Question

Prove that the limit of pointwise convergent sequence of measurable function is measurable.

Answer 1

From Convergence of Limsup and Liminf, it follows that:

$lim_{n\to\infty}f_n=lim_{n\to\infty}sup f_n$

We have Pointwise upper limit of Measurable Functions is measurable.

Hence the result is proved.

Reference: https://proofwiki.org/wiki/Pointwise_Limit_of_Measurable_Functions_is_Measurable