Theory of inclusion-exclusion integral

•A new class of integral that corresponds to a nonadditive measure.•The new integral is defined using an interaction operator.•Several mathematical properties of this integral are demonstrated.•A mathematical model using this integral is proposed and constructed.•The mathematical model is applied to a real dataset. We propose an integral with respect to a nonadditive monotone measure. This integral is a generalization of the Lebesgue integral and also the Choquet integral. It has appropriate properties, and can be extensively applied to real data analysis.