Pointwise Limit of Measurable Functions is Measurable

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Theorem

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

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


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


Proof

From Convergence of Limsup and Liminf, it follows that:

$\displaystyle \lim_{n \to \infty} f_n = \limsup_{n \to \infty} f_n$

We have Pointwise Upper Limit of Measurable Functions is Measurable.


Hence the result.

$\blacksquare$


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