Invariant probability measures for a class of Feller Markov chains

In this paper we consider a Markov chain defined on a locally compact separable metric space which satisfies the Feller property. We introduce a new assumption which generalizes T-chain and irreducibility assumptions, well known in the literature of Markov chains. Under this new assumption, the Foster's criterion is shown to be equivalent to the existence of an invariant probability measure for Feller-Markov chains, which is also equivalent to the existence of a non-singular invariant probability measure.