A Way to Choquet Calculus

In this paper, we deal with the Choquet integral and derivative with respect to fuzzy measures on the nonnegative real line and present a way to Choquet calculus as a new research paradigm. In Choquet calculus, a representation for calculating the continuous Choquet integral is first given by restricting the integrand to a class of nondecreasing and continuous functions and the fuzzy measure to a class of distorted Lebesgue measures. Next, the derivative of functions with respect to distorted Lebesgue measures is defined as the inverse operation of the Choquet integral. Then, elementary properties in Choquet calculus are explored. In addition, we clarify the relation of Choquet calculus with fractional calculus, where the fractional Choquet integral and derivative are newly defined. In addition, we consider differential equations with respect to distorted Lebesgue measures and give their solutions. Finally, we introduce conditional distorted Lebesgue measures and explore their properties.