Let ${E \subset {\bf R}^d}$ be a measurable set of positive measure, and let ${\varepsilon > 0}$. show that there exists a cube ${Q \subset {\bf R}^d}$ of positive sidelength such that ${m(E \cap Q) > (1-\varepsilon) m(Q)}$, without using the Lebesgue differentiation theorem.(Hint: reduce to the case when ${E}$ is bounded, then approximate ${E}$ by an almost disjoint union of cubes.)
Question: It can be shown that the problem can be reduced to $E$ being a bounded open set, and thus can be written as a countable union of almost disjoint cubes. From here, I wonder how to explicitly construct the cube $Q$ with the desired property?