ORTHOGONAL MEASURES AND ABSORBING SETS FOR MARKOV-CHAINS

For a general state space Markov chain on a space (X, B(X)), the exist ence of a Doeblin decomposition, implying the state space can be writt en as a countable union of absorbing 'recurrent' sets and a transient set, is known to be a consequence of several different conditions all implying in some way that there is not an uncountable collection of ab sorbing sets. These include (M) there exists a finite measure which gi ves positive mass to each absorbing subset of X; (G) there exists no u ncountable collection of points (x(alpha)) such that the measures K-th eta(x(alpha), (.)):=(1.-theta)Sigma P-n(x(alpha),(.)) theta(n), are mu tually singular; (C) there is no uncountable disjoint class of absorbi ng subsets of X. We prove that if B(X) is countably generated and sepa rated (distinct elements in X can be separated by disjoint measurable sets), then these conditions are equivalent. Other results on the stru cture of absorbing sets are also developed.