Convergence in measure theorems of nonlinear integrals of functions integrable to the pth power

Abstract

One can find a subadditive measure μ and a sequence {fn}n∈N of nonnegative measurable functions converging in μ-measure to such a function f with the property that for any p>1, fnp no longer converges in μ-measure to fp. Therefore, in nonadditive measure theory an additional and intrinsic discussion is needed to prove integral convergence theorems for functions integrable to the pth power.The purpose of the paper is to propose a method for automatically deriving pth power versions of convergence in measure theorems of nonlinear integrals from already established ones. The proposed method is based on the robustness of some properties of integral functional under the operation f↦fp raising a nonnegative function f to the pth power.