Redirected from Besicovitch - Hausdorff dimension
If M is a metric space, and d > 0 is a real number, then the d-dimensional Hausdorff measure Hd(M) is defined to be the infimum of all m > 0 such that for all r > 0, M can be covered by countably many closed sets of diameter < r and the sum of the d-th powers of these diameters is less than or equal to m.
It turns out that for most values of d, this measure Hd(M) is either 0 or ∞. If d is smaller than the "true dimension" of M, then Hd(M) = ∞; if it is bigger then Hd(M) = 0.
The Hausdorff dimension d(M) is then defined to be the "cutoff point", i.e. the infimum of all d > 0 such that Hd(M) = 0. The Hausdorff dimension is a well-defined real number for any metric space M and we always have 0 ≤ d(M) ≤ ∞.
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