The bounded convergence in measure theorem for nonlinear integral functionals

Abstract

In this paper, we introduce a new notion of the perturbation of nonlinear functionals to formulate a functional form of the convergence theorems for nonlinear integrals in nonadditive measure theory. As its direct consequences, we obtain the bounded convergence in measure theorems for typical nonlinear integrals, which show that the autocontinuity of a nonadditive measure is equivalent to the validity of the bounded convergence in measure theorems for the Choquet, the Sugeno, and the Shilkret integrals as well as their symmetric and asymmetric extensions.