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Expected Maharam-Types and Lyapunov's Theorem for Vector Measures on Banach Spaces

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  • M. Ali Khan
  • Nobusumi Sagara

Abstract

This paper offers a sufficient condition, based on Maharam (1942) and re-emphasized by Hoover-Keisler (1984), for the validity of Lyapunov's (1940) theorem on the range of an atomless vector measure taking values in an infinite-dimensional Banach space that is not necessarily separable nor has the RNP property. In particular, we obtain an extension of a corresponding result due to Uhl (1969). The proposed condition is also shown to be necessary in the sense formalized in Keisler-Sun (2009), and thereby closes a question of long-standing as regards an infinite-dimensional generalization of the theorem. The result is applied to obtain short simple proofs of recent results on the convexity of the integral of a set-valued function, and on the characterization of restricted cores of a saturated economy.

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  • M. Ali Khan & Nobusumi Sagara, 2012. "Expected Maharam-Types and Lyapunov's Theorem for Vector Measures on Banach Spaces," Economics Working Paper Archive 593, The Johns Hopkins University,Department of Economics.
    • Handle: RePEc:jhu:papers:593
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    References listed on IDEAS

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    1. Evren, Özgür & Hüsseinov, Farhad, 2008. "Theorems on the core of an economy with infinitely many commodities and consumers," Journal of Mathematical Economics, Elsevier, vol. 44(11), pages 1180-1196, December.
    2. Grodal, Birgit, 1972. "A Second Remark on the Core of an Atomless Economy," Econometrica, Econometric Society, vol. 40(3), pages 581-583, May.
    3. Vind, Karl, 1972. "A Third Remark on the Core of an Atomless Economy," Econometrica, Econometric Society, vol. 40(3), pages 585-586, May.
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