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We consider average-cost Markov decision processes (MDPs) with Borel stateand action spaces and universally measurable policies. For the nonnegative costmodel and an unbounded cost model, we introduce a set of conditions under whichwe prove the average cost optimality inequality (ACOI) via the vanishingdiscount factor approach. Unlike most existing results on the ACOI, whichrequire compactness/continuity conditions on the MDP, our result does not andcan be applied to problems with discontinuous dynamics and one-stage costs. Thekey idea here is to replace the compactness/continuity conditions used in theprior work by what we call majorization type conditions. In particular, amongothers, we require that for each state, on selected subsets of actions at thatstate, the state transition stochastic kernel is majorized by finite measures,and we use this majorization property together with Egoroff's theorem to provethe ACOI.  We also consider the minimum pair approach for average-cost MDPs and applythe majorization idea. For the case of a discrete action space and strictlyunbounded costs, we prove the existence of a minimum pair that consists of astationary policy and an invariant probability measure induced by the policy.This result is derived by combining Lusin's theorem with another majorizationcondition we introduce, and it can be applied to a class of countable actionspace MDPs in which, with respect to the state variable, the dynamics andone-stage costs are discontinuous.

On Markov Decision Processes with Borel Spaces and an Average Cost Criterion