Not a homework question but an exercise from an past exam.
Let $X, X_1, X_2, \ldots$ be real-valued random variables on a measure space $(\Omega, \mathcal{F}, \mathbb{P})$. Show that $X_n \xrightarrow{n \to \infty} X$ almost surely implies $\mathbb{E}[|X|] \le \liminf_{n \to \infty} \mathbb{E}[| X_n |]$.
I suspect I have to invoke the Lemma of Fatou, which gives $$ \mathbb{E}[\liminf_{n \to \infty} | X_n |] \le \liminf_{n \to \infty} \mathbb{E}[|X_n|] $$ but I don't know how to related the LHS of this inequality with LHS of the inequality I should prove.
I know that the almost sure convergence means that $$ \mathbb{P}(X_n \xrightarrow{n \to \infty} X) = 1, $$ so there exists a null set $\mathcal{N}$ such that $X_n$ converges to $X$ pointwise everywhere outside of $\mathcal{N}$.
Any help (preferably only in the form of hints) is greatly appreciated.
I would think about this in 2 steps:
1) Consider the special case when the assumption $X_n\rightarrow X$ almost surely is strengthened to surely. In that case we can apply Fatou directly and we are done.
2) What if you modify the random variables $X, X_n$ as follows: Define the event $$A =\{\omega \in \Omega : \lim_{n\rightarrow\infty} X_n(\omega) = X(\omega)\}$$
Note that $P[A]=1$. Then define new random variables $\tilde{X}$ and $\tilde{X}_n$ for $n \in \{1, 2, 3, ...\}$ by \begin{align} \tilde{X}(\omega) &= \left\{ \begin{array}{ll} X(\omega) &\mbox{ if $\omega \in A$} \\ 0 & \mbox{ otherwise} \end{array} \right.\\ \tilde{X}_n(\omega) &= \left\{ \begin{array}{ll} X_n(\omega) &\mbox{ if $\omega \in A$} \\ 0 & \mbox{ otherwise} \end{array} \right. \end{align} Now $\tilde{X}_n\rightarrow \tilde{X}$ surely. On the other hand, we have changed the random variables only on a set of measure 0, so the expectations are unchanged.