The weak McShane integral

Mohammed Saadoune; Redouane Sayyad

Czechoslovak Mathematical Journal (2014)

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

Abstract

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We present a weaker version of the Fremlin generalized McShane integral (1995) for functions defined on a σ -finite outer regular quasi Radon measure space ( S , Σ , 𝒯 , μ ) 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 ) 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.

How to cite

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Saadoune, 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 -

References

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