The Denjoy extension of the Riemann and McShane integrals

Park, Jae Myung. "The Denjoy extension of the Riemann and McShane integrals." Czechoslovak Mathematical Journal 50.3 (2000): 615-625. <http://eudml.org/doc/30588>.

@article{Park2000,
abstract = {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.},
author = {Park, Jae Myung},
journal = {Czechoslovak Mathematical Journal},
keywords = {Denjoy-Riemann integral},
language = {eng},
number = {3},
pages = {615-625},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The Denjoy extension of the Riemann and McShane integrals},
url = {http://eudml.org/doc/30588},
volume = {50},
year = {2000},
}

TY - JOUR
AU - Park, Jae Myung
TI - The Denjoy extension of the Riemann and McShane integrals
JO - Czechoslovak Mathematical Journal
PY - 2000
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 50
IS - 3
SP - 615
EP - 625
AB - 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.
LA - eng
KW - Denjoy-Riemann integral
UR - http://eudml.org/doc/30588
ER -