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The question is simply the one in the title; that given $0<\delta< 1$, there is no measurable set $E\subset\mathbb R$ satisfying $\delta< \frac{|E\cap I|}{|I|} < 1-\delta$ for all intervals $I$ of $\mathbb R$.

I want to somehow contradict the fact that $|E| = |E\backslash A| + |E\cap A|$ for all sets $A\subset\mathbb R$. Let $\epsilon>0$ be arbitrary and $A$ be collection of intervals such that $|E\backslash A|<\epsilon$ and $||E|-|A||<\epsilon$. Then $|E\cap A| = |E|-|E\backslash A|$, so $$\frac{|E\cap A|}{|A|} = \frac{|E|}{|A|} - \frac{|E\backslash A|}{|A|} > \frac{|A|-\epsilon}{|A|} - \frac{\epsilon}{|A|} \to 1\hspace{10pt}\text{as $\epsilon \searrow 0$},$$ contradicting $|E\cap A|/|A|<1-\delta$ for some fixed $\delta>0$. Is this argument correct? Can I choose $A$ as such?

Nico
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  • $|E\cap A| = |E| - |E\setminus A|,$ not $|E| + |E\setminus A|,$ so I think your first step is fallacious. –  Jul 10 '21 at 16:22
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    Here is a related question. – David Mitra Jul 10 '21 at 16:24
  • @DavidMitra How do you attach a link to "Here" in a comment? I have tried several times but somehow it does not work for me in a comment. Thanks – p_square Jul 10 '21 at 16:26
  • Enclose the link name in brackets (brackets are "[" and "]") then right after the url in parentheses. – David Mitra Jul 10 '21 at 16:27
  • @MichaelBarz Fixed, but the term goes to zero so that is not terribly irrelevant. – Nico Jul 10 '21 at 18:35
  • @downvoter what is the issue? I showed my work and asked for verification, what merits the downvote? And this is not a duplicate btw – Nico Jul 10 '21 at 18:36
  • @DavidMitra I looked over your link and it essentially answers the question, but I'm looking for proof verification (forgot the tag at first). Thank you for the link regardless! – Nico Jul 10 '21 at 19:23
  • If $A$ is a $collection$ of intervals then $|E\cap A|$ makes no sense..... To solve the Q, use the Lebesgue Density Lemma. – DanielWainfleet Jul 10 '21 at 21:55
  • On 2nd thought you can bypass the Lebesgue Density Lemma. I think you have the right idea, but your presentation is garbled. – DanielWainfleet Jul 10 '21 at 22:13

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