Pointwise Lower Limit of Measurable Functions is Measurable

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

Let $\left({f_n}\right)_{n \in \N}$, $f_n: X \to \overline{\R}$ be a sequence of $\Sigma$-measurable functions.

Then the pointwise lower limit $\displaystyle \liminf_{n \to \infty} f_n: X \to \overline{\R}$ is also $\Sigma$-measurable.

Proof
By definition of lower limit, we have:


 * $\displaystyle \liminf_{n \to \infty} f_n = \sup_{m \mathop \in \N} \ \inf_{n \ge m} f_n$

The result follows from combining:


 * Pointwise Infimum of Measurable Functions is Measurable
 * Pointwise Supremum of Measurable Functions is Measurable