We consider the problem of maximizing the expectation of the discounted total reward in Markovian decision processes with arbitrary state space and compact action space varying with the state. We get the existence theorem for a $(p, \epsilon)$-optimal stationary policy, and the relation between the optimality of a policy and the optimality equation. Assuming the action space is a compact subset of $n$-dimensional Euclidean space, the existence of an optimal stationary policy is established, and an algorithm is obtained for finding the optimal policy. The last two facts are based on the Borel implicit function lemma given in this paper.
Nagata Furukawa. "Markovian Decision Processes with Compact Action Spaces." Ann. Math. Statist. 43 (5) 1612 - 1622, October, 1972. https://doi.org/10.1214/aoms/1177692393