Fatou's Lemma for Measures

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 limit inferior is taken in the extended real numbers $\overline \R$.