Pointwise Upper Limit of Measurable Functions is Measurable

From ProofWiki
Jump to navigation Jump to search

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 upper limit $\displaystyle \limsup_{n \to \infty} f_n: X \to \overline{\R}$ is also $\Sigma$-measurable.


Proof

By definition of upper limit, we have:

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


The result follows from combining:

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

$\blacksquare$


Sources