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 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×S →D+
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:
- du,v(x) = 0 for all x ⇔ u = v (u, v ∈ S).
- du,v(x) = dv,u(x) for all x and for every u, v ∈ S.
- du,v(x) = 1 and dv,w(y) = 1 ⇒ du,w(x + y) = 1 for u, v, w ∈ S and x, y ∈ R.
