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

 

Purification and saturation


Authors: Peter Loeb and Yeneng Sun
Journal: Proc. Amer. Math. Soc. 137 (2009), 2719-2724
MSC (2000): Primary 28A25, 03H05, 28E05, 91A06; Secondary 26E35
Published electronically: February 4, 2009
MathSciNet review: 2497484
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Abstract: This paper illustrates the general technique established in 1984 by Hoover and Keisler for extending certain types of results from atomless Loeb measure spaces to measure spaces that we shall call ``nowhere countably generated''. The Hoover-Keisler technique is applied here to further extend the authors' 2006 generalization of a theorem of Dvoretzky, Wald and Wolfowitz on the purification of measure-valued maps. The authors' 2006 result was first extended to these more general spaces by K. Podczeck in 2007; he used new results in functional analysis produced for that purpose. This paper demonstrates that, in general, such extensions follow from the Hoover-Keisler technique. Moreover, adaptations of counterexamples from earlier papers show that the extension obtained here holds only for nowhere countably generated spaces.


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Additional Information

Peter Loeb
Affiliation: Department of Mathematics, University of Illinois, Urbana, Illinois 61801
Email: loeb@math.uiuc.edu

Yeneng Sun
Affiliation: Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543
Email: matsuny@nus.edu.sg

DOI: http://dx.doi.org/10.1090/S0002-9939-09-09818-9
PII: S 0002-9939(09)09818-9
Keywords: Loeb measures, nowhere countably generated measure spaces, saturation, purification.
Received by editor(s): December 14, 2007
Received by editor(s) in revised form: July 7, 2008, and October 29, 2008
Published electronically: February 4, 2009
Communicated by: Julia Knight
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.