Let $m$ be the Lebesgue measure on $\Bbb R$, then find $E$ Borel, such that for all $a<b, a,b\in\Bbb R$, $$m(E\cap(a,b))>0,\,m(E^c\cap(a,b))>0. $$
I couldn't find one. But after some failed attempts I know that $E$ must fulfil certain properties:
1). $E$ must be neither open nor closed. Moreover, neither $E$ nor $E^c$ can have any interior point.
2). Both $E$ and $E^c$ must be dense in $\Bbb R$.
3). Neither $E$ and $E^c$ can have measure zero.
So my constructions from the weirdest monsters like the Cantor set, the rationals, or even the sequence of "exponentially decreasing" neighbourhoods of rationals all failed. It's been really hard for me to fantasise anything weirder.
