Generalized subadditivity and superadditivity of monotone measures

The concepts of generalized finite subadditivity and generalized finite superadditivity with respect to a pair of monotone measures are proposed. As special cases, they cover the concepts of subadditivity and superadditivity of monotone measures, respectively. By means of these new structural characteristics of monotone measures, we show the generalized sub(super-)additivity of the pan-integrals, and describe the generalized order relations among the Choquet, pan- and concave integral in general context concerning an ordered pair of monotone measures. The generalized Hölder and Minkowski inequalities of the pan-integrals are also presented. The previous results related to the Choquet, pan- and concave integral for sub(super-)additive monotone measures are recovered.