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 $g(x,k):={\mathbf 1}_{\{k=1\}}$ 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 $(X,\mu,T)$.