The Vitali convergence in measure theorem of nonlinear integrals

Abstract

The Vitali convergence in measure theorem for the abstract Lebesgue integral is fundamental in Lebesgue integration theory and yields the bounded convergence theorem and the dominated convergence theorem as its applications. In this paper, in a unified way using the perturbation method the Vitali convergence in measure theorem is established for nonlinear integrals such as the Choquet, Šipoš, Sugeno, and Shilkret integrals, and their symmetric and asymmetric extensions. It is derived from the Fatou and the reverse Fatou type lemmas for perturbative nonlinear integral functionals.