Ultrafilters and Birkhoff averages

Question

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?