Show that the outer regularity from Corollary 18.10 coincides with the usual notion, i.e.

\[\overline{\mathcal{H}}^{\phi}(A)=\inf \left\{\mathcal{H}^{\phi}(U): U \supset A, U \text { open }ight\},\]

provided that there exists some open set \(U \supset A\) with finite Hausdorff measure. Use the example \(\mathcal{H}^{0}\) (counting measure) to show that this condition is essential.

Data from corollary 18.10

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Corollary 18.10 Let (X, d) be a metric space. Hausdorff measure H is outer regular in the sense that for each ACX there is a G-set G (i.e. a countable intersection of open sets) such that H (A)= H (G).