Geodesic
The Denjoy extension of the Riemann and McShane integrals
Czechoslovak Mathematical Journal, Tome 50 (2000) no. 3, pp. 615-625 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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.
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.
Classification : 26A42, 28A25, 28A50, 28B05 
@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},
     year = {2000},
     volume = {50},
     number = {3},
     mrnumber = {1777481},
     zbl = {1079.28502},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2000_50_3_a13/}
}
TY  - JOUR
AU  - Park, Jae Myung
TI  - The Denjoy extension of the Riemann and McShane integrals
JO  - Czechoslovak Mathematical Journal
PY  - 2000
SP  - 615
EP  - 625
VL  - 50
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/CMJ_2000_50_3_a13/
LA  - en
ID  - CMJ_2000_50_3_a13
ER  - 
%0 Journal Article
%A Park, Jae Myung
%T The Denjoy extension of the Riemann and McShane integrals
%J Czechoslovak Mathematical Journal
%D 2000
%P 615-625
%V 50
%N 3
%U http://geodesic.mathdoc.fr/item/CMJ_2000_50_3_a13/
%G en
%F 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/

[1] J. Diestel, J. J. Uhl: Vector Measures. Amer. Math. Soc., Providence, R. I., 1977. | MR

[2] D. H. Fremlin: The Henstock and McShane integrals of vector-valued functions. Illinois J. Math. 38 (1994), 471–479. | DOI | MR | Zbl

[3] D. H. Fremlin, J. Mendoza: On the integration of vector-valued functions. Illinois J. Math. 38 (1994), 127–147. | DOI | MR

[4] R. A. Gordon: The Denjoy extension of the Bochner, Pettis, and Dunford integrals. Studia Math. 92 (1989), 73–91. | DOI | MR | Zbl

[5] R. A. Gordon: The McShane integral of Banach-valued functions. Illinois J. Math. 34 (1990), 557–567. | DOI | MR | Zbl

[6] R. A. Gordon: Riemann integration in Banach spaces. Rocky Mountain J. Math. 21 (1991), 923–949. | DOI | MR | Zbl

[7] R. A. Gordon: The Integrals of Lebesgue, Denjoy, Perron, and Henstock. Amer. Math. Soc., 1994. | MR | Zbl

[8] R. A. Gordon: Differentiation in Banach spaces. preprint.

[9] J. M. Park: Bounded convergence theorem and integral operator for operator valued measures. Czechoslovak Math. J. 47(122) (1997), 425–430. | DOI | MR | Zbl

[10] B. J. Pettis: Differentiation in Banach spaces. Duke Math. J. 5 (1939), 254–269. | DOI | MR | Zbl