If a nonnegative measurable f is not in L^1, then the ergodic averages diverge a.e. I have been trying to prove the following implication but have not succeeded yet.
Let $(X,\mathscr{B}, \mu,T)$ be an ergodic system, and let $f:X\rightarrow [0,\infty)$ be a measurable function with
$$
\limsup\frac{1}{n}\sum_{k=0}^n f(T^kx)<\infty.
$$
Then show that $f\in L^1(X,\mathscr{B},\mu)$.
I tried to integrate and then possibly use the reverse Fatou’s lemma to bring the limsup out of the integral. But, this is not quite true. I can’t think of any other way to approach the proof. Any hint would be highly welcome.

By the Ergodic Theorem we have, for almost all $x$, $\int (f\wedge N) d\mu =\lim \frac  1n \sum\limits_{k=0}^{n}(f\wedge N)(T^{k}(x))\leq \lim \sup  \frac  1n \sum\limits_{k=0}^{n}f( T^{k}(x))$ for each $N$. Take sup over $N$.
