Maharam-Types and Lyapunov's Theorem for Vector Measures on Locally Convex Spaces with Control Measures

This paper presents an equivalence between (i) the Lyapunov property under which a vector measure with values in a sequentially complete, separable locally convex Hausdorff space (lcHs) has a weakly compact and convex range, (ii) the thinness property of subsets of Bochner integrable functions due to Kingman-Robertson (1968) and (iii) the saturation property due to Maharam (1942) and Hoover-Keisler (1984). It also considers the case of a non-separable range space, and presents versions of the Lyapunov theorem for a quasicomplete lcHs based either on the Egorov property or the notion of Maharam-types. The results are applied to two canonical objects in convex analysis: the integral and the distribution of a multifunction.