Average optimal switching of a Markov chain with a Borel state space

We extend results on average per unit time optimality criterion in a switching model from a countable state space to a Borel state space. In the model we consider, a controller selects an increasing sequence of stopping times with respect to a Markov chain, and gets rewards and pays costs at them in an alternating order. The rewards and costs depend on the state of the chain. We find the optimal average gain and construct an optimal strategy. The basic tool is a variational problem with two obstacles that appears also in Dynkin games.