Fuzzy measures defined by addition of fuzzy measures

Given a measurable space (X, A) and two fuzzy measures μ₁ and μ2 defined on A, the addition of μ₁ and μ2 determines another set function on A. Such a new set function is a fuzzy measure, too. We shall discuss the relation between the new fuzzy measure and the original fuzzy measures. We shall see that the new fuzzy measure preserves some structural characteristics, such as subadditivity, several kinds of continuity, null-additivity, weak null-additivity, pseudo generating property, property (S), strong order continuity, regularity, of the original fuzzy measures. By means of these inherited characteristics of fuzzy measures, we show the generalized versions of Egoroff-like theorem, Lebesgue-like theorem, Riesz-like theorem and Lusin-like theorem, which are associated with two fuzzy measures, respectively. It is shown that the Choquet integral with respect to addition of two fuzzy measures is equal to the sum of two Choquet integrals for the original fuzzy measures.