Leavable gambling problems with unbounded utilities

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.