Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space and let $X, X_{1}, X_{2}, \ldots$ be random variables. Write an essay in which you discuss the statement: if $X_{n} \rightarrow X$ almost everywhere, then $\mathbb{E}\left(X_{n}\right) \rightarrow \mathbb{E}(X)$. You should include accounts of monotone, dominated, and bounded convergence, and of Fatou's lemma.

[You may assume without proof the following fact. Let $(\Omega, \mathcal{F}, \mu)$ be a measure space, and let $f: \Omega \rightarrow \mathbb{R}$ be non-negative with finite integral $\mu(f) .$ If $\left(f_{n}: n \geqslant 1\right)$ are non-negative measurable functions with $f_{n}(\omega) \uparrow f(\omega)$ for all $\omega \in \Omega$, then $\mu\left(f_{n}\right) \rightarrow \mu(f)$ as $n \rightarrow \infty$.]