Birkhoff Ergodic Theorem for sequences of functions

Question

The Birkhoff Ergodic Theorem provides us with almost sure convergence of the series $$\frac{1}{n}\sum_{j=0}^{n-1}f\circ T^j$$ provided that $f\in L^1$ and $T$ is measure preserving. Can this same type of convergence be guaranteed if the observable $f$ is dependent on the index $j$ with $f_j\in L^1$ for every $j$? Or example if $$f=f_j = \prod_{k=0}^{j-1}a_k(x).$$ I suspected the answer to this question would be yes since both $f\circ T^j$ and $f_j\circ T^j$ would be dependent on $j$ however I my only concern here would be that the $j$ dependence appears in the function that has nothing to do with $x$.

Answer 1

If the sequence $f_n$ converges almost everywhere and in $L^1$ to a function $f$, then $$ \lim_{n\to\infty}\frac{1}{n}\sum_{j=0}^{n-1}f_{n-j}\circ T^j =\lim_{n\to\infty}\frac{1}{n}\sum_{j=0}^{n-1}f\circ T^j $$ almost everywhere and in $L^1$ (the right-hand side is well defined because of Birkhoff's ergodic theorem).

For a proof you can see for example Corollary 1.9 in Mañé's book. Note that the argument, unsurspringly, uses Birkhoff's ergodic theorem, but it is not really immediate.