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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The gap between probability and prevalence: Loneliness in vector spaces


Author: Maxwell B. Stinchcombe
Journal: Proc. Amer. Math. Soc. 129 (2001), 451-457
MSC (1991): Primary 28C20, 60B11; Secondary 90B40
Published electronically: July 27, 2000
MathSciNet review: 1694881
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Abstract:

The best available definition of a subset of an infinite dimensional, complete, metric vector space $V$ being ``small'' is Christensen's Haar zero sets, equivalently, Hunt, Sauer, and Yorke's shy sets. The complement of a shy set is a prevalent set. There is a gap between prevalence and likelihood. For any probability $\mu$ on $V$, there is a shy set $C$ with $\mu(C) = 1$. Further, when $V$ is locally convex, any i.i.d. sequence with law $\mu$ repeatedly visits neighborhoods of only a shy set of points if the neighborhoods shrink to $0$ at any rate.


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Additional Information

Maxwell B. Stinchcombe
Affiliation: Department of Economics, University of Texas at Austin, Austin, Texas 78712-1173
Email: maxwell@eco.utexas.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-00-05543-X
PII: S 0002-9939(00)05543-X
Received by editor(s): March 1, 1999
Received by editor(s) in revised form: April 19, 1999
Published electronically: July 27, 2000
Communicated by: Claudia Neuhauser
Article copyright: © Copyright 2000 American Mathematical Society