We consider the classical Foster-Lyapunov condition for the existence of an
invariant measure for a Markov chain when there are no continuity or irred
ucibility assumptions. Provided a weak uniform countable additivity conditi
on is satisfied, we show that there are a finite number of orthogonal invar
iant measures under the usual drift criterion, and give conditions under wh
ich the invariant measure is unique. The structure of these invariant measu
res is also identified. These conditions are of particular value for a larg
e class of non-linear time series models. (C) 2001 Elsevier Science B.V. Al
l rights reserved.