On maximal ranges of vector measures for subsets and purification of transition probabilities
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- by Peng Dai and Eugene A. Feinberg
- Proc. Amer. Math. Soc. 139 (2011), 4497-4511
- DOI: https://doi.org/10.1090/S0002-9939-2011-10860-8
- Published electronically: April 25, 2011
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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.References
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Bibliographic 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
- 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
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 139 (2011), 4497-4511
- MSC (2010): Primary 60A10, 28A10
- DOI: https://doi.org/10.1090/S0002-9939-2011-10860-8
- MathSciNet review: 2823095