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Let $G$ be compact (and Hausdorff) group, $\mu$ be Haar measure on $G$. Is it always true that $(G,\mu)$ is a standard probability space (Lebesgue-Rokhlin space)? It is so if (a priori not iff) the topology of $G$ is metrizable, but this is probably not necessary.

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Well, some silly counter-examples: A finite group. Or $\{0,1\}^I$ for some very, very large set $I$. –  Matthew Daws May 11 '11 at 19:10
    
well, finite does not count, I had to mention it the second example is ok, thank you –  Fedor Petrov May 11 '11 at 19:54
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