On integration in Banach spaces and total sets

Abstract

Let $X$ be a Banach space and $\Gamma \subseteq X^{\ast }$ 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 $⟨x^{\ast },f⟩$ for $x^{\ast }\in \Gamma$. We show that $\Gamma$-integrability and Pettis integrability are equivalent whenever $X$ has Plichko's property (${\mathcal{D}}^{\prime }$) (meaning that every $w^{\ast }$-sequentially closed subspace of $X^{\ast }$ is $w^{\ast }$-closed). This property is enjoyed by many Banach spaces including all spaces with $w^{\ast }$-angelic dual as well as all spaces which are $w^{\ast }$-sequentially dense in their bidual. A particular case of special interest arises when considering $\Gamma =T^{\ast }(Y^{\ast })$ 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 (${\mathcal{D}}^{\prime }$) and $Y$ has the Radon-Nikodým property, then $X$ has the weak Radon-Nikodým property. This extends earlier results by Delbaen (for separable $X$) and Diestel and Uhl (for weakly $\mathcal{K}$-analytic $X$).