The weak topology on of a vector measure

Let ν be a countably additive measure defined on a measurable space ( Ω , Σ ) and taking values in a Banach space X. Let 1 < p < ∞ . In this paper we study some aspects of the weak topology on the Banach lattice L p ( ν ) of all (equivalence classes of) measurable real-valued functions on Ω which are pth power integrable with respect to ν. We show that every subspace of L p ( ν ) is weakly compactly generated and has weakly compactly generated dual. We prove that a bounded net ( f α ) in L p ( ν ) is weakly convergent to f ∈ L p ( ν ) if and only if ∫ A f α d ν → ∫ A f d ν weakly in X for every A ∈ Σ . Finally, we also provide sufficient conditions ensuring that the set of functionals { f ↦ ∫ Ω f g d 〈 x ∗ , ν 〉 : g ∈ B L q ( ν ) , x ∗ ∈ B X ∗ } ⊂ B L p ( ν ) ∗ is a James boundary, where 1 / p + 1 / q = 1 .