Pseudo-addition of monotone measures and integrals

Let (X, A) be a fixed measurable space and two monotone measures μ₁ and μ₂ two monotone measures defined on A, we consider the pseudo-addition μ₁ ⊕ μ₂ of these two monotone measures. It is a new monotone measure λ. Such a monotone measure is absolutely continuous with respect to μ₁ and μ₂. Not only that, the new monotone measure preserves most of desirable structural characteristics of the monotone measures μ₁ and 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 μ₁ and μ₂, respectively.