Universal integrals based on copulas

A hierarchical family of integrals based on a fixed copula is introduced and discussed. The extremal members of this family correspond to the inner and outer extension of integrals of basic functions, the copula under consideration being the corresponding multiplication. The limits of the members of the family are just copulabased universal integrals as recently introduced in Klement et al. (IEEE Trans Fuzzy Syst 18:178–187, 2010). For the product copula, the family of integrals considered here contains the Choquet and the Shilkret integral, and it belongs to the class of decomposition integrals proposed in Even and Lehrer (Econ Theory, 2013) as well as to the class of superdecomposition integrals introduced in Mesiar et al. (Superdecomposition integral, 2013). For the upper Fréchet-Hoeffding bound, the corresponding E. P. Klement (B) Department of Knowledge-Based Mathematical Systems, Johannes Kepler University, Linz, Austria e-mail: [email protected] R. Mesiar Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Slovak University of Technology, Bratislava, Slovakia R. Mesiar ÚTIA AV ČR, Prague, Czech Republic e-mail: [email protected] F. Spizzichino Department of Mathematics, University of Rome “La Sapienza”, Rome, Italy e-mail: [email protected] A. Stupňanová Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Slovak University of Technology, Bratislava, Slovakia e-mail: [email protected]