I'm trying to understand the tricky proof of the ergodic theorem (Birkhoff 1931). My reference is "Ward,Einsiedler - Ergodic theory (with a view towards number Theory)" section 1.6:
Consider the sequence of functions $$a_n(x)=\frac{1}{n}\sum_{i=0}^{n-1}f(T^i(x))$$ where $T$ is a measure preserving transformation and $f\in L^1_\mu$ (suppose real valued) . I've understand the part of the proof where it is proved that the pointwise limit $f^\ast $does exist but I've problems to show that $f^\ast\in L^1_\mu $. In particular, the following passage is obscure:
Lets define $g_n(x)=|a_n(x)|$, why it is true that $$\int g_n\,d\mu\le\int|f|\,d\mu$$
for every $n$?
(The rest of the proof follows by the Fatou's lemma)
From the triangle inequality, $$\lVert a_n\rVert_1=\frac 1n\left\lVert\sum_{i=0}^{n-1}f\circ T^i\right\rVert_1\leqslant \frac 1n\sum_{i=0}^{n-1}\left\lVert f\circ T^i\right\rVert_1.$$ To conclude, notice that since $T$ is measure preserving, then $$\tag{*}\int g\mathrm d\mu=\int g \circ T\mathrm d\mu$$ for any $g$ which can be expressed as a linear combination of characteristic functions. By an approximation argument, we extend (*) to any integrable function $g$ and we conclude that $$\lVert a_n\rVert_1\leqslant \lVert f\rVert_1.$$