# Fatou's Lemma for Measures

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*This proof is about Fatou's Lemma in the context of measures. For other uses, see Fatou's Lemma.*

## Theorem

Let $\struct {X, \Sigma, \mu}$ be a measure space.

Let $\sequence {E_n}_{n \mathop \in \N} \in \Sigma$ be a sequence of $\Sigma$-measurable sets.

Then:

- $\displaystyle \map \mu {\liminf_{n \mathop \to \infty} E_n} \le \liminf_{n \mathop \to \infty} \map \mu {E_n}$

where:

- $\displaystyle \liminf_{n \mathop \to \infty} E_n$ is the limit inferior of the $E_n$
- the right hand side limit inferior is taken in the extended real numbers $\overline \R$.

### Corollary

Let $\mu$ be a finite measure.

Then:

- $\displaystyle \map \mu {\limsup_{n \mathop \to \infty} E_n} \le \limsup_{n \mathop \to \infty} \map \mu {E_n}$

## Proof

## Source of Name

This entry was named for Pierre Joseph Louis Fatou.

## Sources

- 2005: René L. Schilling:
*Measures, Integrals and Martingales*... (previous) ... (next): $\S 9$: Problem $9 \ \text{(ii)}$