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 $\forall n\in \mathbb {N}$