I don't understand the final formula. Birkhoff's theorem says that the sample mean of some random variable taken on a map $T^{i - 1}$ (index is $i$ in the sum, from 1 to $n$) (ergodic and measure preserving) approaches almost surely the mean of $X$. However, in the proof, there is no map $T$ at all, just that one random variable? The sum has index $i$, yet there is no $i$ being summed over, only one random variable independent of $i$?
In the proof, $\hat{X}_1$ is the first coordinate projection, $P$ is our probability measure, and $\mathbb{X}(P)$ is the distribution of a process $(X_n)$ on a measurable space of infinite sequences and an associated sigma algebra generataed by the coordinate projections.