Show that the \(d\)-dimensional Hausdorff measure of a bounded open set \(U \subset \mathbb{R}^{d}\) is finite and positive: \(0<\mathscr{H}^{d}(U)<\infty\). In particular, \(\operatorname{dim} U=d\).

Since Hausdorff measure is monotone, it is enough to show that for an axis-parallel cube \(Q \subset \mathbb{R}^{d}\) we have \(0<\mathscr{H}^{d}(Q)<\infty\). Without loss, assume that \(Q=[0,1]^{d}\). The upper bound follows by covering \(Q\) with small cubes of side length \(1 / n\). The lower bound follows since every \(\delta\)-cover \(\left(E_{j}ight)_{j \geqslant 1}\) of \(Q\) gives rise to a cover by cubes whose side length is at most \(2 \delta\).