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 c₀, 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 c₀.

Keywords: Henstock-Kurzweil, McShane and McShane-Pettis integrals, bounded variation, V B, AC and ACG functions

Quaestiones Mathematicae 35(2012), 11-21