I need help constructing a Borel set $A$ on $\mathbb{R}$ with the following property:
For every open interval $I$,
$$0<m(A \cap I)< m(I)$$
A obviously needs to be dense in $\mathbb{R}$ and it also must have empty interior, but honestly I don't know what to do from here on.
Even the generalized cantor set doesn't have positive measure for all the sub intervals it is constructed on.....
