On finite long run costs and rewards in infinite Markov chains

Conditions for the finiteness of long run costs and rewards associated with infinite recurrent Markov chains that may be discrete or continuous in time are considered. Without resorting to results from the theory of Markov processes on general state spaces we provide instructive proofs in the course of which we derive auxiliary results that are of interest in themselves. Potential applications of the finiteness conditions are outlined in order to elucidate their high practical relevance.