Convergence theorems for sequences of Choquet integrals

The Choquet integral with respect to nonadditive monotone set functions, including imprecise probabilities and fuzzy measures, is a generalization of the classical Lebesgue integral. It is one kind of nonlinear functionals defined on a subspace of all real valued measurable functions. In this paper, several different types of convergence, including the mean convergence that is based on the Choquet integral, for sequences of measurable functions are considered, and the corresponding convergence theorems for sequence of Choquet integrals are demonstrated. Particularly, the theorem of convergence in measure is presented in a form of ''necessary and sufficient condition'' by using the structural characteristics of nonnegative monotone set functions. As an application of convergence theorems, the stability of a class of nonlinear integral systems is discussed.