
Journal of Convex Analysis 22 (2015), No. 3, 647672 Copyright Heldermann Verlag 2015 MaharamTypes and Lyapunov's Theorem for Vector Measures on Locally Convex Spaces with Control Measures M. Ali Khan Dept. of Economics, The Johns Hopkins University, Baltimore, MD 21218, U.S.A. akhan@jhu.edu Nobusumi Sagara Dept. of Economics, Hosei University, 4342 Aihara Machida, Tokyo 1940298, Japan nsagara@hosei.ac.jp 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 KingmanRobertson (1968) and (iii) the saturation property due to Maharam (1942) and HooverKeisler (1984). It also considers the case of a nonseparable range space, and presents versions of the Lyapunov theorem for a quasicomplete lcHs based either on the Egorov property or the notion of Maharamtypes. The results are applied to two canonical objects in convex analysis: the integral and the distribution of a multifunction. Keywords: Saturation property, Lyapunov's theorem, locally convex space, thin sets, integral, distribution, multifunction, RadonNikodym property, control measure, Maharamtype. MSC: 28B05, 46G10; 28B20, 46B22 [ Fulltextpdf (225 KB)] for subscribers only. 