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Questions about Hausdorff measures, their variants (such as spherical Hausdorff measures) and generalisations.

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Continuity of Hausdorff measure on level sets

As Leo Moos suggested in the comments, in any dimension $d$, this is a simple consequence of the implicit function theorem. The implicit function theorem implies that every point $x\in\phi^{-1}(0)$ ha …
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