A unified approach to the monotone convergence theorem for nonlinear integrals

We present a unified approach to the monotone convergence theorem for nonlinear integrals such as the Choquet, the Šipoš, the Sugeno, and the Shilkret integral. A nonlinear integral may be viewed as a nonlinear functional defined on a set of pairs of a nonadditive measure and a measurable function. We thus formulate our general type of monotone convergence theorem for such a functional. The key tool is a perturbation of functional that manages not only the monotonicity of the functional but also the small change of the functional value arising as a result of adding small amounts to a measure and a function in the domain of the functional. Our approach is also applicable to the Lebesgue integral when a nonadditive measure is σ-additive.