Probabilistic metric space

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A probabilistic metric space is a generalization of metric spaces where the distance is no longer defined on positive real numbers, but on 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 sup(F(x)) = 1 for x∈R.

The ordered pair (S,F) is said to be a probabilistic metric space if S is a nonempty set and F: S×S → D+ (F(p, q) is denoted by F_p,q for every (p, q) ∈ S × S) satisfies the following conditions:

• F_u,v(x) = 1 for every x > 0 ⇔ u = v (u, v ∈ S).
• F_u,v = F_v,u for every u, v ∈ S.
• F_u,v(x) = 1 and F_v,w(y) = 1 ⇒ F_u,w(x + y) = 1 f or u, v, w ∈ S and x, y ∈ R+.