What does Fatou's Lemma really say?

Question

Fatou's lemma says that

if $f_n:X \rightarrow [0,\infty]$ are measurable,then

$$\liminf_{n\rightarrow \infty}\left(\int_X f_n \,\mathrm{d} \mu\right) \geq \int_X \liminf_{n\rightarrow \infty} f_n \,\mathrm{d}\mu$$

I like to know what this Lemma really says. That is, how can I express in words (rather informally) what this lemma actually says?

Answer 1

$$ (0,1),\quad(1,0),\quad(0,1),\quad(1,0),\quad\ldots $$ The terms of a sequence are alternately $(0,1)$ and $(1,0)$. In either case, the sum of the two components of each pair is $1$, so the lim inf of the sum of the two is $1$. But the lim inf of the sequence of pairs is $(0,0)$, and the sum of the two components of that pair is $0$.

In other words, for every value of $x$, $f_n(x)$ may be small for infinitely many $n$, but the values of $n$ for which $f_n(a)$ is small are not the same ones for which $f_n(b)$ is small.

Answer 2

If $E_i$ is measurable then $$ \lim\ \inf E_i =\{x \mid x\in E_i\ \text{ except finitely many } i \} $$

Then we have $$ \mu (\lim\ \inf E_i)\leq \lim\ \inf \mu(E_i)\ \ast$$

If $E_i$ is a measurable, then $f_i=\chi_{E_i}$ is measurable. So $$\lim\ \inf \mu(E_i) = \lim\ \inf \int_X f_i $$

And if $E:=\lim\ \inf E_i$ then $$ \lim\ \inf f_i= \chi_E $$

Hence Fatou is a generalization of $\ast$.