Approximating fuzzy measures with distorted probabilities

Fuzzy measures, also called capacities, non-additive measures, and monotonic games, are an interesting mathematical object and have been used in a large number of real-world applications. They are useful in problems where we need to aggregate information. In this case, we often use them in combination with fuzzy integrals. Sugeno and Choquet integrals are classical examples of fuzzy integrals. Fuzzy measures are set functions, and as such, when the reference set is finite, they are defined by 2 n values where n represents the cardinality of the reference set. This makes difficult both to define them and to interpret them. In this work, we propose to approximate fuzzy measures in terms of distorted probabilities as a way to better understand the structure and properties of the measure. This approach can be used when fuzzy measures are learned from examples. We show, however, that, in general, the approximation will not represent all the properties of an arbitrary fuzzy measure.