When L1 of a vector measure is an AL–space

We consider the space of real functions which are integrable with respect to a countably additive vector measure with values in a Banach space. In a previous paper we showed that this space can be any order continuous Banach lattice with weak order unit. We study a priori conditions on the vector measure in order to guarantee that the resulting Lι is order isomorphic to an AL-space. We prove that for separable measures with no atoms there exists a Co-valued measure that generates the same space of integrable functions. Introduction. Given a vector measure v we consider the space of classes of real functions which are integrable with respect to v in the sense of Lewis [L-l], denoted by Lι{y). In [C, Theorem 8] we showed that every order continuous Banach lattice with weak unit can be obtained as Lι(u) for a suitable vector measure v. In particula