  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.
