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I've been trying to solve this problem for days. Here is the problem:

Suppose that $f$ is any complex-valued measurable function defined in the complex plane, and prove that there is a sequence of holomorphic polynomials $P_n$ such that $lim_{n\to\infty}P_n(z)=f(z)$ for almost every $z$ (with respect to two-dimensional Lebesgue measure)

Since continuous functions are very "close" to measurable functions, I tried to approximate the given function with continuous one, and then to approximate the continuous function with polynomials. However, This method does not seem to work.

I've also googled a lot to find something relevant, but found nothing.

Any help will be appreciated.

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 Maybe try to prove it is not possible for the measurable function $\overline{z}$ . – Gerald Edgar Nov 26 2011 at 17:21
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 To approximate a continuous function with polynomials, fill a big square with small disjoint squares that form an almost tiling (the spaces between them are much smaller than their size), take a function that is constant on each small square and use Runge. – fedja Nov 26 2011 at 17:26
This is obviously homework, voting to close. – Igor Rivin Nov 26 2011 at 18:55
The way you phrase this makes it sound like an exercise - if so, where did you find it? If it's not an exercise, is there a special case that you particularly need in the course of some research? – Yemon Choi Nov 26 2011 at 18:59

closed as off topic by Qiaochu Yuan, Gerald Edgar, fedja, Igor Rivin, Yemon Choi Nov 26 2011 at 19:04

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