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In my notes this statement is left unproven. I want to show that for any measurable set $E \subset \mathbb{R}$ with $\theta(E)>0$, there exists an interval $(a,b)$ that covers $E$ arbitrarily closely. From the definition of the Lebesgue Outer Measure:

$$ \exists \bigcup_i(a_i,b_i) \supseteq E : \sum_i(b_i-a_i) \leq \frac{1}{1-\epsilon}\theta(E) $$

And then $\theta(E)=\theta(\cup_i E\cap(a_i,b_i))$, but I'm not sure this helps. Can anyone show me how to finish this off?

Tom Offer
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