Fatou’s lemma

If $f_1,f_2,\mathrm{\dots }$ is a sequence of nonnegative measurable functions in a measure space $X$, then

$${\int }_X\underset{n\to \mathrm{\infty }}{lim\; inf}{f}_n\le \underset{n\to \mathrm{\infty }}{lim\; inf}{\int }_X{f}_n$$