Let $\Omega = (0,1]$ with the Borel sigma algebra and the Lebesgue measure dx. Let,
$$X_n(w) = \begin{cases} n, & \text{for $ 0< w \le \frac {1}{n}$ } \\ 0, & \text{otherwise } \\ \end{cases}$$
Can I claim that $\lim\mathbb{E}[X_n]= \mathbb{E}[\lim X_n]$? I do not think so, indeed: $X_n$ converges almost surely to 0, so that $\mathbb{E}[\lim X_n]=0$. While, $\lim\mathbb{E}[X_n]= \lim(n \frac1n))=1$.
However I may apply the Fatou's lemma and claim that $0= \mathbb{E}[\lim \inf X_n]\le \lim \inf E[X_n]=1$. Am I right?