Fuzzy Measures Defined by Fuzzy Integral and their Absolute Continuity

Abstract

Given a measurable space (X,F), a fuzzy measure μ on (X,F), and a nonnegative functionfonXthat is measurable with respect toF, we can define a new set function ν on (X,F) by the fuzzy integral. It is known that ν is a lower semicontinuous fuzzy measure on (X,F) and, moreover, if μ is finite, then ν is a finite fuzzy measure as well. In this paper, we generalize in several different ways the concept of absolute continuity of set functions, as defined in classical measure theory. In addition, we investigate the relationship among these generalizations by using the structural characteristics of set functions such as null-additivity and autocontinuity, and determine which types of absolute continuity of fuzzy measures are possessed by the fuzzy measure (or the lower semicontinuous fuzzy measure) obtained by the fuzzy integral.