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- W4301166311 abstract "Let $X$ be a Banach space and $Gamma subseteq X^*$ a total linear subspace. We study the concept of $Gamma$-integrability for $X$-valued functions $f$ defined on a complete probability space, i.e. an analogue of Pettis integrability by dealing only with the compositions $langle x^*,f rangle$ for $x^*in Gamma$. We show that $Gamma$-integrability and Pettis integrability are equivalent whenever $X$ has Plichko's property ($mathcal{D}'$) (meaning that every $w^*$-sequentially closed subspace of $X^*$ is $w^*$-closed). This property is enjoyed by many Banach spaces including all spaces with $w^*$-angelic dual as well as all spaces which are $w^*$-sequentially dense in their bidual. A particular case of special interest arises when considering $Gamma=T^*(Y^*)$ for some injective operator $T:X to Y$. Within this framework, we show that if $T:X to Y$ is a semi-embedding, $X$ has property ($mathcal{D}'$) and $Y$ has the Radon-Nikod'{y}m property, then $X$ has the weak Radon-Nikod'{y}m property. This extends earlier results by Delbaen (for separable $X$) and Diestel and Uhl (for weakly $mathcal{K}$-analytic $X$)." @default.
- W4301166311 created "2022-10-04" @default.
- W4301166311 creator A5038550012 @default.
- W4301166311 date "2018-06-26" @default.
- W4301166311 modified "2023-09-28" @default.
- W4301166311 title "On integration in Banach spaces and total sets" @default.
- W4301166311 doi "https://doi.org/10.48550/arxiv.1806.10049" @default.
- W4301166311 hasPublicationYear "2018" @default.
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