On the definition of probabilistic metric spaces by means of fuzzy measures

Abstract

Metric spaces are defined in terms of a space and a metric, or distance. Probabilistic metric spaces are a useful extension of metric spaces where the distance is a distribution instead of a number. In this way, we can take into account uncertainty. Then, the triangle inequality is replaced by a condition based on triangle functions on the distributions. In this paper we introduce F-spaces. This is a new type of probabilistic metric spaces which is based on fuzzy measures (also known as non-additive measures and capacities). We prove some properties that describe which families of fuzzy measures are compatible with which type of triangle functions. Then, we show how we can use Sugeno, Choquet integrals, and, in general, any other fuzzy integral as a tool for building these spaces. We show how these results can be used to compute distances between functions. We illustrate the example comparing three types of means when applied to a set of databases. The example uses Sugeno λ-measures to illustrate the theoretical results presented in the paper.