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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)


Leavable gambling problems with unbounded utilities

Authors: A. Maitra, R. Purves and W. Sudderth
Journal: Trans. Amer. Math. Soc. 320 (1990), 543-567
MSC: Primary 60G40; Secondary 03E15, 03E35, 62L15, 90D60, 93E20
MathSciNet review: 989581
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Abstract: The optimal return function $ U$ of a Borel measurable gambling problem with a positive utility function is known to be universally measurable. With a negative utility function, however, $ U$ may not be so measurable. As shown here, the measurability of $ U$ for all Borel gambling problems with negative utility functions is equivalent to the measurability of all PCA sets, a property of such sets known to be independent of the usual axioms of set theory. If the utility function is further required to satisfy certain uniform integrability conditions, or if the gambling problem corresponds to an optimal stopping problem, the optimal return function is measurable. Another return function $ W$ is introduced as an alternative to $ U$. It is shown that $ W$ is always measurable and coincides with $ U$ when the utility function is positive.

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Additional Information

PII: S 0002-9947(1990)0989581-5
Keywords: Measurable gambling, réduite, optimal stopping, measurable strategies, analytic sets, PCA sets
Article copyright: © Copyright 1990 American Mathematical Society

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