# Fatou's Lemma

## Statement

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

${\displaystyle \int \liminf _{n\rightarrow +\infty }f_{n}\leq \liminf _{n\rightarrow +\infty }\int f_{n}}$. [1]

## Proof[2]

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

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

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

Furthermore, by definition we have ${\displaystyle g_{n}\leq f_{n},\forall n\in \mathbb {N} }$, implying that ${\displaystyle \int g_{n}\leq \int f_{n}}$.

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

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

## References

1. Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, second edition, §2.2
2. Craig, Katy. MATH 201A Lecture 14. UC Santa Barbara, Fall 2020.