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

Define $g_{n}:=\inf _{k\geq n}f_{k}$ for all $n\in \mathbb {N}$.

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 $g_{n}\leq f_{n},\forall n\in \mathbb {N}$, implying that $\int g_{n}\leq \int f_{n}$.

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

$\int \liminf _{n\rightarrow +\infty }f_{n}=\lim _{n\rightarrow +\infty }\int g_{n}=\liminf _{n\rightarrow +\infty }\int g_{n}\leq \liminf _{n\rightarrow +\infty }\int f_{n}$.