I have been trying to find a Borel measurable set $B$ in [0,1] such that the intersection of $B$ with any open interval has non-zero Lebesgue measure (i.e. is dense in measure), and such that its complement $B^c$ is also dense in measure.
I have come up with the set $B=\{\mathbb{Q} \cup S\} $ where $S$ is the set containing any rational $r$ in the inteval in an infinite sum of the form
$$ r +\frac{a_1}{\sqrt{2}}+\frac{a_2}{\sqrt{3}}+\frac{a_3}{\sqrt{5}}+\cdots $$
where $a_i\in\{-1,0,1\}$ and we do not need finitely many non zero $a_i$.
This set contains non-algebraic numbers and therefore is at least a subset of the uncountable part of the irrationals. I believe it is indeed uncountable because for each rational number it can be combined with an infinite combination of the inverse square roots. In fact I'm fairly sure that the possible combinations for rational numbers in that interval with such square roots is the power set of the natural numbers.
My question is, how would I be able to calculate the measure of this set, or any intersection of this set with an open interval of [0,1]?