbers R into [0, 1] such that

sup F(x) = 1

where the supremum is taken over all x in R.

The ordered pair (S,d) is said to be a probabilistic metric space if S is a nonempty set and

d: S×SD+

In the following, d(p, q) is denoted by dp,q for every (p, q) ∈ S × S and is a distribution function dp,q(x). The distance-distribution function satisfies the following conditions:

See also

References


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