  This paper studies the approximation of optimal control policies by quantized
(discretized) policies for a very general class of Markov decision processes
(MDPs). The problem is motivated by applications in networked control systems,
computational methods for MDPs, and learning algorithms for MDPs. We consider
the finite-action approximation of stationary policies for a discrete-time
Markov decision process with discounted and average costs under a weak
continuity assumption on the transition probability, which is a significant
relaxation of conditions required in earlier literature. The discretization is
constructive, and quantized policies are shown to approximate optimal
deterministic stationary policies with arbitrary precision. The results are
applied to the fully observed reduction of a partially observed Markov decision
process, where weak continuity is a much more reasonable assumption than more
stringent conditions such as strong continuity or continuity in total
variation.
