Journal of Convex Analysis 22 (2015), No. 3, 647--672
Copyright Heldermann Verlag 2015
Maharam-Types 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 194-0298, 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 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.
Keywords: Saturation property, Lyapunov's theorem, locally convex space, thin sets, integral, distribution, multifunction, Radon-Nikodym property, control measure, Maharam-type.
MSC: 28B05, 46G10; 28B20, 46B22
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