(i) Show that \(\lambda^{1}((a, b))=b-a\) for all \(a, b \in \mathbb{R}, a \leqslant b\).

[ approximate \((a, b)\) by half-open intervals and use Proposition 4.3 (vi), (vii).]

(ii) Let \(H \subset \mathbb{R}^{2}\) be a hyperplane which is perpendicular to the \(x_{1}\)-direction (that is to say, \(H\) is a translate of the \(x_{2}\)-axis). Show that \(H \in \mathscr{B}\left(\mathbb{R}^{2}ight)\) and \(\lambda^{2}(H)=0\).

[ note that \(H \subset y+\cup_{k \in \mathbb{N}} A_{k}\) for some \(y\) and \(A_{k}=\left[-\epsilon 2^{-k}, \epsilon 2^{-k}ight) \times[-k, k)\).]

(iii) State and prove the \(\mathbb{R}^{n}\)-analogues of parts (i) and (ii).

Data from proposition 4.3

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Let (X, A,p) be a measure space and A, B, An, BEA, NEN. Then (1) ANB=0 (AJB) = (A) + (B) (ii) ACB (A)