« Previous |
11 - 12 of 12
|
Next »
Number of results to display per page
Search Results
12. The weak McShane integral
- Creator:
- Saadoune, Mohammed and Sayyade, Redouane
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- Pettis integral, McShane integral, weak McShane integral, and uniform integrability
- Language:
- English
- Description:
- 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.
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public