Let \(\lambda:=\left.\lambda^{1}ight|_{[0,1]}\) be Lebesgue measure on \(([0,1], \mathscr{B}[0,1])\). Show that for every \(\epsilon>0\) there is a dense open set \(U \subset[0,1]\) with \(\lambda(U) \leqslant \epsilon\).

[take an enumeration \(\left(q_{i}ight)_{i \in \mathbb{N}}\) of \(\mathbb{Q} \cap(0,1)\) and make each \(q_{i}\) the centre of a small open interval.]