On integration in Banach spaces and total sets

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 $X$).