Markov operators with a unique invariant measure

LetMbe the set of all finite Borel measures on a Polish space X. Let P be a Markov operator onMand π the transition function corresponding to P. Set Γ (x) = supp π(x, ·), x ∈ X. It is proved that, if P admits a unique invariant measure μ∗, then μ∗(D) = 0 or μ∗( ∞n=0 Γ n(D)) = 1 for every Borel set D such that Γ (D) ⊂ D. Moreover, if P is nonexpansive, then a trajectory of every Markov chain corresponding to P and starting from suppμ∗ is dense in suppμ∗. The last statement fails if we drop nonexpansivity condition.  2002 Elsevier Science (USA). All rights reserved.