Hausdorff Measure

A family of outer measures on a metric space parameterized by a nonnegative real s, constructed by covering sets with countably many sets of small diameter and taking the infimum of the sum of diameters^s (Carathéodory construction); for each s this yields the s-dimensional Hausdorff measure, which generalizes length, area, and volume and identifies fractal dimensions.

For each nonnegative real s, the Hausdorff s-measure is an outer measure defined on a metric space by taking the limit as δ→0 of the infimum of sums ∑ diam(U_i)^s over countable covers {U_i} with diam(U_i)≤δ; it generalizes length, area, and volume and detects sizes at non-integer (fractal) dimensions.

A family of outer measures on a metric space defined for each dimension parameter d≥0 by covering sets with arbitrarily small diameter and summing the dth power of those diameters (possibly with a constant); these outer measures generalize length, area and volume and detect fractal scaling.