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