| Title:
|
The weak McShane integral (English) |
| Author:
|
Saadoune, Mohammed |
| Author:
|
Sayyad, Redouane |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
64 |
| Issue:
|
2 |
| Year:
|
2014 |
| Pages:
|
387-418 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We present a weaker version of the Fremlin generalized McShane integral (1995) for functions defined on a $\sigma $-finite outer regular quasi Radon measure space $(S,\Sigma ,\mathcal {T},\mu )$ into a Banach space $X$ and study its relation with the Pettis integral. In accordance with this new method of integration, the resulting integral can be expressed as a limit of McShane sums with respect to the weak topology. It is shown that a function $f$ from $S$ into $X$ is weakly McShane integrable on each measurable subset of $S$ if and only if it is Pettis and weakly McShane integrable on $S$. On the other hand, we prove that if an $X$-valued function is weakly McShane integrable on $S$, then it is Pettis integrable on each member of an increasing sequence $(S_\ell )_{\ell \geq 1}$ of measurable sets of finite measure with union $S$. For weakly sequentially complete spaces or for spaces that do not contain a copy of $c_0$, a weakly McShane integrable function on $S$ is always Pettis integrable. A class of functions that are weakly McShane integrable on $S$ but not Pettis integrable is included. (English) |
| Keyword:
|
Pettis integral |
| Keyword:
|
McShane integral |
| Keyword:
|
weak McShane integral |
| Keyword:
|
uniform integrability |
| MSC:
|
26A39 |
| MSC:
|
26E20 |
| MSC:
|
28B05 |
| MSC:
|
46G10 |
| idZBL:
|
Zbl 06391501 |
| idMR:
|
MR3277743 |
| DOI:
|
10.1007/s10587-014-0108-7 |
| . |
| Date available:
|
2014-11-10T09:40:55Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144005 |
| . |
| Reference:
|
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| Reference:
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| Reference:
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| Reference:
|
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| Reference:
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