[proofplan] This is the ergodic specialization of the Birkhoff pointwise ergodic theorem. Birkhoff gives almost-everywhere convergence to an invariant integrable function. Ergodicity forces that invariant function to be constant, and the probability normalization identifies the constant as the integral of $f$. [/proofplan] [step:Apply Birkhoff's pointwise ergodic theorem] By the [Birkhoff Ergodic Theorem](/theorems/518), the averages \begin{align*} A_Nf(x):=\frac1N\sum_{n=0}^{N-1}f(T^n x) \end{align*} converge for $\mu$-almost every $x$ to an invariant function $\bar f\in L^1(X,\mu)$ satisfying \begin{align*} \bar f\circ T=\bar f \end{align*} $\mu$-almost everywhere. [/step] [step:Use ergodicity to identify the limit] Since $T$ is ergodic, every invariant integrable function is constant almost everywhere. Hence there is a constant $c\in\mathbb{R}$ such that \begin{align*} \bar f=c \end{align*} $\mu$-almost everywhere. The same theorem identifies the constant in the finite-measure ergodic case as the spatial average. Since $\mu(X)=1$, \begin{align*} c=\int_X f\,d\mu. \end{align*} Therefore \begin{align*} A_Nf(x)\to\int_X f\,d\mu \end{align*} for $\mu$-almost every $x$. [/step]