(a) Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space and let $\theta: \Omega \rightarrow \Omega$ be measurable. What is meant by saying that $\theta$ is measure-preserving? Define an invariant event and an invariant random variable, and explain what is meant by saying that $\theta$ is ergodic.

(b) Let $m$ be a probability measure on $(\mathbb{R}, \mathcal{B})$. Let

$$\Omega=\mathbb{R}^{\mathbb{N}}=\left\{x=\left(x_{1}, x_{2}, \ldots\right): x_{i} \in \mathbb{R} \text { for } i \geqslant 1\right\}$$

let $\mathcal{F}$ be the smallest $\sigma$-field of $\Omega$ with respect to which the coordinate maps $X_{n}(x)=x_{n}$, for $x \in \Omega, n \geqslant 1$, are measurable, and let $\mathbb{P}$ be the unique probability measure on $(\Omega, \mathcal{F})$ satisfying

$$\mathbb{P}\left(X_{i} \in A_{i} \text { for } 1 \leqslant i \leqslant n\right)=\prod_{i=1}^{n} m\left(A_{i}\right)$$

for all $A_{i} \in \mathcal{B}, n \geqslant 1$. Define $\theta: \Omega \rightarrow \Omega$ by $\theta(x)=\left(x_{2}, x_{3}, \ldots\right)$ for $x=\left(x_{1}, x_{2}, \ldots\right)$.

(i) Show that $\theta$ is measurable and measure-preserving.

(ii) Define the tail $\sigma$-field $\mathcal{T}$ of the coordinate maps $X_{1}, X_{2}, \ldots$, and show that the invariant $\sigma$-field $\mathcal{I}$ of $\theta$ satisfies $\mathcal{I} \subseteq \mathcal{T}$. Deduce that $\theta$ is ergodic. [Any general result used must be stated clearly but the proof may be omitted.]

(c) State Birkhoff's ergodic theorem and explain how to deduce that, given independent identically-distributed integrable random variables $Y_{1}, Y_{2}, \ldots$, there exists $\nu \in \mathbb{R}$ such that

$$\frac{1}{n}\left(Y_{1}+Y_{2}+\cdots+Y_{n}\right) \rightarrow \nu \quad \text { a.e. as } \quad n \rightarrow \infty .$$