Many authors have studied the problem of obtaining an optimal plan for a dynamic programming problem or a stochastic game. In several of these works a maximal selection theorem of Dubins and Savage for u.s.c. (upper-semi-continuous) maps has played a key role. Our purpose is to relax the requirements that the maps be u.s.c. and still obtain maximal or at least nearly maximal selections.
"Remarks on Measurable Selections." Ann. Statist. 2 (4) 845 - 847, July, 1974. https://doi.org/10.1214/aos/1176342775