The Birkhoff Ergodic Theorem states:


Let $(X,\mathcal{B},m)$ be a finite or sigma finite measure space. Suppose $T:(X,\mathcal{B},m)\to (X,\mathcal{B},m)$ is measure-preserving and $f\in L^1(m)$. Then 


$$\lim\_{n\to \infty} \frac{1}{n} \sum\_{i=0}^{n-1} f(T^i(x))$$


converges a.e. to a function $g\in L^1(m)$. 


Can we say anything if $(X,\mathcal{B},m)$ is larger than sigma finite? I know in applications these sort of spaces are just "too large", but I can't come up with a proof or find a counter-example (i.e. $f$ and $T$ satisfying the hypothesis such that the yielded $g \not \in L^1(m)$.


I've also posted this here: 


<https://math.stackexchange.com/questions/785707/birkhoff-ergodic-theorem-counterexample>



Birkhoff Ergodic Theorem or Counterexample
The support $S:=\{f\neq0\}$ of an integrable function $f$ is in any case $\sigma$-finite, and so is the $T$-invariant set $$X':=\limsup\_{j\to\infty} T^{-j}(S):=\bigcap\_{k\ge0}\bigcup\_{j\ge k}T^{-j}(S). $$ Clearly the time average is zero in the complement of $X'$ (indeed, for these the points $f(T^j(x))$ eventually vanishes), and we are reduced to the $\sigma$-finite case of the measure-preserving transformation $ T\_{|X'}:X'\to X'$.


**Rmk:** Note that both $T(X')\subset X'$ and $T^{-1}(X')\subset X'$ hold true, and are needed, in order that $ T\_{|X'}$ be a true measure-preserving transformation on $ X'$, so that the reduction argument holds plainly.


