Probabilistic metric space

In mathematics, probabilistic metric spaces are a generalization of metric spaces where the distance no longer takes values in the non-negative real numbers $R \geq 0$ , but 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 R into [0, 1] such that max(F) = 1).

Then given a non-empty set S and a function F: S × S → D+ where we denote F(p, q) by F_p,q for every (p, q) ∈ S × S, the ordered pair (S, F) is said to be a probabilistic metric space if:

For all u and v in S, $u = v$ if and only if $F u, v (x) = 1$ for all x > 0.
For all u and v in S, $F u, v = F v, u$ .
For all u, v and w in S, $F u, v (x) = 1$ and $F v, w (y) = 1 \Rightarrow F u, w (x + y) = 1$ for $x, y > 0$ .