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 | References | Similar Articles | Additional Information

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:
https://doi.org/10.1090/S0002-9939-09-09818-9

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.