Fatou's Lemma

Definition

Have measure $\mu$ on $X$, and $f_n:X\to [0,\infty ]$ measurable. Then $\int {\lim }_{n\to \infty }\inf f_{n}d\mu \le {\lim }_{n\to \infty }\inf \int f_{n}d\mu$

What is $\lim \inf$?

Definition of infimum (it is basically the opposite of a supremum).

$\{x_n\}\subset [0,\infty ]$
${\lim }_{n\to \infty }\inf x_n={\sup }_m{\inf }_{n\ge m}{x}_n$

${\inf }_{n\ge m}=y_m\le y_{m+1}\le \dots$