The Relationships between Choquet Integral, Pan-integral, Upper Integral and Lower Integral

Various types of integrals with respect to fuzzy measure on finite sets in the literature (Grabisch et al., Fuzzy Measures and Integrals. Theory and Applications, Heidelberg, Germany, 2000; Wang et al., Int J Intell Syst 2006; 21:1073-1092; Wang and Klir, Int J Uncertainty, Fuzziness knowl-based Syst 1997; 5(2):163-175; Wang and Klir, Fuzzy Measures Theory, New York; Plenum, 1992) have been proposed, including Choquet integral, pan-integral, the upper integral, and the lower integral. In this paper, we present the relationships between these different types of integrals. The sufficient and necessary conditions for the equality between these integrals are discussed. (C) 2011 Wiley Periodicals, Inc.