Let be a metric space, be a subset of , and a number . The -dimensional Hausdorff measure of , , is the infimum of positive numbers such that for every , can be covered by a countable family of closed sets, each of diameter less than , such that the sum of the th powers of their diameters is less than . Note that may be infinite, and need not be an integer.
Hausdorff Measure
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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.Referenced on Wolfram|Alpha
Hausdorff MeasureCite this as:
Weisstein, Eric W. "Hausdorff Measure." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HausdorffMeasure.html