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Dear Andreas Blass: The example you presented may be seen as a sequence $[f(T^n(x))]_n$, where $f: X \to R$ and $T: X \to X$ are measurable (are even better than that). For those $f$ and $T$, the Birkhoff (arithmetical) average of $f$ [that is, $f_n(x)=1/n \sum_{j=1}^{n-1} f (T^j(x))$] converges for Lebesgue almost every $x$. Do you know an example (where measurability, in spite of $f$ and $T$ being measurable, is lost through a non-principal ultrafilter limit) such that the sequence $(f_n)_n$ is a Birkhoff average?

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@DavideGiraudo Most likely this one. – Harald Hanche-Olsen Jan 21 '13 at 15:34
Here is the link with Andreas Blass' example:… – user59396 Jan 23 '13 at 17:38

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