Pointwise Lower Limit of Measurable Functions is Measurable

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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

$\blacksquare$


Sources