proof of Fatou’s lemma

Let $f(x)=\liminf_{n\to\infty}f_{n}(x)$ and let $g_{n}(x)=\inf_{k\geq n}f_{k}(x)$ so that we have

f(x)=\sup_{n}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}f\,d\mu=\lim_{n\to\infty}\int_{X}g_{n}\,d\mu.

On the other hand, being $g_{n}\leq f_{n}$ , we conclude by observing

\lim_{n\to\infty}\int_{X}g_{n}\,d\mu=\liminf_{n\to\infty}\int_{X}g_{n}\,d\mu% \leq\liminf_{n\to\infty}\int_{X}f_{n}\,d\mu.