Whether stationary families of strategies are uniformly adequate for a leavable, analytically measurable, nonnegative gambling problem whose optimal return function is everywhere finite is a question which remains open. It is, however, given an affirmative answer if, for example, the fortune space is Euclidean and all nontrivial, available gambles are absolutely continuous with respect to Lebesgue measure.
"On Stationary Strategies for Absolutely Continuous Houses." Ann. Probab. 7 (3) 461 - 476, June, 1979. https://doi.org/10.1214/aop/1176995047