On integration in Banach spaces and total sets

TL;DR

This paper explores a generalized form of Pettis integrability in Banach spaces using total subspaces of the dual, establishing equivalences under certain properties and extending classical results to broader classes of spaces.

Contribution

It introduces the concept of $ ext{ extGamma}$-integrability, proves its equivalence to Pettis integrability under Plichko's property, and extends Radon-Nikodým property results to new settings.

Findings

$ ext{ extGamma}$-integrability is equivalent to Pettis integrability under property ($ ext{ extD}'$)

Spaces with $w^*$-angelic duals satisfy the property ($ ext{ extD}'$)

Extension of Radon-Nikodým property results to semi-embeddings and $w^*$-sequentially dense spaces.

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 $X$).