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Joint measurability and the one-way Fubini property for a continuum of independent random variables

Authors: Peter J. Hammond and Yeneng Sun
Journal: Proc. Amer. Math. Soc. 134 (2006), 737-747
MSC (2000): Primary 28A05, 60G07; Secondary 03E20, 03H05, 28A20
Published electronically: July 18, 2005
MathSciNet review: 2180892
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Abstract: As is well known, a continuous parameter process with mutually independent random variables is not jointly measurable in the usual sense. This paper proposes an extension of the usual product measure-theoretic framework, using a natural ``one-way Fubini'' property. When the random variables are independent even in a very weak sense, this property guarantees joint measurability and defines a unique measure on a suitable minimal $ \sigma $-algebra. However, a further extension to satisfy the usual (two-way) Fubini property, as in the case of Loeb product measures, may not be possible in general. Some applications are also given.

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Additional Information

Peter J. Hammond
Affiliation: Department of Economics, Stanford University, 579 Serra Mall, Stanford, California 94305–6072

Yeneng Sun
Affiliation: Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543 – and – Department of Economics, National University of Singapore, 1 Arts Link, Singapore 117570

Keywords: Loeb product measures, product-measurable sets, continuum of independent random variables, joint measurability problem, one-way Fubini property
Received by editor(s): July 31, 2002
Received by editor(s) in revised form: October 8, 2004
Published electronically: July 18, 2005
Additional Notes: Part of this work was done when the first author was visiting Singapore in November 1999 and when the second author was visiting Stanford in July 2002.
Communicated by: Carl G. Jockusch, Jr.
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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