Does there exist a Lebesgue measurable set $E$ in the plane $\Bbb{R}^2$ such that for every countable sequence $(Q_n)_{n \in \Bbb{N}}$ of rectangles with sides parallel to the coordinates axes the Lebesgue measure of the symmetric difference $E \Delta \bigcup_{n = 1}^\infty Q_n$ is never zero?
In other words I am wondering whether measurable sets can be decomposed as countable union of rectangles modulo sets of measure zero. I believe that is not true and I am looking for a counterexample.