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Background

This question was asked a few minutes ago and then deleted after another user exhibited what he believed to be a duplicate but I fail to see the link between the two.

Here is the statement of the deleted question, $m$ being the Lebesgue measure :

Can we find $A,B \subset \mathbb R$ such that $m(A \cap I)$ and $m(B \cap I)$ are positive for each open interval $I$ and $A \cap B $ is the empty set ?

And here is the other question, which happens to have a positive answer. I copy it on this page as well :

Starting from a countable basis of $\mathbb R$, I am asked to construct a Borel set such that $0<m(E∩I)<m(I)$ for every non empty segment $I$.

Can someone explain how a positive answer to the second question answers the first ?

nicomezi
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1 Answers1

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Take $A=E$ and $B=E^{c}$. If $I$ is any open interval $m(A\cap I) >0$ so $A\cap I$ is non-empty. This proves that $A$ is dense. Similarly $B$ is also dense.

Kavi Rama Murthy
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  • Dont bother, I got it. $m(B \cap I)>0$ holds since we have strict inequalities. Well, at least to me, the link was not obvious at all. Thank you. – nicomezi Jun 03 '20 at 09:23