# The weak McShane integral

Mohammed Saadoune; Redouane Sayyad

Czechoslovak Mathematical Journal (2014)

- Volume: 64, Issue: 2, page 387-418
- ISSN: 0011-4642

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topSaadoune, Mohammed, and Sayyad, Redouane. "The weak McShane integral." Czechoslovak Mathematical Journal 64.2 (2014): 387-418. <http://eudml.org/doc/262021>.

@article{Saadoune2014,

abstract = {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 $(S_\ell )_\{\ell \ge 1\}$ 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.},

author = {Saadoune, Mohammed, Sayyad, Redouane},

journal = {Czechoslovak Mathematical Journal},

keywords = {Pettis integral; McShane integral; weak McShane integral; uniform integrability; generalized McShane integral; weak McShane integral; Pettis integral; uniform integrability},

language = {eng},

number = {2},

pages = {387-418},

publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},

title = {The weak McShane integral},

url = {http://eudml.org/doc/262021},

volume = {64},

year = {2014},

}

TY - JOUR

AU - Saadoune, Mohammed

AU - Sayyad, Redouane

TI - The weak McShane integral

JO - Czechoslovak Mathematical Journal

PY - 2014

PB - Institute of Mathematics, Academy of Sciences of the Czech Republic

VL - 64

IS - 2

SP - 387

EP - 418

AB - 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 $(S_\ell )_{\ell \ge 1}$ 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.

LA - eng

KW - Pettis integral; McShane integral; weak McShane integral; uniform integrability; generalized McShane integral; weak McShane integral; Pettis integral; uniform integrability

UR - http://eudml.org/doc/262021

ER -

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