The Denjoy extension of the Riemann and Mcshane integrals

Abstract

In this paper we study the Denjoy-Riemann and Denjoy-McShane integrals of functions mapping an interval [a, b] into a Banach space X. It is shown that a Denjoy-Bochner integrable function on [a, b] is Denjoy-Riemann integrable on [a, b], that a Denjoy-Riemann integrable function on [a, b] is Denjoy-McShane integrable on [a, b] and that a Denjoy-McShane integrable function on [a, b] is Denjoy-Pettis integrable on [a, b]. In addition, it is shown that for spaces that do not contain a copy of c 0, a measurable Denjoy-McShane integrable function on [a, b] is McShane integrable on some subinterval of [a, b]. Some examples of functions that are integrable in one sense but not another are included.