An elegant characterization of optimal strategies for gambling problems was given by Dubins and Savage in the finitely additive setting of their book How to Gamble If You Must. An exposition of their ideas is given here in a measurable, countably additive framework. With the additional measurability assumptions, it becomes possible to treat a larger class of payoff functions. Also, necessary and sufficient conditions are given for a strategy to be $\epsilon$-optimal, a problem not considered by Dubins and Savage.
William D. Sudderth. "On the Dubins and Savage Characterization of Optimal Strategies." Ann. Math. Statist. 43 (2) 498 - 507, April, 1972. https://doi.org/10.1214/aoms/1177692630