$\mu$-everywhere existence of a limit.

Question

Let $(X, \mathcal{B}, \mu, T)$ be a measure preserving system and $f_n$ a sequence of dominated functions in $L^1$ which converges almost everywhere to $f \in L^1$. Prove that $\frac{1}{n}\sum_{k=0}^{n-1}f_k(T^k(x)) \rightarrow \tilde{f(x)}$ almost everywhere, where $\tilde{f(x)}$ as in Birkhoffs pointwise theorem (https://en.wikipedia.org/wiki/Ergodic_theory#Ergodic_theorems).
Actually, I want to prove something less general, but I found this generalization in Ricardo Mane's textbook and I decided to give it a shot. The problem is I'm not quite sure how to approach it, and doing it so by standard machine method doesn't seem to me as a good idea, since our claim is for $L^1$ functions and not just measurable. Any hints appreaciated.

Answer 1

Considering $\widetilde{f_n}:=\left|f_n-f\right|$ instead of $f_n$, we can assume that $f_n$ is non-negative and that $f=0$. Observe that for any $x$, each positive integer $N$ and $n\geqslant N$, $$\frac 1n\sum_{k=0}^{n-1}f_k\left(T^kx\right) \leqslant \frac 1n\sum_{k=0}^{R-1}f_k\left(T^kx\right)+ \frac 1n\sum_{k=R}^{n-1}\sup_{i\geqslant R}f_i\left(T^kx\right).$$ Applying Birkhoff's pointwise ergodic theorem, we infer that for almost every $x$, $$\limsup_{n\to +\infty}\frac 1n\sum_{k=0}^{n-1}f_k\left(T^kx\right) \leqslant \mathbb E\left[\sup_{i\geqslant R}f_i \mid\mathcal I\right]\left(x\right),$$ where $\mathcal I$ denotes the $\sigma$-algebra of $T$-invariant sets. Now conclude by monotone convergence.