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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 it. 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)

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I remember it by thinking of how it's used. We have some function that is defined as the limit (or lim inf) of a sequence of functions, and we'd like to get a bound on its integral. –  Jair Taylor Nov 24 '12 at 3:55
Thanks Nate for the clarification. I corrected it in the question. –  Learner Nov 24 '12 at 5:03

4 Answers 4

up vote 56 down vote accepted

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.

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+1 for graphical explanation. I think nothing like what you did here could illustrate Fatou's lemme. –  Babak S. Nov 23 '12 at 17:41
awesome answer. i also kept forgetting fatou's lemma.but this illustration cleared my intuition –  Koushik Dec 31 '12 at 3:48
I used to have trouble remember the direction of the inequality until I saw this post. +1 –  Daniel Montealegre Oct 20 '13 at 21:34

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

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I do the same except let fn be an indicator that x >= n so that lim inf of the integral is infinity. –  Dason Nov 23 '12 at 4:40

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.

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

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While this is a bit extreme, I tend to keep useful information at hand if I know I'll be needing it. –  Mariano Suárez-Alvarez Nov 24 '12 at 4:37
Have you actually done this for an important theorem? –  Potato Mar 28 '13 at 1:54
Of course not :-) Not even for an unimportant one! –  Mariano Suárez-Alvarez Mar 28 '13 at 2:06

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