Upper Derivatives of Set Functions Represented as the Choquet Indefinite Integral

Abstract

This paper shows that, for a set function ν represented as the Choquet indefinite integral of a function f with respect to a set function μ, the upper derivative of ν at a measurable set A with respect to a measure m is, under a certain condition, equal to the difference calculated by subtracting the product of the negative part f ^− −. and the lower derivative of μ at the whole set with respect to m from the product of the positive part f ⁺ and the upper derivative of μ at A with respect to m.