TITLE: What does Fatou's Lemma really say?
QUESTION [12 upvotes]: 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?

REPLY [3 votes]: 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 $\ast$.