Sequentially continuous non-monotonic choquet integrals

Abstract

Schmeidler (Proc. Amer. Math. Soc. 97 (2) (1986) 255–261) established an integral representation theorem through the Choquet integral for functionals satisfying monotonicity and a weaker condition than additivity, namely comonotonic additivity. Murofushi–Sugeno–Machida (Fuzzy Sets and Systems 64 (1994) 73–86) generalize this representation to the case of bounded variation functionals omitting the monotonicity condition. We give an alternative approach which is based on sequential continuity and tolerates non-monotonicity. This later condition is equivalent to σ -additivity in measure theory.