Density averaging for Markov processes and the invariant measures problem

A necessary and sufficient condition is given for the existence of a finite invariant measure equivalent to a given reference measure for a discrete time, general state Markov process. The condition is an extension of one given by D. Maharam in the deterministic case and involves an averaging method (called by Maraham ‘density averaging’) applied to the Radon-Nikodym derivatives with respect to the reference measure of the usual sequence of measures induced by the Markov process acting on the fixed reference