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Suppose that $f: \mathbb{R} \to \mathbb{R}$ is a bounded measurable function.

Is it possible to find a sequence of functions $\{f_n \}_n: \mathbb{R} \to \mathbb{R}$ in $C^{\infty}_c( \mathbb{R})$ such that $f_n(x) \uparrow f(x)$, for every $x \in \mathbb{R}$?

This seems to be true without the assumption that the convergence is from below, by standard results from the literature. However, I am really not sure about this case.

Richard
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1 Answers1

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No. It can be shown that a pointwise limit of a sequence of continuous functions is continuous on a dense set. [This is a consequence of Baire Category Theorem]. However, a bounded measurable function can be discontinuous at every point.

Kavi Rama Murthy
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  • Heyy, really like your answer, if you could help with an exercise, would really appreciate it :) https://math.stackexchange.com/questions/3096022/normed-vector-space-schauder-basis-exercise – Homaniac Feb 04 '19 at 08:44