On the Differentiation of Henstock and McShane Integrals

Abstract

It is well known that the derivative of the primitive of 1-dimensional Henstock integral exists almost everywhere. Point-interval pairs used in the derivative are Henstock point-interval pairs, which are consistent with point-interval pairs used in the Henstock integral. Note that “almost everywhere” is a set of points, more precisely, the derivative does not exist on a set of points with measure zero. We can transform a set of Henstock point-interval pairs to a set of points with measure zero because of Vitali’s covering theorem. For 1-dimensional McShane integrals, [Formula: see text]-dimensional McShane and Henstock integrals, covering theorems of Vitali’s type cannot be applied. In this paper, we shall discuss differentiation of [Formula: see text]-dimensional McShane and Henstock integrals.