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Extension of Lyapunov's convexity theorem to subranges


Authors: Peng Dai and Eugene A. Feinberg
Journal: Proc. Amer. Math. Soc. 142 (2014), 361-367
MSC (2010): Primary 60A10, 28A10
DOI: https://doi.org/10.1090/S0002-9939-2013-11864-2
Published electronically: September 20, 2013
MathSciNet review: 3119209
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Abstract: Consider a measurable space with a finite vector measure. This measure defines a mapping of the $ \sigma $-field into a Euclidean space. According to Lyapunov's convexity theorem, the range of this mapping is compact and, if the measure is atomless, this range is convex. Similar ranges are also defined for measurable subsets of the space. We show that the union of the ranges of all subsets having the same given vector measure is also compact and, if the measure is atomless, it is convex. We further provide a geometrically constructed convex compactum in the Euclidean space that contains this union. The equality of these two sets, which holds for two-dimensional measures, can be violated in higher dimensions.


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Additional Information

Peng Dai
Affiliation: Department of Applied Mathematics and Statistics, Stony Brook University, Stony Brook, New York 11794-3600
Email: Peng.Dai@outlook.com

Eugene A. Feinberg
Affiliation: Department of Applied Mathematics and Statistics, Stony Brook University, Stony Brook, New York 11794-3600
Email: Eugene.Feinberg@stonybrook.edu

DOI: https://doi.org/10.1090/S0002-9939-2013-11864-2
Keywords: Atomless vector measure, Lyapunov's convexity theorem, purification of transition probabilities.
Received by editor(s): February 21, 2011
Received by editor(s) in revised form: February 28, 2012
Published electronically: September 20, 2013
Additional Notes: This research was partially supported by NSF grants CMMI-0900206 and CMMI-0928490.
Communicated by: Mark M. Meerschaert
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.