When<i>L</i>¹of a vector measure is an AL-space

Abstract

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 L 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.