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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality 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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Joint measurability and the one-way Fubini property for a continuum of independent random variables
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by Peter J. Hammond and Yeneng Sun PDF
Proc. Amer. Math. Soc. 134 (2006), 737-747 Request permission

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
  • Email: peter.hammond@stanford.edu
  • 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
  • Email: matsuny@nus.edu.sg
  • 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.
  • © Copyright 2005 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 134 (2006), 737-747
  • MSC (2000): Primary 28A05, 60G07; Secondary 03E20, 03H05, 28A20
  • DOI: https://doi.org/10.1090/S0002-9939-05-08016-0
  • MathSciNet review: 2180892