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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Purification and saturation
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by Peter Loeb and Yeneng Sun PDF
Proc. Amer. Math. Soc. 137 (2009), 2719-2724 Request permission


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:
  • Yeneng Sun
  • Affiliation: Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543
  • Email:
  • 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
  • © Copyright 2009 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 137 (2009), 2719-2724
  • MSC (2000): Primary 28A25, 03H05, 28E05, 91A06; Secondary 26E35
  • DOI:
  • MathSciNet review: 2497484