User:OdedSchramm/hm

In mathematics a Hausdorff measure assigns a number in $$[0,\infty]$$ to every metric space. The zero dimensional Hausdorff measure of a metric space is the number of points in the space (if the space is finite) or $$\infty$$ if the space is infinite. The one dimensional Hausdorff measure of a metric space which is an imbedded path in $$\R^n$$ is proportional to the length of the path. Likewise, the two dimensional Hausdorff measure of a subset of $$\R^2$$ is proportional to the area of the set. Thus, the concept of the Hausdorff measure generalizes counting, length, area. It also generalizes volume. In fact, there are $$d$$-dimensional Hausdorff measures for any $$d\ge 0$$ which is not necessarily an integer. These measures are useful for studying the size of fractals.

Definition
Fix some $$d\ge 0$$ and a metric space $$X$$. Let $$S\subset X$$ be any subset of $$X$$. For $$\delta>0$$ let
 * $$H^d_\delta(S):=\inf\Bigl\{\sum_i r_i^d : \text{there is a collection of balls with radii }r_i\in(0,\delta)\text{ which cover }S\Bigr\}. $$

Note that $$H^d_\delta(S)$$ is monotone decreasing in $$\delta$$ since the larger $$\delta$$ is, the more collections of balls are permitted. Thus, the limit $$\lim_{\delta\to 0}H^d_\delta(S)$$ exists. Set
 * $$ H^d(S):=\sup_{\delta>0} H^d_\delta(S)=\lim_{\delta\to 0}H^d_\delta(S).$$

This is the $$d$$-dimensional Hausdorff measure of $$S$$.

Properties of Hausdorff measures
The Hausdorff measures $$H^d$$ are outer measures. Moreover, all Borel subsets of $$X$$ are $$H^d$$ measureable. In particular, the theory of outer measures implies that $$H^d$$ is countably additive on the Borel σ-field.