Published in The Annals of Probability, Volume 11, Issue 2, May 1, 1983, pages 442-450. Copyright © 1983 Institute of Mathematical Statistics. The original publication is available at http://dx.doi.org/10.1214/aop/1176993609.
NOTE: At the time of publication, the author Theodore P. Hill was not yet affiliated with Cal Poly.
Comparisons are made between the expected returns using measurable and non-measurable stop rules in discrete-time stopping problems. In the independent case, a natural sufficient condition ("preservation of independence") is found for the expected return of every bounded non-measurable stopping function to be equal to that of a measurable one, and for that of every unbounded non-measurable stopping function to be arbitrarily close to that of a measurable one. For non-negative and for uniformly-bounded independent random variables, universal sharp bounds are found for the advantage of using non-measurable stopping functions over using measurable ones. Partial results for the dependent case are obtained.