The Denjoy extension of the Riemann and McShane integrals
Czechoslovak Mathematical Journal, Tome 50 (2000) no. 3, pp. 615-625
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In this paper we study the Denjoy-Riemann and Denjoy-McShane integrals of functions mapping an interval $\left[ a,b\right] $ into a Banach space $X.$ It is shown that a Denjoy-Bochner integrable function on $ \left[ a,b\right] $ is Denjoy-Riemann integrable on $\left[ a,b\right] $, that a Denjoy-Riemann integrable function on $\left[ a,b\right] $ is Denjoy-McShane integrable on $\left[ a,b\right] $ and that a Denjoy-McShane integrable function on $\left[ a,b\right] $ is Denjoy-Pettis integrable on $\left[ a,b\right].$ In addition, it is shown that for spaces that do not contain a copy of $c_{0}$, a measurable Denjoy-McShane integrable function on $\left[ a,b\right] $ is McShane integrable on some subinterval of $\left[ a,b\right].$ Some examples of functions that are integrable in one sense but not another are included.
@article{CMJ_2000__50_3_a13,
author = {Park, Jae Myung},
title = {The {Denjoy} extension of the {Riemann} and {McShane} integrals},
journal = {Czechoslovak Mathematical Journal},
pages = {615--625},
publisher = {mathdoc},
volume = {50},
number = {3},
year = {2000},
mrnumber = {1777481},
zbl = {1079.28502},
language = {en},
url = {http://geodesic.mathdoc.fr/item/CMJ_2000__50_3_a13/}
}
Park, Jae Myung. The Denjoy extension of the Riemann and McShane integrals. Czechoslovak Mathematical Journal, Tome 50 (2000) no. 3, pp. 615-625. http://geodesic.mathdoc.fr/item/CMJ_2000__50_3_a13/