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.^[1]

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 $x, y > 0$ .^[2]

^ Sherwood, H. (1971). "Complete probabilistic metric spaces". Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete. 20 (2): 117–128. doi:10.1007/bf00536289. ISSN 0044-3719.
^ Schweizer, Berthold; Sklar, Abe (1983). Probabilistic metric spaces. North-Holland series in probability and applied mathematics. New York: North-Holland. ISBN 978-0-444-00666-0.

[1] Sherwood, H. (1971). "Complete probabilistic metric spaces". Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete. 20 (2): 117–128. doi:10.1007/bf00536289. ISSN 0044-3719.

[2] Schweizer, Berthold; Sklar, Abe (1983). Probabilistic metric spaces. North-Holland series in probability and applied mathematics. New York: North-Holland. ISBN 978-0-444-00666-0.

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