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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.

Nobusumi Sagara
Dept. of Economics, Hosei University, 4342 Aihara Machida, Tokyo 194-0298, Japan

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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