Pseudo-addition of monotone measures and integrals
Xiaoli Hu, Mingjing Hou, Shunyang Wei, Jun Li · 2016
Let (X, A) be a fixed measurable space and two monotone measures μ1and μ2two monotone measures defined on A, we consider the pseudo-addition μ1⊕ μ2of these two monotone measures. It is a new monotone measure λ. Such a monotone measure is absolutely continuous with respect to μ1and μ2. Not only that, the new monotone measure preserves most of desirable structural characteristics of the monotone measures μ1and μ2, such as subadditivity, null-additivity, etc. We shall present the generalized versions of convergence theorems for sequence of measurable functions, such as the Egoroff theorem and the Lusin theorem. These theorems are associated with two monotone measures on the measurable space. The previous results are covered by the results presented here. We shall show that each of usual nonlinear intrgrals, the Sugeno integral, the Choquet integral, and the Shilkret integral with respect to μ1⊕ μ2, is equal to the ⊕-sum of two same kinds of integrals with respect to μ1and μ2, respectively.