Ergodic decomposition

Let X be a compact metric space and f:X\to X be a homeomorphism. Let \mathcal{M}(f) be the set of f-invariant measures and \mathcal{E}(f) be the set of f-invariant ergodic measures.

Let \mu be an invariant measure. A distribution \tau on \mathcal{E}(f) is said to be the ergodic decomposition of \mu if for each continuous function \phi:X\to\mathbb{R}, the following holds:

\int_{\mathcal{E}(f)}(\int_X \phi d\nu) d\tau(\nu)=\int_X \phi d\mu.

The following approach is attributed to R. Mane.
Let G_\nu be the set of generic points of \nu and G=\bigcup_{\nu\in \mathcal{E}(f)}G_\nu. Then G is a Borel subset of X and of full probability. Consider the map \beta:G\to \mathcal{E}(f),x\mapsto\nu_x=\lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}\delta_{f^kx}.

Proposition: The map \beta:G\to \mathcal{E}(f) is Borel.

So the pushforwad \tau=\beta_*(\mu) is a Borel distribution on \mathcal{E}(f). Moreover for each continuous map \Phi:\mathcal{M}(X)\to \mathbb{R}, we have

In particular letting \Phi(\nu)=\int_X \phi d\nu, we have \Phi(\beta(x))=\int\phi d\nu_x=\phi^*(x) and

\int_{\mathcal{E}(f)}(\int_X \phi d\nu)d\tau(\nu)=\int_G \phi^*(x)d\mu(x)=\int_X\phi(x)d\mu(x), where \phi^* is the Birkhoff average and the last equality follows from Brikhoff ergodic theorem. So \tau=\beta_*(\mu) is the ergodic decomposition of \mu.

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