Continuous m-dimensional distorted probabilities

Abstract

Fuzzy measures, also known as non-additive measures, monotonic games, and capacities, have been used in many contexts. For example, in economics, risk analysis, in computer science, computer vision and machine learning and, in general, in mathematics. However, when looking at applications, one of the problems that still needs to be solved is how the measure should be defined in an easy and intuitive way. When the reference set is finite, a few families of measures have been established, e.g. distorted probabilities, k-additive and decomposable measures. But, when the reference set is infinite, the only family is distorted probabilities. In this paper we give a definition for m-dimensional distorted probabilities in the case that the reference set is not finite, and we study some properties of this family. We also give a definition for hierarchically decomposable m-dimensional distorted probabilities that relates to another family of measures defined for the finite case.