Geodesic
The weak McShane integral
Czechoslovak Mathematical Journal, Tome 64 (2014) no. 2, pp. 387-418 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
DOI : 10.1007/s10587-014-0108-7
Classification : 26A39, 26E20, 28B05, 46G10
Keywords: Pettis integral; McShane integral; weak McShane integral; uniform integrability 
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Saadoune, Mohammed; Sayyad, Redouane. The weak McShane integral. Czechoslovak Mathematical Journal, Tome 64 (2014) no. 2, pp. 387-418. doi: 10.1007/s10587-014-0108-7

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