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 -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

DOI:
https://doi.org/10.1090/S0002-9939-05-08016-0

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.