Finish the proof of Theorem 18.12 (i) and show that every set \(A \subset \mathbb{R}^{n}\) is indeed \(\overline{\mathcal{H}}^{0}\)-measurable.

Data from theorem 18.12

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Theorem 18.12 Let Hs, s20 be Hausdorff measure on (R", B(R")). (i) H is the counting measure on P(R"). (ii) H is one-dimensional Lebesgue measure on B(R). (iii) H (R")=0 ifs>n. (iv) (4) 0. (vi) H(f(A))