Construct a Borel set $E$ in $\mathbb{R}^1$ such that $$0<m(E \cap I)<m(I)$$ for every nonempty segment $I$. Is it possible to have $m(E)<\infty$ for such a set?
I wonder how can I construct a set like this, though I've read the solution. Can someone tell me what leads to the construction of this set and what's the idea? I'll appreciate it. (The measure $m$ is the Lebesgue measure on $\mathbb{R}^1$)
