What are Some Tricks to Remember Fatou's Lemma?

Question

For a sequence of non-negative measurable functions $f_n$, Fatou's lemma is a statement about the inequality

$$\int \liminf_{n\rightarrow \infty} f_n \mathrm{d}\mu \leq \liminf_{n\rightarrow \infty}(\int f_n \mathrm{d} \mu)$$

or alternatively (for sequences of real functions dominated by some integrable function)

$$\limsup_{n\rightarrow \infty}(\int f_n \mathrm{d} \mu) \leq \int \limsup_{n\rightarrow \infty} f_n \mathrm{d}\mu$$

I keep forgetting the direction of these two inequalities. I know that using the concepts repeatedly is the best way to remember them.

But I am interested about learning intuitive tricks that people use to quickly remember them.

(For instance, to remember the direction of Jensen's inequality, I just picture a convex function and a line intersecting it.)

Answer 1

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.

Answer 2

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

Answer 3

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.

Answer 4

Write the statement of the lemma on a credit-card sized piece of paper and carry it around for a month.