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

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.

Answer 5

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.

Answer 6

Hey the way I remember it is by visualizing it, here is how I do it:

Given a Measure Space $(X,\Sigma,m)$ and a sequence of positive measurable ( ͡° ͜ʖ ͡°). Let $E\in\Sigma$ be a measurable set, we have the Fatou's Lenny: $$ \int_E \liminf\text{( ͡° ͜ʖ ͡°)}\,dm\le \liminf\int_E\text{( ͡° ͜ʖ ͡°)}\,dm $$

Answer 7

1. The application $f \mapsto \int f$ is lower semicontinuous (for the a.e. convergence of positive functions).

2. Finding an upper bound for a $\liminf$ is not as useful as finding a lower bound for a $\liminf$.

Answer 8

A mnemonic which helps me to remember it is that integral behaves similarly as sum (see fore example here) and we have $$\liminf x_n + \liminf y_n \le \liminf (x_n+y_n)$$ and $$\sum\limits_{k=1}^n \liminf x^{(k)} \le \liminf \sum\limits_{k=1}^n x^{(k)},$$ where each $x^{(k)}$ is a sequence.

If we mechanically replace sequence by a function and sum by an integral, we get $$\int\liminf f_k \le \liminf \int f_k.$$

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The above inequality is a part of the following chain of inequalities for $\limsup$ and $\liminf$ $$\liminf x_n+\liminf y_n \le \liminf (x_n+y_n) \le \liminf x_n + \limsup y_n \le \limsup (x_n+y_n) \le \limsup x_n+\limsup y_n$$ which I have encountered quite often, so I remember it.

See fore example this post and other questions shown there among linked questions.

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I should point out that the two inequalities are of a different nature. (The inequality for the sum is not a special case of the inequality for the integral.) If I wrote the inequalities more precisely, the inequality for sums is $$\sum\limits_{k=1}^n \liminf_{j\to\infty} x^{(k)}_j \le \liminf_{j\to\infty} \sum\limits_{k=1}^n x^{(k)}_j,$$ whereas in Fatou's theorem we have $$\int\liminf_{k\to\infty} f_k \le \liminf_{k\to\infty} \int f_k.$$ So limit inferior is taken with respect to different variables. But as a mnemonic, this could still work.

Answer 9

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