# 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

 $\lim_{n\to\infty}\int_{X}g_{n}\,d\mu=\liminf_{n\to\infty}\int_{X}g_{n}\,d\mu% \leq\liminf_{n\to\infty}\int_{X}f_{n}\,d\mu.$
Title proof of Fatou’s lemma ProofOfFatousLemma 2013-03-22 13:29:59 2013-03-22 13:29:59 paolini (1187) paolini (1187) 4 paolini (1187) Proof msc 28A20