Orthogonal measures and absorbing sets for Markov chains

For a general state space Markov chain on a space (X, B(X)), the existence of a Doeblin decomposition, implying the state space can be written 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 absorbing sets. These include (M) there exists a finite measure which gives positive mass to each absorbing subset of X; (G) there exists no uncountable collection of points (x(alpha)) such that the measures K-theta(x(alpha), (.)):=(1.-theta)Sigma P-n(x(alpha),(.)) theta(n), are mutually singular; (C) there is no uncountable disjoint class of absorbing subsets of X. We prove that if B(X) is countably generated and separated (distinct elements in X can be separated by disjoint measurable sets), then these conditions are equivalent. Other results on the structure of absorbing sets are also developed.