# Reference request: stationary measures as convex combinations of ergodic measures

Does anyone know a good reference for the fact that a stationary probability measure is a convex combination of the stationary and ergodic probability measures?

I have found some references for the fact that the ergodic measures are the extreme points of the set of stationary measures, but I don't see how it follows directly from this that all the stationary measures are convex combinations of the extreme points (although maybe I'm missing something obvious).

I also have figured out a quick constructive way to show this fact. If $\mu$ is a stationary probability measure and $\mathcal{I}$ is the shift-invariant $\sigma$-field then we can write $$\mu(\cdot) = \int_\Omega \mu( \cdot \, | \mathcal{I} )(\omega) \, \mu(d\omega),$$ and it's easy to see that for almost every $\omega$ the conditional probability $\mu(\cdot \, | \mathcal{I})(\omega)$ is an ergodic probability measure. I think this is a nice short argument, but I haven't been able to find this or any other reference in the books I've looked at so far.

• In the context of ergodic versus non-ergodic invariant measures of a topological dynamical system this follows from a suitable version of Choquet's theorem, which is a general result describing how a points in a compact convex set can be written as an integral with respect to a measure which gives full measure to the set of extreme points. It is possible that something similar to this works in the abstract measurable setting but I am not sure of the details. This question may be somewhat relevant: mathoverflow.net/questions/124066/… – Ian Morris Sep 19 '13 at 17:20
• Choquet's theorem works in a very general settings (std. probability space?), maybe under the name Choquet-Bishop-deLeeuw, or the "disintegration theorem". Check the little book by Phelps called "Lectures on Choquet's theorem" – Asaf Sep 19 '13 at 17:51
• And another remark about terminology - a stationary measure (in a Bernoulli system) is nothing more than an invariant measure. – Asaf Sep 19 '13 at 17:54
• I wrote a survey on mixing and ergodicity that appears in the (enormous) Springer Encyclopedia of Complexity. I think it's on my web page too. This has an argument along the lines that you're mentioning. I don't think it's as easy as you're saying though. You could also look at Dan Rudolph's book. Again, ergodic decomposition isn't so easy there... – Anthony Quas Sep 19 '13 at 18:03
• For what it's worth, the proof of ergodic decomposition in Rudolph's book works in the generality of Lebesgue spaces – Anthony Quas Sep 19 '13 at 18:13