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Let $g$ be a measurable function on $[0,1]$. Suppose that $g$ is finite almost everywhere and let $\mu$ be the Lebesgue measure. Then for any $\epsilon >0$, there is a polynomial $h$, such that $ \mu\left(\{x: |g(x)-h(x)|>\epsilon \}\right) < \epsilon.$

Well, I know that there is a continuous function, say, $f(x)$ such that $|g(x)-f(x)|<\epsilon$ except on a set of measure less than $\epsilon.$ Now, since polynomials are a continuous functions, can I take $h(x)=f(x)$, and thus proving the above? If not, how to I go about proving it?

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    That's $f$ine; it's just more than you need.2012-03-02

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No, nothing at the outset guarantees that the $f$ you wind up with is a polynomial. The Weierstrass approximation theorem however does.

First choose a continuous function $f$ such that $|g(x)-f(x)|<\epsilon/2$ for all $x\in[0,1]\setminus A$ where the measure of $A$ is less than $\epsilon$. Then use the aforementioned theorem to find a polynomial $p$ such that $|f(x)-p(x)|<\epsilon/2$ for all $x\in[0,1]$. Then $p$ is the desired polynomial.

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    @Joe Yes. Just use the triangle inequality.2012-03-02