Is Birkhoff's pointwise/individual ergodic theorem for $L\_\infty.$ Clearly, it is true if the measure space is finite? What about the measure space not finite?



Is Birkhoff's ergodic theorem true for $L_\infty$?
Here is a counter-example to the question I think you're asking. Let $X$ be the space of two-sided 0-1 valued sequences. Let $x$ the sequence with $x\_n=0$ for all $n\le 0$; $x\_n=1$ if $2^k<n\le 2^{k+1}$ for any even $k\ge 0$ and $x\_n=0$ if $2^k<n\le 2^{k+1}$ for any odd $k\ge 0$. Then form an infinite measure by $\mu=\sum\_{j\in\mathbb Z}\delta\_{\sigma^j(x)}$. It's easy to check that if $f(z)=z\_0$, averages fail to converge for $\mu$-a.e. $z$. Notice this isn't a great counter-example because the measure $\mu$ fails to be conservative. There are conservative counter-examples also (based on renewal sequences).


