Fatou's Lemma and Almost Sure Convergence (Pt. 1) I have a question regarding Fatou's Lemma and a sequence of random variables converging almost surely.  Fatou's Lemma states

If $\forall n \in \mathbb{N}, \,\, X_{n} \ge 0$ and $\displaystyle X = \liminf_{n \rightarrow \infty} X_{n}$, then $\displaystyle\mathbb{E}[ \liminf_{n \rightarrow \infty}\: X_{n}] \le \liminf_{n \rightarrow \infty}\: \mathbb{E}[ X_{n}]$

Suppose we also know that $X_{n} \rightarrow X$ almost surely.  How can we connect this to the requirements of Fatou's Lemma?  It seems to me that the Lemma asks for pointwise convergence, a wholly different beast.  

Suppose you know that $X_n \to X$ a.s.  To avoid confusion, let's write $Y$ for $\liminf X_n$.  Since for a convergent sequence, the limit and liminf are equal, we have $X = Y$ a.s., so $E[X] = E[Y]$, and by Fatou $E[Y] \le \liminf E[X_n]$.
