Fatou's Lemma for Measures

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
If $\mu$ is a finite measure, then:


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