Fatou's Lemma for Measures

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:

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

where:

$\ds \liminf_{n \mathop \to \infty} E_n$ 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:

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

Proof

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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 $9 \ \text{(ii)}$