There is a simple reduction of the Birkhoff ergodic Theorem for $L^1$ functions to the bounded case using Kakutani-Rokhlin towers, that I learned  from H. Furstenberg and B. Weiss decades ago. Given $f \in L^1(X)$ we may assume it is nonnegative, and then (by subtracting the fractional part and adding 1) that it takes values in the positive integers. Consider the subset
$$Y=\{(x,k): 1 \le k \le f(x) \}$$
of $X \times {\mathbb N}$, endowed with the product $\sigma$-algebra, the probability measure $\nu$ determined by  $$\nu(A \times{k})=\frac{\mu(A)}{\int_X f \, d\mu}$$ 
for measurable sets $A \subset X$ where $\min_A f \ge k$, and the transformation $S$ defined by  $S(x,k)=(x,k+1)$ if $k<f(x)$ and $S(x,k)=(T(x),1)$ if $k=f(x)$.

 Then the ergodic theorem for the indicator function
$h(x,k):={\mathbf 1}_{\{k=1\}}$ in $(Y,\nu,S)$ (along the sequence of return times to the base $\{k=1\}$ of the tower) implies the ergodic theorem for $f$ in $(X,\mu,T)$.

Indeed, consider the Birkhoff sums
$$R_n(x):=\sum_{j=0}^{n-1}f(T^jx) \,.$$
For each $x \in X$, we have
$$\{\ell \ge 0 : h(S^\ell(x,1))=1 \}=  \{R_j(x) \; : \: j \ge 0\} \,$$
whence
$$\lim_{n \to \infty} \frac{n}{R_n(x)}=
\lim_{n \to \infty} \frac{ \sum_{\ell=0}^{R_n(x)-1}h(S^\ell(x,a)) }{R_n(x)}$$