**Proposition.**  Let $f$ be a bounded measurable function on $[0,1]$.  Then there is a sequence of $C^\infty$ functions which converges to $f$ almost everywhere.

*Proof (by flyswatter)*.  Take the convolution of $f$ with a sequence of standard mollifiers.

*Proof (by nuke)*.  By <a href="http://en.wikipedia.org/wiki/Carleson_theorem">Carleson's theorem</a> the Fourier series of $f$ is such a sequence.