Probabilistic Metric Space

A probabilistic metric space is a generalization of metric spaces where the distance is no longer valued in positive 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, left continuous mapping from the real numbers R into 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×S →D+

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

• d_u,v(x) = 1 for all x > 0 ⇔ u = v (u, v ∈ S).
• d_u,v(x) = d_v,u(x) for all x and for every u, v ∈ S.
• d_u,v(x) = 1 and d_v,w(y) = 1 ⇒ d_u,w(x + y) = 1 for u, v, w ∈ S and x, y ∈ R.