Probabilistic Geometry

Abstract

We generalize the concept of a metric space by associating a cumulative distribution function, Δ_ab, with every ordered pair (a, b) of elements of a set S. The value Δ_ab(x) may be interpreted as the probability that the distance from a to b be < x. But S need not be a metric space in the ordinary sense. The distribution functions and the association of these functions with the pairs of elements of S are all that is assumed. For instance, a probabilistic metric consisting of five elements is a 5-by-5 matrix of distribution functions—as an ordinary metric space consisting of five points is a 5-by-5 matrix of numbers.