Probabilistic Metric Space
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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. 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 n...
Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. 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).
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