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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!

Mark V.
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There is a proof in this PDF.

Brian M. Scott
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  • But in that PDF the intervals are not disjoint. – Twnk Sep 11 '20 at 22:43
  • @Twink: It’s straightforward to prove that any union of finitely many open intervals is the union of finitely many pairwise disjoint open intervals, namely, its connected components. – Brian M. Scott Sep 11 '20 at 23:22