Monotone set functions defined by Choquet integral

Abstract

Given a nonnegative monotone set function and a nonnegative measurable function on a measurable space, the Choquet integral determines a new nonnegative monotone set function that is absolutely continuous with respect to the original one (in a generalized sense for monotone set functions). This new set function preserves almost all desirable structural characteristics of the original monotone set function, such as continuity, subadditivity, superadditivity, null-additivity, converse-null-additivity, autocontinuity, converse-autocontinuity, uniform autocontinuity, uniform converse-autocontinuity, and fuzzy multiplicativity. Such a construction is a useful method to define sound fuzzy measures in various applications.