Say $f_n\geq0\ \forall n$ are measurable in the sense that $\{x\ :\ f(x)<\alpha\}\in\mathcal M\ \forall\alpha\in\Bbb R$ where $\mathcal M$ is the set of Lebesgue measurable subsets of $\Bbb R$. And $f_n\rightarrow f$ Is $f :\Bbb R\mapsto\Bbb R$ necessarily a measurable function in the same sense?
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1In which norm do you want $f_n \rightarrow f$ to hold? – cdwe Jun 05 '18 at 17:41
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Just pointwise convergence $f_n\rightarrow f\ \iff \lim\limits_{n\rightarrow\infty}f_n(x)=f(x)$ – John Cataldo Jun 05 '18 at 17:47