  We consider average-cost Markov decision processes (MDPs) with Borel state
and action spaces and universally measurable policies. For the nonnegative cost
model and an unbounded cost model with a Lyapunov-type stability character, we
introduce a set of new conditions under which we prove the average cost
optimality inequality (ACOI) via the vanishing discount factor approach. Unlike
most existing results on the ACOI, our result does not require any compactness
and continuity conditions on the MDPs. Instead, the main idea is to use the
almost-uniform-convergence property of a pointwise convergent sequence of
measurable functions as asserted in Egoroff's theorem. Our conditions are
formulated in order to exploit this property. Among others, we require that for
each state, on selected subsets of actions at that state, the state transition
stochastic kernel is majorized by finite measures. We combine this majorization
property of the transition kernel with Egoroff's theorem to prove the ACOI.
