What are Some Tricks to Remember Fatou's Lemma?

I like to think of the following pictures. The first two are $\int f_1$ and $\int f_2$ respectively, but even the smaller of these is larger than the area in the third picture, which is $\int \inf f_n$. Of course, Fatou's lemma is more subtle since we're talking about the limit infimum rather than just the minimum, but for the purpose of intuition this helps to make sure the inequalities go the right way.

I like to remember this by example; specifically let $f_n = \chi_{[n,n+1]}$. Then $\lim \inf f_n = 0$, and $\lim \inf \int f_n = 1$.

When you pass to the limit, you can lose mass (by pushing it off to infinity, as in Thomas Belulovich's example), but the inequality in Fatou's lemma says you cannot gain mass.