(a) Suppose that $(E, \mathcal{E}, \mu)$ is a finite measure space and $\theta: E \rightarrow E$ is a measurable map. Prove that $\mu_{\theta}(A)=\mu\left(\theta^{-1}(A)\right)$ defines a measure on $(E, \mathcal{E})$.

(b) Suppose that $\mathcal{A}$ is a $\pi$-system which generates $\mathcal{E}$. Using Dynkin's lemma, prove that $\theta$ is measure-preserving if and only if $\mu_{\theta}(A)=\mu(A)$ for all $A \in \mathcal{A}$.

(c) State Birkhoff's ergodic theorem and the maximal ergodic lemma.

(d) Consider the case $(E, \mathcal{E}, \mu)=([0,1), \mathcal{B}([0,1)), \mu)$ where $\mu$ is Lebesgue measure on $[0,1)$. Let $\theta:[0,1) \rightarrow[0,1)$ be the following map. If $x=\sum_{n=1}^{\infty} 2^{-n} \omega_{n}$ is the binary expansion of $x$ (where we disallow infinite sequences of $1 \mathrm{~s}$ ), then $\theta(x)=$ $\sum_{n=1}^{\infty} 2^{-n}\left(\omega_{n-1} \mathbf{1}_{n \in E}+\omega_{n+1} \mathbf{1}_{n \in O}\right)$ where $E$ and $O$ are respectively the even and odd elements of $\mathbb{N}$.

(i) Prove that $\theta$ is measure-preserving. [You may assume that $\theta$ is measurable.]

(ii) Prove or disprove: $\theta$ is ergodic.