I am trying to come up with a counterexample to this theorem under the assumption that the space is not sigma finite. I tried working with the power set of the real numbers with the measure $\mu(A) = \sum_{a\in A} \lambda(a)$, where $a\subset \mathbb{R}$ and $A\subset \mathcal{P}(\mathbb{R})$.
Specifically the theorem states:
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 $g\in L^1(m)$.