Fatou's Lemma for Measures/Corollary

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. Let $\mu$ be a finite measure.

Then:


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

where:


 * $\displaystyle \limsup_{n \to \infty} E_n$ is the limit superior of the $E_n$
 * the right-hand side limit superior is taken in the extended real numbers $\overline{\R}$.