Necessary and sufficient conditions for non-singular invariant probability measures for Feller Markov chains

In this paper, we present necessary and sufficient conditions for the existence of a non-singular invariant probability measure for a Feller Markov chain taking values on a locally compact separable metric space. The necessary and sufficient condition is written in terms of the Foster's criterion with an extra requirement. Furthermore, we extend an assumption recently presented by the authors Costa and Dufour, Statist. Probab. Lett. 50 (3) (2000) 13-21, named T2 condition, which generalizes T-chain and irreducibility assumptions for Feller Markov chains on a locally compact separable metric space, and show that under this assumption the extra requirement on the Foster's criterion can be eliminated.