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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On maximal ranges of vector measures for subsets and purification of transition probabilities


Authors: Peng Dai and Eugene A. Feinberg
Journal: Proc. Amer. Math. Soc. 139 (2011), 4497-4511
MSC (2010): Primary 60A10, 28A10
Published electronically: April 25, 2011
MathSciNet review: 2823095
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Abstract: Consider a measurable space with an atomless finite vector measure. This measure defines a mapping of the $ \sigma$-field into a Euclidean space. According to the Lyapunov convexity theorem, the range of this mapping is a convex compactum. Similar ranges are also defined for measurable subsets of the space. Two subsets with the same vector measure may have different ranges. We investigate the question whether, among all the subsets having the same given vector measure, there always exists a set with the maximal range of the vector measure. The answer to this question is positive for two-dimensional vector measures and negative for higher dimensions. We use the existence of maximal ranges to strengthen the Dvoretzky-Wald-Wolfowitz purification theorem for the case of two measures.


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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@stonybrook.edu

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

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-10860-8
PII: S 0002-9939(2011)10860-8
Keywords: Lyapunov convexity theorem, maximal subset, purification of transition probabilities.
Received by editor(s): June 1, 2010
Received by editor(s) in revised form: June 16, 2010, and October 20, 2010
Published electronically: April 25, 2011
Additional Notes: This research was partially supported by NSF grants CMMI-0900206 and CMMI-0928490.
Communicated by: Richard C. Bradley
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.