Consider regular continued fraction expansions of numbers in $(0,1)$, as generated by the Gauss map $Tx=\frac1x\pmod1$. I managed to show, using ergodic theory, that the geometric mean of the digits $(a_n)_{n\in\mathbb N}$ converges to Khinchin's constant. In some sources it is stated that the arithmetic mean $\frac{a_1+\cdots+a_n}{n}$ diverges as $n\to\infty$ for $\lambda$-a.a. $x\in(0,1)$. I tried proving this in a similar fashion as convergence of the geometric mean is established (see linked Wikipedia page), but here we get for the ergodic average, writing $a_k=a_1(T^{k-1}x),$ where $a_1(x)=n$ for $x\in[\frac1{n+1},\frac1n)$,$$\lim_{n\to\infty}\frac1n\sum_{k=1}^na_k(x)=\lim_{n\to\infty}\frac1n\sum_{k=0}^{n-1}a_1(T^kx)=\int_0^1a_1=\sum_{n\geq1}\int_{\frac1{n+1}}^{\frac1n}n\ \mathrm dx=\sum_{n\geq1}\frac1{n+1}=\infty.$$ This would follow, $\lambda$-a.e., using the Birkhoff Ergodic Theorem, but this theorem assumes that $a_1\in L^1$, which is clearly not the case by the above display.
I do not see how to justify the application of the Birkhoff Ergodic Theorem, and also cannot think of another approach. Any help is much appreciated.
N.B.: on these slides, which I found online, it even seems that this same approach is used (slide 15). Again, I do not see why we can apply Birkhoff Ergodic Theorem when $a_1\notin L^1$, since $\int a_1=\infty$.