Convergence theorems for sequences of Choquet integrals and the stability of nonlinear integral systems

Abstract

In the last 20 years, the theoretical as well as practical significance of nonadditive set functions and nonlinear integrals has increasingly been recognized. The Choquet integral with respect to nonadditive monotone set functions is one kind of nonlinear functionals defined on a subspace of all real-valued measurable functions. Unlike the fuzzy integral, which uses the maximum and minimum operators, the Choquet integral is defined via the common addition and multiplication and, therefore, it is a generalization of the classical Lebesgue integral. The convergence of sequences of measurable functions and relevant convergence theorems for sequences of fuzzy integrals have already been investigated by Wang (1984) and Wang and Klir (1992). In an analogous way, we investigate the convergence of sequences of Choquet integrals in this paper. This investigation is, perhaps, even more relevant to practical applications. As an application of convergent theorems, we investigate the stability of a class of nonlinear systems that can be identified by nonnegative monotone set functions with the Choquet integral.