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I have the following question:

Let $X$ be a compact Hausdorff topological space with a continuous group action from a topological amenable group $G$. Assume I have finitely many closed invariant subsets $Y_1,...,Y_k\subseteq X$.

If I am not mistaken, the set $\hat{Y}:=\cup_{j=1}^k Y_j$ is also invariant and closed. I was wondering whether every Radon invariant probability measure $\mu$ on $\hat{Y}$ can be written as $\mu= \sum_{j=1}^k p_j \nu_j$, where $\sum_{j=1}^kp_j=1$, $p_1,...,p_k\in [0,1]$ and each $\nu_j$ is a Radon invariant probability measure on $Y_j$?

Since a convex combination of Radon invariant probability measures is an invariant probability measure, such a convex combination is an invariant probability measure on $\hat{Y}$. I have a feeling that if this is true, then one should be able to naturally construct such probability measures from a given measure. So I was wondering whether there exist counter-examples which I'm not familiar with.

Keen-ameteur
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1 Answers1

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This is precisely the content of Ergodic Decomposition which can be stated as follows:

Let $ (X, \mathcal{B}, \mu, T) $ be a measure-preserving system, where $ X $ is a compact metric space, $\mathcal{B} $ is the Borel $\sigma$-algebra, and $\mu $ is a Radon measure. Then for $\mu$-a.e. $y$ there exists an invariant ergodic Radon probability measure $\lambda_y $ such that for every $ f \in L^1(\mu) $, $ \int_X f \, d\mu = \int_{X} \left( \int_X f(x) \, d\lambda_y(x) \right) d\mu(y) $ (and the map $ y \rightarrow \int_X f d\lambda_y$ is T-inv and measurable).

Now the ergodic components are exactly the invariant sets, and this theorem tells you any measure on the space is a convex combination of ergodic ones, ie Radon measures on your sets $Y_i$. This is proved using Choquet's theorem and appears in any standard ergodic theory textbook.

Derek
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