A Henstock-Kurzweil integrable vector-valued function which is not McShane integrable on any portion

Abstract

We give an example of a function from [a, b] into c ₀, which is Henstock-Kurzweil integrable on [a, b] and not McShane integrable on any nondegenerate subinterval of [a, b]. As a result, any Henstock-Kurzweil integrable function from [a, b] into a Banach space X is McShane integrable on some nondegenerate subinterval of [a, b] if and only if X contains no isomorphic copy of c ₀.