(i) Let $(E, \mathcal{E}, \mu)$ be a measure space. What does it mean to say that a function $\theta: E \rightarrow E$ is a measure-preserving transformation?

What does it mean to say that $\theta$ is ergodic?

State Birkhoff's almost everywhere ergodic theorem.

(ii) Consider the set $E=(0,1]^{2}$ equipped with its Borel $\sigma$-algebra and Lebesgue measure. Fix an irrational number $a \in(0,1]$ and define $\theta: E \rightarrow E$ by

$$\theta\left(x_{1}, x_{2}\right)=\left(x_{1}+a, x_{2}+a\right)$$

where addition in each coordinate is understood to be modulo 1 . Show that $\theta$ is a measurepreserving transformation. Is $\theta$ ergodic? Justify your answer.

Let $f$ be an integrable function on $E$ and let $\bar{f}$ be the invariant function associated with $f$ by Birkhoff's theorem. Write down a formula for $\bar{f}$ in terms of $f$. [You are not expected to justify this answer.]