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.....

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    $\begingroup$ $\mathbb{R}$ is a Baire Space and $\mathbb{Q}$ is countable $\endgroup$ – Jonas Gomes Oct 25 '14 at 23:02