Let $(E, \mathcal{E}, \mu)$ be a measure space with $\mu(E)<\infty$ and let $\theta: E \rightarrow E$ be measurable.

(a) Define an invariant set $A \in \mathcal{E}$ and an invariant function $f: E \rightarrow \mathbb{R}$.

What is meant by saying that $\theta$ is measure-preserving?

What is meant by saying that $\theta$ is ergodic?

(b) Which of the following functions $\theta_{1}$ to $\theta_{4}$ is ergodic? Justify your answer.

On the measure space $([0,1], \mathcal{B}([0,1]), \mu)$ with Lebesgue measure $\mu$ consider

$$\theta_{1}(x)=1+x, \quad \theta_{2}(x)=x^{2}, \quad \theta_{3}(x)=1-x$$

On the discrete measure space $\left(\{-1,1\}, \mathcal{P}(\{-1,1\}), \frac{1}{2} \delta_{-1}+\frac{1}{2} \delta_{1}\right)$ consider

$$\theta_{4}(x)=-x$$

(c) State Birkhoff's almost everywhere ergodic theorem.

(d) Let $\theta$ be measure-preserving and let $f: E \rightarrow \mathbb{R}$ be bounded.

Prove that $\frac{1}{n}\left(f+f \circ \theta+\ldots+f \circ \theta^{n-1}\right)$ converges in $L^{p}$ for all $p \in[1, \infty)$.