Hausdorff measure

Summary

The $s$-dimensional Hausdorff measure of a subset of a metric space measures its $s$-dimensional size, for any real $s \ge 0$. It is built from covers by small sets, each counted by its diameter to the power $s$. For whole numbers $s = n$ it extends length, area and volume to rough sets, and in $\R^n$ it agrees with Lebesgue measure up to a constant. It defines Hausdorff dimension and is the basic measure of fractal geometry.