Fuzzy numbers and fuzzification of the Choquet integral
Fuzzy Sets and Systems
Applying fuzzy measures and nonlinear integrals in data mining
Fuzzy Sets and Systems
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Fuzzy measures and the Choquet integral are generalizations of classical measures and the Lebesgue integral, respectively. Given a fuzzy measure and a nonnegative measurable function on a measurable space, the Choquet integral determines a new fuzzy measure that is absolutely continuous with respect to the original one (in a generalized sense for fuzzy measures). This new fuzzy measure preserves almost all desirable structural characteristics of the original fuzzy measure, such as subadditivity, superadditivity, null-additivity, converse-null-additivity, autocontinuity, converse-autocontinuity, uniform autocontinuity, uniform converse-autocontinuity, and fuzzy multiplicativity. As a notable exception, fuzzy additivity is not necessarily preserved. Such a construction is a useful method to define sound fuzzy measures or revise fuzzy measures in various applications.