On integration in banach spaces and total sets

Abstract

Let X be a Banach space and Γ ⊆ 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 () (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 () 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 -analytic X).

Mathematics Subject Classification (2010):

• 46B22
• 46G10

Key words:

• Pettis integral
• Γ-integral
• Radon-Nikodým property
• weak Radon-Nikodým property
• weakly Lindelöf determined Banach space
• property (D′)
• semi-embedding