The proof of Birkhoff ergodic theorem in the book of Peter Walters; An introduction to Ergodic Theory.
Page 39.
The second case when $m(X)=+\infty$.
After the sentence (The function $H_N$ ...) I couldn't understand many things.
Why is the set $E_{\alpha,\beta}$ contained in the union ?
Why is the $$\alpha m(C)\leq\int|f|dm$$
Why since $X$ is $\sigma -$ finite then $m(E_{\alpha,\beta})<+\infty$?
The inclusion $E_{\alpha,\beta}\subset \bigcup_{N=0}^{+\infty}\left\{H_N\gt 0\right\}$ follows from the fact that if $x\in E_{\alpha,\beta}$, then $$ \limsup_{n\to +\infty}\frac 1n\sum_{i=0}^{n-1}f\circ T^i\left(x\right)\gt \alpha $$ which means that $$ \limsup_{n\to +\infty}\frac 1n\sum_{i=0}^{n-1}\left(f-\alpha\right)\circ T^i\left(x\right)\gt 0 $$ and in particular, there should exists some $N$ such that $\left(f-\alpha\right)\circ T^N\left(x\right)\gt 0$ which implies that $\left(f-\alpha\mathbf 1_C\right)\circ T^N\left(x\right)\gt 0$ hence $H_N(x)\gt 0$.
The inequality obtained after an application of the maximal ergodic theorem reads $$ \alpha\cdot m\left(C\cap \left\{x: H_N(x)\gt 0\right\}\right)\leqslant \int_{\left\{ H_N \gt 0\right\}}f\mathrm dm. $$ The right hand side do not exceed $\int_{\left\{ H_N \gt 0\right\}}\left\lvert f\right\rvert\mathrm dm\leqslant\int_{X}\left\lvert f\right\rvert\mathrm dm $ hence $$ \alpha\cdot m\left(C\cap \left\{x: H_N(x)\gt 0\right\}\right)\leqslant \int_{X}\left\lvert f\right\rvert\mathrm dm. $$ Since the sequence of sets $\left(C\cap \left\{x: H_N(x)\gt 0\right\}\right)_{N\geqslant 1}$ is non-decreasing and the union of these sets is $C$, we can take the limit as $N\to \infty$ to get $$ \alpha\cdot m\left(C\right)\leqslant \int_{X}\left\lvert f\right\rvert\mathrm dm. $$
To conclude, we use $\sigma$-finiteness of $E_{\alpha,\beta}$ to write this set as the non-decreasing union of a sequence $\left(C_k\right)_{k\geqslant 1}$ of set of finite measure. Since for all $k$, $$ m\left(C_k\right)\leqslant \alpha^{-1}\int_{X}\left\lvert f\right\rvert\mathrm dm $$ it follows that $$ m\left(E_{\alpha,\beta}\right)\leqslant \alpha^{-1}\int_{X}\left\lvert f\right\rvert\mathrm dm. $$