Generalization of inclusion–exclusion integral for nondiscrete monotone measure space

Abstract We generalize the inclusion–exclusion integral for nondiscrete monotone measure spaces. The inclusion–exclusion integral for a monotone measure space on a finite space is discussed in previous papers. This is an integral with respect to a nonadditive monotone measure, which includes the Lebesgue and Choquet integrals. We are concerned with the monotonicity of the integral, and a sufficient condition for the monotonicity of the t-norm based inclusion–exclusion integral is given by introducing the complete monotonicity of the t-norm based interaction operator. Moreover concrete examples of the interaction operators that satisfy the complete monotonicity are shown.