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Does anyone have a good proof of Littlewood's first principle?

Let $E$ be a measurable subset of $\mathbb{R}$ of finite measure, and let $\epsilon > 0$. Can anyone provide a rigorous proof that there is an open set $O$ which is the union of a finite number of pairwise disjoint open bounded intervals such that $m(O \setminus E) + m(E \setminus O) < \epsilon$.

Any references or answers would be greatly appreciated!

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So I imagine it's obvious to anyone who actually knows what the answer is, but you might want to specify what $E$ is supposed to be... –  Ben Millwood May 13 '12 at 0:13
    
See here for some thoughts. –  leo May 13 '12 at 0:47

1 Answer 1

There is a proof in this PDF.

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