Hausdorff Measure

Let $X$ be a Metric Space, $A$ be a Subset of $X$, and $d$ a number $\geq 0$. 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 $d$th Powers of their diameters is less than $y$. Note that $H^d(A)$ may be infinite, and $d$ need not be an Integer.

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.