FUZZY MEASURES DEFINED BY FUZZY INTEGRAL AND THEIR ABSOLUTE CONTINUITY

Given a measurable space (X,J), a fuzzy measure mu on (X,J), and a non negative function f on X that is measurable with respect to J, we can define a new set function nu on (X,J) by the fuzzy integral. It is kno wn that nu is a lower semicontinuous fuzzy measure on (X,J) and, moreo ver, if mu is finite, then nu is a finite fuzzy measure as well. In th is paper, we generalize in several different ways the concept of absol ute continuity of set functions, as defined in classical measure theor y. In addition, we investigate the relationship among these generaliza tions by using the structural characteristics of set functions such as null-additivity and autocontinuity, and determine which types of abso lute continuity of fuzzy measures are possessed by the fuzzy measure ( or the lower semicontinuous fuzzy measure) obtained by the fuzzy integ ral. (C) 1996 Academic Press, Inc.