Fatou's Lemma

Statement

Suppose $\{f_{n}\}$ is a sequence of non-negative measurable functions, $f_{n}:X\to [0,+\infty ]$. Then: $\int \liminf _{n\rightarrow +\infty }f_{n}\leq \liminf _{n\rightarrow +\infty }\int f_{n}$.

Proof

For any $n\in \mathbb {N}$, let $g_{n}=\inf _{k\geq n}f_{k}$.

By definition, $\liminf _{n\rightarrow +\infty }f_{n}=\lim _{n\rightarrow +\infty }(inf_{k\geq n}f_{k})=\lim _{n\rightarrow +\infty }g_{n}$. And $g_{n}\leq g_{n+1},\forall n\in \mathbb {N}$, so by Monotone Convergence Theorem, $\lim _{n\rightarrow +\infty }\int g_{n}=\int \lim _{n\rightarrow +\infty }g_{n}=\int \liminf _{n\rightarrow +\infty }f_{n}$.

Furthermore, by definition we have $g_{n}\leq f_{n}$, then $\int g_{n}\leq \int f_{n}$.

Since $\lim _{n\rightarrow +\infty }$ exists, taking $\liminf _{n\rightarrow +\infty }$ of both sides: $\int \liminf _{n\rightarrow +\infty }f_{n}=\lim _{n\rightarrow +\infty }=\liminf _{n\rightarrow +\infty }\int g_{n}\leq \liminf _{n\rightarrow +\infty }\int f_{n}$.