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Hausdorff Measure


Let X be a metric space, A be a subset of X, and d a number >=0. The d-dimensional Hausdorff measure of A, H^d(A), is the infimum of positive numbers y such that for every r>0, A can be covered by a countable family of closed sets, each of diameter less than r, such that the sum of the dth powers of their diameters is less than y. Note that H^d(A) may be infinite, and d need not be an integer.


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References

Federer, H. Geometric Measure Theory. New York: Springer-Verlag, 1969.Ott, E. Chaos in Dynamical Systems. Cambridge, England: Cambridge University Press, p. 103, 1993.Rogers, C. A. Hausdorff Measures, 2nd ed. Cambridge, England: Cambridge University Press, 1999.

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Hausdorff Measure

Cite this as:

Weisstein, Eric W. "Hausdorff Measure." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HausdorffMeasure.html

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