Let \(\lambda=\lambda^{1}\) be Lebesgue measure on \((\mathbb{R}, \mathscr{B}(\mathbb{R}))\). Show that \(N \in \mathscr{B}(\mathbb{R})\) is a null set if, and only if, for every \(\epsilon>0\) there is an open set \(U_{\epsilon} \supset N\) such that \(\lambda\left(U_{\epsilon}ight)

[ sufficiency is trivial, for necessity use \(\lambda^{*}\) constructed in Theorem 6.1 from \(\left.\lambdaight|_{\mathscr{O}}\) and observe that \(\left.\lambdaight|_{\mathscr{B}\left(\mathbb{R}^{n}ight)}=\left.\lambda^{*}ight|_{\mathscr{B}\left(\mathbb{R}^{n}ight)}\). This gives the required open cover.]

Data from theorem 6.1

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(Carathodory) Let S P(X) be a semi-ring and u: S [0, ] a pre-measure, i.e. a set function with (i) (0) = 0; (ii) (Sn)neNCS, disjoint and S=USES (S) = (Sn). NEN NEN