# Is there a set $E$ such that for every open interval $I$, $0<m(I\cap E)<m(I)$? [duplicate]

Is it possible for there to be a measurable set $$E\subseteq\mathbb{R}$$ such that for every open interval $$I$$, $$0 (where $$m$$ is the Lebesgue measure of a set)?

The first thing I notice is that this set must be dense, since if it were not dense we would be able to find an interval with no members of $$E$$. It of course also must be uncountable in size, since it cannot have zero measure itself. It is also pretty clear that $$\mathbb{R}\setminus E$$ must also have this property if $$E$$ has this property.

Aside from that I'm stumped. I've tried a couple of toy sets but they all turned out to not have this property. Then I tried to show no set could have this property but I hit a brick wall pretty quickly.