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According to [Royden], there is a theorem;

Let $E$ be a measurable set of finite outer measure. Then, for each $\epsilon$, there is a finite disjoint collection of open intervals $\{I_k\}^n_{k=1}$ for which if $\Delta=\bigcup^n_{k=1} I_k$, then $m^*(E-\Delta)+m^*(\Delta -E)<\epsilon$

The theorem demands the finite outer measure of $E$ to apply exision property. My question is whether this theorem holds without the finite outer measure of $E$. If there is any counterexample, I want to know...!

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  • $\begingroup$ How about $E = \cup_{k\in\Bbb{Z}} [k, k+\frac{1}{2}]$? $\endgroup$ – Sangchul Lee May 19 '17 at 11:01
  • $\begingroup$ I got it, thanks! $\endgroup$ – Herace May 19 '17 at 14:14

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