In Walters' An Introduction to Ergodic Theory, page 153, Remark (2), he writes

If $E(X,T)$ denotes the set of extreme points of $M(X,T)$ then for each $\mu \in M(X,T)$ there is a unique measure $\tau$ on the Borel subsets of the compact metrisable space $M(X,T)$ such that $\tau(E(X,T))=1$ and $\forall f \in C(X)$ $$\int_{X} f(x) \ d\mu(x) = \int_{E(X,T)} \left( \int_{X} f(x) \ dm(x)\right) \ d\tau(m).$$ We write $\mu = \int_{E(X,T)} m \ d\tau(m)$ and call this the ergodic decomposition of $\mu$.

Here $X$ is a compact metric space, $T:X\to X$ is a continuous, $M(X,T)$ is the space of $T$-invariant ($\mu\circ T^{-1} = \mu$) Borel probability measures, and $C(X)$ is the space of continuous, real-valued functions on $X$.

Why is $E(X,T)$ a Borel set? In this paper, the author points out on page 5 that the set of extremal points is not Borel in general. He then claims that he'll give sufficient conditions that guarantee set of extremal will be Borel, but doesn't seem to. Does he give such conditions? If so, where?

By saying "we write $\mu = \int_{E(X,T)} m \ d\tau(m)$", Walters seems to be introducing shorthand notation. Indeed, since the integrand $f(m)=m$ is not real- (nor complex-) valued, that integral doesn't make sense. Why is the notation $\mu = \int_{E(X,T)} m \ d\tau(m)$ appropriate? In particular, I suspect that experts have some intuition about this decomposition that fits this notation, but I cannot see what it is.

  1. Let $K$ be a compact convex metrizable set in a topological vector space. Then the set of extremal points is a $G_\delta$ (a countable intersection of open sets).

    Indeed, fix a compatible metric $d$ on $K$. Let $$F_n = \left\{x \in K\,:\, \text{there are } y,z \in K\text{ such that }x = \frac{1}{2}(y+z)\text{ and }d(y,z) \geq \frac{1}{n}\right\}.$$ Then $F_n$ is closed and a point is non-extremal if and only if it is in $F = \bigcup_n F_n$. Thus the set of extremal points $\operatorname{ex}{K} = K \smallsetminus F$ is a $G_\delta$, in particular it is Borel.

    Observe that the set of Borel probability measures on $X$ is compact convex and metrizable in its weak$^\ast$ topology and that the $T$-invariant measures are a closed convex subset.

  2. The integral is to be understood in the barycenter sense: Let $K$ be a compact convex set in a locally convex topological vector space $E$. Then for each Borel probability measure $\mu \in M(K)$ on $K$ there is a unique point $b_\mu \in K$ such that for all continuous linear functionals $\varphi \in E'$ we have $\varphi(b_\mu) = \int_K (\varphi|_K)(k) \,d\mu(k)$: Uniqueness follows easily from Hahn-Banach and existence follows from approximating $\mu$ by convex combinations of point measures and using compactness of $K$.

    Added: The barycenter map $b: M(K) \to K, \mu \mapsto b_\mu$ should be thought of as a generalization of convex combinations of points in $K$: If $k \in K$ is a point and $\mu = \delta_k$ is the point measure at $k$ then $b_\mu = k$ since $\int_K \varphi \,d\delta_k = \varphi(k)$. If $\mu = t_1 \delta_{k_1} + \cdots + t_n \delta_{k_n}$ is a convex combination of point measures then $b_\mu = t_1 k_1 + \cdots + t_n k_n$ because $\varphi(t_1 k_1 + \cdots + t_n k_n) = \int_K \varphi\,d\mu$ for all continuous linear functionals $\varphi \in E'$.

    Uniqueness of the barycenter: Given $\mu \in M(K)$ there can be at most one point $b$ such that $\varphi(b) = \int_K \varphi\,d\mu$ for all continuous linear functionals $\varphi \in E'$ since for every other point $b'$ we can find a continuous linear functional $\varphi$ such that $\varphi(b') \neq \varphi(b) = \int \varphi\,d\mu$ by Hahn-Banach.

    Existence of the barycenter: If $\mu \in M(K)$ is an arbitrary probability measure then $\mu$ is a weak$^{\ast}$-limit of a net $\{\mu_i\}_{i \in I}$ of convex combinations of point measures with corresponding barycenters $b_i = b_{\mu_i} \in K$. By passing to a subnet if necessary we may assume that $b_i \to b$ in $K$. For every continuous linear functional $\varphi \in E'$ we then have on the one hand $\varphi(b_i) \to \varphi(b)$ and on the other hand $\varphi(b_i) = \int_K \varphi\,d\mu_i \to \int_{K} \varphi\,d\mu$ by weak$^\ast$-convergence, so $\varphi(b) = \int_{K} \varphi\,d\mu$ and thus we can put $b_\mu = b$.

For a thorough introduction to all these ideas I recommend having a look at Phelps's Lectures on Choquet's theorem, Springer LNM 1757.

  • $\begingroup$ Should "...for each Borel probability measure $\mu$ on $K$..." be "...for each Borel probability measure $\mu$ on $X$..."? $\endgroup$ – Quinn Culver Dec 29 '11 at 20:07
  • $\begingroup$ No. I've expanded a bit, I hope it is clearer now. Note that I didn't really address ergodic decomposition itself, which is the special case $K = M(X,T)$. But I recommend to try and understand the construction in the abstract before specializing to case of interest. I can add more explanations if needed. $\endgroup$ – t.b. Dec 30 '11 at 2:30
  • $\begingroup$ This is a great answer. Thank you. $\endgroup$ – Quinn Culver Dec 30 '11 at 21:12
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    $\begingroup$ The integral mentioned can also be understood in the Bochner sense, and generally you can integrate (nice) Banach space-valued functions on some measure space. In this case the Banach space is the space of finite, signed Borel measures on $X$ with the total variation norm and the measure space is the set of $T$-invariant Borel probability measures, with the weak-* topology, the Borel $\sigma$-algebra and the measure $\tau$. The function to be integrated is just the indicator function restricted to the subset of ergodic $T$-invariant probability measures. $\endgroup$ – Mark Dec 31 '11 at 2:29
  • $\begingroup$ Minor correction of the above comment: "identity function", not "indicator function". $\endgroup$ – Mark Dec 31 '11 at 21:56

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