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.

Here's the worst possible way to keep straight which direction the inequality goes: In a typical programming language strings are sorts alphabetically, so that for example "cat" < "dog", since "c" < "d". Observe that $$\text{IL < LI.}$$"Integral(limit) < Limit(integral)".

This is in my opinion extremely awful, because it has nothing to do with the math, just relies on a quirk of English spelling. However I have a great deal of respect for the person who pointed this out to me some years ago. He's a very smart guy and he says with a straight face that this is how he's always remembered it - so I thought I'd mention it for the benefit of anyone out there who happens to be like him.

I remember the chain of inequalities $$\int \liminf_{n\rightarrow \infty} f_n \mathrm{d}\mu \leq \liminf_{n\rightarrow \infty}\int f_n \mathrm{d} \mu \leq \limsup_{n\rightarrow \infty}\int f_n \mathrm{d} \mu \leq \int\limsup_{n\rightarrow \infty} f_n \mathrm{d} \mu $$ the middle of which is easy to remember and the rest can be easily deduced.