(Dominated convergence for sets). Let ( $X, \mathcal{B}, \mu$ ) be a measure space. Let $E_1, E_2, \ldots$ be a sequence of $\mathcal{B}$-measurable sets that converge to another set $E$, in the sense that $1_{E_n}$ converges pointwise to $1_E$.
(i) Show that $E$ is also $\mathcal{B}$-measurable.
(ii) If there exists a $\mathcal{B}$-measurable set $F$ of finite measure (i.e. $\mu(F)<\infty)$ that contains all of the $E_n$, show that $\lim _{n \rightarrow \infty} \mu\left(E_n\right)= \mu(E)$. (Hint: Apply downward monotonicity to the sets $\left.\bigcup_{n>N}\left(E_n \Delta E\right).\right)$
(iii) Show that the previous part of this exercise can fail if the hypothesis that all the $E_n$ are contained in a set of finite measure is omitted.