# Pointwise Lower Limit of Measurable Functions is Measurable

 It has been suggested that this article or section be renamed: Pointwise Limit Inferior of Measurable Functions is Measurable -- see talk page. One may discuss this suggestion on the talk page.

## Theorem

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

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

Then the pointwise lower limit:

$\displaystyle \liminf_{n \mathop \to \infty} f_n: X \to \overline \R$

is also $\Sigma$-measurable.

## Proof

By definition of limit inferior, we have:

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

The result follows from combining:

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

$\blacksquare$