My question refers to the yesterday's question (see here) of John Learner and goes as follows:

Can we deduce the existence of an ergodic measure if we know that an invariant measure exists, but the state space is noncompact? Let say it is complete and separable. To be more precise, let us assume that we have a Markov operator $P$ (a linear operator on the space of bounded measurable functions on complete and separable $\Omega$). We consider the discrete-time Markov chain corresponding to $P$. We assume that some $P$-invariant measure $\mu$ exists on $\Omega$, i.e. there is a measure $\mu$ such that for any bounded measurable function $g$ on $\Omega$, we have $$\int_{\Omega}Pg\,d\mu=\int_{\Omega}g\,d\mu.$$ Can we deduce that an ergodic invariant measure exists? (And can we deduce the existence of two distinct ergodic invariant measures from the existence of two distinct invariant measures?)

My doubt concernes the existence of an extremal point of the set of all invariant measures. I know that the set of all ergodic invariant measures forms the subset of extremal points of the set of invariant measures. So by the Krein–Milman theorem it is nonempty provided if we assume that $\Omega$ is compact. Is there any generalization of this theorem which would be useful in my situation?

  • 2
    $\begingroup$ Complete and separable -> Polish -> Borel equivalent to $[0,1]$ (except when countable, but that case is obvious) -> as good as compact for all measure theory purposes. $\endgroup$ – fedja Aug 16 '13 at 21:57

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