Take the tour ×  Here is a quote from Lectures on Ergodic Theory
by Halmos:  I cannot resist the temptation of concluding these
comments with an alternative "proof" of the ergodic theorem. If
$f$ is a complex valued function on the nonnegative integers,
write $\int f(n)dn=\lim > \frac{1}{n}\sum_{j=1}^nf(n)$ and
whenever the limit exists call such functions integrable. If $T$
is a measure preserving transformation on a space $X$, then $$ >
\int\int|f(T^nx)|dndx=\int\int|f(T^nx)|dxdn=\int\int|f(x)|dxdn=\
int|f(x)|dx<\infty. > $$ Hence, by "Fubini's theorem" (!),
$f(T^nx)$ is an integrable function of its two arguments, and
therefore, for almost every fixed $x$ it is an integrable
function of $n$. Can any of this nonsense be made meaningful?
Can any of this nonsense be made meaningful using nonstandard
analysis? I know that Kamae gave a short proof of the ergodic
theorem using nonstandard analysis in A simple proof of the
ergodic theorem using nonstandard analysis, Israel Journal of
Mathematics, Vol. 42, No. 4,