A note on asymptotics of discounted value function and strong 0-discount optimality

The paper considers a Markov decision process with a Borel state space, bounded rewards, and a bounded transition density satisfying a simultaneous Doeblin-Doob condition. An asymptotics for the discounted value function related to the existence of stationary strong 0-discount optimal policies is extended from the case of finite action sets to the case of compact action sets and continuous in action rewards and transition densities.