On the Differentiation of Henstock and McShane Integrals

• Main Authors: Chew, Tuan Seng, Cabral, Emmanuel A, Benitez, Julius V
• Format: text
• Published: Archīum Ateneo 2021
• Subjects: Henstock-Kurzweil integral; derivative of an integral; Vitali covering theorem; Mathematics
• Online Access: https://archium.ateneo.edu/mathematics-faculty-pubs/165; https://www.worldscientific.com/doi/epdf/10.1142/S2591722621400019
• Institution: Ateneo De Manila University

It is well known that the derivative of the primitive of one-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 one-dimensional McShane integrals; n-dimensional McShane and Henstock integrals; covering theorems of the Vitali type cannot be applied. In this paper; we shall discuss differentiation of n-dimensional McShane and Henstock integrals."