Let \(\nu \) be a countably additive vector measure defined on a \(\sigma \) -algebra and taking values in a Banach space. In this paper we deal with the following three properties for the Banach lattice \(L_1(\nu )\) of all \(\nu \) -integrable real-valued functions: the Dunford-Pettis property, the positive Schur property and being lattice-isomorphic to an AL-space. We give new results and we also provide alternative proofs of some already known ones.
