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.