| Title:
|
The Denjoy extension of the Riemann and McShane integrals (English) |
| Author:
|
Park, Jae Myung |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
50 |
| Issue:
|
3 |
| Year:
|
2000 |
| Pages:
|
615-625 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| MSC:
|
26A42 |
| MSC:
|
28A25 |
| MSC:
|
28A50 |
| MSC:
|
28B05 |
| idZBL:
|
Zbl 1079.28502 |
| idMR:
|
MR1777481 |
| . |
| Date available:
|
2009-09-24T10:36:16Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/127597 |
| . |
| Reference:
|
[1] J. Diestel, J. J. Uhl: Vector Measures.Amer. Math. Soc., Providence, R. I., 1977. MR 0453964 |
| Reference:
|
[2] D. H. Fremlin: The Henstock and McShane integrals of vector-valued functions.Illinois J. Math. 38 (1994), 471–479. Zbl 0797.28006, MR 1269699, 10.1215/ijm/1255986726 |
| Reference:
|
[3] D. H. Fremlin, J. Mendoza: On the integration of vector-valued functions.Illinois J. Math. 38 (1994), 127–147. MR 1245838, 10.1215/ijm/1255986891 |
| Reference:
|
[4] R. A. Gordon: The Denjoy extension of the Bochner, Pettis, and Dunford integrals.Studia Math. 92 (1989), 73–91. Zbl 0681.28006, MR 0984851, 10.4064/sm-92-1-73-91 |
| Reference:
|
[5] R. A. Gordon: The McShane integral of Banach-valued functions.Illinois J. Math. 34 (1990), 557–567. Zbl 0685.28003, MR 1053562, 10.1215/ijm/1255988170 |
| Reference:
|
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| Reference:
|
[7] R. A. Gordon: The Integrals of Lebesgue, Denjoy, Perron, and Henstock.Amer. Math. Soc., 1994. Zbl 0807.26004, MR 1288751 |
| Reference:
|
[8] R. A. Gordon: Differentiation in Banach spaces.preprint. |
| Reference:
|
[9] J. M. Park: Bounded convergence theorem and integral operator for operator valued measures.Czechoslovak Math. J. 47(122) (1997), 425–430. Zbl 0903.46040, MR 1461422, 10.1023/A:1022403232211 |
| Reference:
|
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