proof of Fatou’s lemma

Let $f(x)={lim\; inf}_{n\to \mathrm{\infty }}{f}_n(x)$ and let $g_n(x)={inf}_{k\ge n}{f}_k(x)$so that we have

$$f(x)=\underset{n}{sup}{g}_n(x).$$

As $g_n$ is an increasing sequence of measurable nonnegative functions we can apply the monotone convergence Theorem to obtain

$${\int }_{X}f𝑑\mu =\underset{n\to \mathrm{\infty }}{lim}{\int }_X{g}_n𝑑\mu .$$

On the other hand, being $g_n\le f_n$, we conclude by observing

$$\underset{n\to \mathrm{\infty }}{lim}{\int }_X{g}_n𝑑\mu =\underset{n\to \mathrm{\infty }}{lim\; inf}{\int }_X{g}_n𝑑\mu \le \underset{n\to \mathrm{\infty }}{lim\; inf}{\int }_X{f}_n𝑑\mu .$$