A Note on the Representation of the Choquet Integral Based on the Derivative of Fuzzy Measure

Abstract

This paper gives another proof of Sugeno's theorem [Theorem 1, Fuzzy Sets and Systems 222 (2013) 1–17] to calculate the Choquet integral of monotone functions on the real line. Sugeno’s original theorem assumes that the integrand is continuous. However the presented new proof shows that the conclusion of the theorem is satisfied without the assumption of continuity.