(i) State and prove Fatou's lemma. State and prove Lebesgue's dominated convergence theorem. [You may assume the monotone convergence theorem.]

In the rest of the question, let $f_{n}$ be a sequence of integrable functions on some measure space $(E, \mathcal{E}, \mu)$, and assume that $f_{n} \rightarrow f$ almost everywhere, where $f$ is a given integrable function. We also assume that $\int\left|f_{n}\right| d \mu \rightarrow \int|f| d \mu$ as $n \rightarrow \infty$.

(ii) Show that $\int f_{n}^{+} d \mu \rightarrow \int f^{+} d \mu$ and that $\int f_{n}^{-} d \mu \rightarrow \int f^{-} d \mu$, where $\phi^{+}=\max (\phi, 0)$ and $\phi^{-}=\max (-\phi, 0)$ denote the positive and negative parts of a function $\phi$.

(iii) Here we assume also that $f_{n} \geqslant 0$. Deduce that $\int\left|f-f_{n}\right| d \mu \rightarrow 0$.