# Probabilistic metric space

Probabilistic metric space

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 "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" ⇔ "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.

* Probability metric

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