# Article

Full entry | PDF   (0.3 MB)
Summary:
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.
References:
[1] J. Diestel, J. J. Uhl: Vector Measures. Amer. Math. Soc., Providence, R. I., 1977. MR 0453964
[2] D. H. Fremlin: The Henstock and McShane integrals of vector-valued functions. Illinois J. Math. 38 (1994), 471–479. MR 1269699 | Zbl 0797.28006
[3] D. H. Fremlin, J. Mendoza: On the integration of vector-valued functions. Illinois J. Math. 38 (1994), 127–147. MR 1245838
[4] R. A. Gordon: The Denjoy extension of the Bochner, Pettis, and Dunford integrals. Studia Math. 92 (1989), 73–91. MR 0984851 | Zbl 0681.28006
[5] R. A. Gordon: The McShane integral of Banach-valued functions. Illinois J. Math. 34 (1990), 557–567. MR 1053562 | Zbl 0685.28003
[6] R. A. Gordon: Riemann integration in Banach spaces. Rocky Mountain J. Math. 21 (1991), 923–949. DOI 10.1216/rmjm/1181072923 | MR 1138145 | Zbl 0764.28008
[7] R. A. Gordon: The Integrals of Lebesgue, Denjoy, Perron, and Henstock. Amer. Math. Soc., 1994. MR 1288751 | Zbl 0807.26004
[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 10.1023/A:1022403232211 | MR 1461422 | Zbl 0903.46040
[10] B. J. Pettis: Differentiation in Banach spaces. Duke Math. J. 5 (1939), 254–269. DOI 10.1215/S0012-7094-39-00523-5 | MR 1546122 | Zbl 0021.32602

Partner of