Dunford-Pettis type properties in L₁ of a vector measure

Let ν be a countably additive vector measure defined on a σ-algebra and taking values in a Banach space. In this paper we deal with the following three properties for the Banach lattice L₁(ν) of all ν-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.