Fatou's Lemma for Measures

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This proof is about Fatou's Lemma for measures. For other uses, see Fatou's Lemma.

Theorem

Let $\left({X, \Sigma, \mu}\right)$ be a measure space.

Let $\left({E_n}\right)_{n \in \N} \in \Sigma$ be a sequence of $\Sigma$-measurable sets.


Then:

$\displaystyle \mu \left({\liminf_{n \to \infty} E_n}\right) \le \liminf_{n \to \infty} \mu \left({E_n}\right)$

where:

$\displaystyle \liminf_{n \to \infty} E_n$ 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:

$\displaystyle \mu \left({\limsup_{n \to \infty} E_n}\right) \ge \limsup_{n \to \infty} \mu \left({E_n}\right)$


Proof


Source of Name

This entry was named for Pierre Joseph Louis Fatou.


Sources