M. Viana, K. Oliveira. Foundations of Ergodic Theory. Chapter $5$, Ergodic Decomposition. Page $143$.
In this case, is not every subset measurable ?
I mean $\hat B$ consists of all subsets ?
2026-03-26 16:07:07.1774541227
M. Viana, K. Oliveira. Foundations of Ergodic Theory. Chapter $5$, Ergodic Decomposition. Page $143$
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No. Consider $M=\mathbb{R}$ and $\mathcal{P}\cong [0,1)$ where I've declared $x\sim y$ if $x-y\in \mathbb{Z}$. Then, you can check that $\hat{\mathcal{B}}$ is simply the ordinary Borel algebra on $[0,1)$ (it stems from the fact that the quotient map is a quotient map in the topological sense, when $\mathcal{P}$ is identified with the circle).