State and prove Birkhoff's almost-everywhere ergodic theorem.

[You need not prove convergence in $L_{p}$ and the maximal ergodic lemma may be assumed provided that it is clearly stated.]

Let $\Omega=[0,1), \mathcal{F}=\mathcal{B}([0,1))$ be the Borel $\sigma$-field and let $\mathbb{P}$ be Lebesgue measure on $(\Omega, \mathcal{F})$. Give an example of an ergodic measure-preserving map $\theta: \Omega \rightarrow \Omega$ (you need not prove it is ergodic).

Let $f(x)=x$ for $x \in[0,1)$. Find (at least for all $x$ outside a set of measure zero)

$$\lim _{n \rightarrow \infty} \frac{1}{n} \sum_{i=1}^{n}\left(f \circ \theta^{i-1}\right)(x) .$$

Briefly justify your answer.