Abstract

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.

Author information

Contact details are reproduced from the original publication and may be historical.

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

Keywords

  • Saturation property
  • Lyapunov's theorem
  • locally convex space
  • thin sets
  • integral
  • distribution
  • multifunction
  • Radon-Nikodym property
  • control measure
  • Maharam-type

Mathematics Subject Classification

28B05, 46G10; 28B20, 46B22

Suggested citation

M. A. Khan, N. Sagara. “Maharam-Types and Lyapunov's Theorem for Vector Measures on Locally Convex Spaces with Control Measures.” Journal of Convex Analysis 22 (2015), No. 3, 647–672. https://doi.org/10.68381/jca22033

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Published by Heldermann Verlag, 2015. Rights now held by Banach Press.