On integration in Banach spaces and

Abstract

Let X be a Banach space and ΓC X^*  a total linear subspace. We study the concept of Γ-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^* , f⟩ for x^*∈ Γ . We show that Γ-integrability and Pettis integrability are equivalent whenever X has Plichko's property (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 Γ = T^*  (Y^*) for some injective operator T : X→Y. Within this framework, we show that if T : X → Y is a semi-embedding, X has property (D′ ) and Y has the Radon-Nikodym property, then X has the weak Radon-Nikodym property. This extends earlier results by Delbaen (for separable X) and Diestel and Uhl (for weakly K-analytic X).

Key words: Pettis integral, Γ-integral, Radon-Nikodym property, weak Radon-Nikodym property, weakly Lindelof determined Banach space,  property (D′), semi-embedding.