Let \(\lambda\) denote Lebesgue measure on \(\mathbb{R}^{n}\) and \(u \in \mathcal{L}^{1}(\lambda)\).

(i) For every \(\epsilon>0\) there is a set \(B \in \mathscr{B}\left(\mathbb{R}^{n}ight), \lambda(B)<\infty\) with \(\sup _{B}|u|<\infty\) and \(\int_{B^{c}}|u| d \lambda<\epsilon\).

(ii) Use (i) to show that \(\lim _{\lambda(B) ightarrow 0} \int_{B}|u| d \lambda=0\).