Drift conditions and invariant measures for Markov chains

We consider the classical Foster–Lyapunov condition for the existence of an invariant measure for a Markov chain when there are no continuity or irreducibility assumptions. Provided a weak uniform countable additivity condition is satisfied, we show that there are a finite number of orthogonal invariant measures under the usual drift criterion, and give conditions under which the invariant measure is unique. The structure of these invariant measures is also identified. These conditions are of particular value for a large class of non-linear time series models.