I am looking for a proof for the following statement; Given a Markov kernel P, and let I be the set of all invariant probability measures for P, and let E be subset of I, being the collection of ergodic measures, then I is convex and E is the set of its extremal points. Further for every invariant measure $\mu$ there exists a probability measure $Q_{\mu}$ on E such that
$$\mu(A)= \int_{E} \nu(A) Q_{\mu}(d\nu) $$ In addition, any two distinct elements of E are mutually singular