Dunford-Pettis type properties in $L_1$ of a vector measure

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.