Let $(X, \mathcal{A})$ be a measurable space. Let $T: X \rightarrow X$ be a measurable map, and $\mu$ a probability measure on $(X, \mathcal{A})$.

(a) State the definition of the following properties of the system $(X, \mathcal{A}, \mu, T)$ :

(i) $\mu$ is T-invariant.

(ii) $T$ is ergodic with respect to $\mu$.

(b) State the pointwise ergodic theorem.

(c) Give an example of a probability measure preserving system $(X, \mathcal{A}, \mu, T)$ in which $\operatorname{Card}\left(T^{-1}\{x\}\right)>1$ for $\mu$-a.e. $x$.

(d) Assume $X$ is finite and $\mathcal{A}$ is the boolean algebra of all subsets of $X$. Suppose that $\mu$ is a $T$-invariant probability measure on $X$ such that $\mu(\{x\})>0$ for all $x \in X$. Show that $T$ is a bijection.

(e) Let $X=\mathbb{N}$, the set of positive integers, and $\mathcal{A}$ be the $\sigma$-algebra of all subsets of $X$. Suppose that $\mu$ is a $T$-invariant ergodic probability measure on $X$. Show that there is a finite subset $Y \subseteq X$ with $\mu(Y)=1$.