Dedicated to the late K. Floret Abstract. Associated with every vector measure m taking its values in a Frechet space X is the space L1(m) of all m-integrable functions. It turns out that L1(m) is always a Frechet lattice. We show that possession of the AL-property for the lattice L1(m) has some remarkable consequences for both the underlying Fr´ echet space X and the integration operator f 7! R f dm.