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Joint measurability and the one-way Fubini property for a continuum of independent random variables
Author(s):
Peter
J.
Hammond;
Yeneng
Sun
Journal:
Proc. Amer. Math. Soc.
134
(2006),
737-747.
MSC (2000):
Primary 28A05, 60G07;
Secondary 03E20, 03H05, 28A20
Posted:
July 18, 2005
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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:
10.1090/S0002-9939-05-08016-0
PII:
S 0002-9939(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
Posted:
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 of article:
Copyright
2005,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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