A note on derivatives of functions with respect to fuzzy measures

This paper deals with the Choquet integral on the non-negative real line. First it gives a representation of the Choquet integral of a non-negative, continuous and increasing function with respect to a fuzzy measure. Next, restricting fuzzy measures to a class of distorted Lebesgue measures, it considers Choquet integral equations. In order to solve Choquet integral equations, a concept of the derivatives of functions with respect to fuzzy measures is introduced. For distorted Lebesgue measures, it is shown that Choquet integral equations are formulated as Volterra integral equations of the first kind. The differentiability of functions with respect to fuzzy measures is also discussed. It further shows a relation of a Choquet integral equation with the Abel integral equation. Finally this paper introduces simple differential equations with respect to fuzzy measures and gives their solutions.