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 $\mathbb {R} ^{n}$ is proportional to the length of the path. Likewise, the two dimensional Hausdorff measure of a subset of $\mathbb {R} ^{2}$ 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 $d\geq 0$ which is not necessarily an integer. These measures are useful for studying the size of fractals.

Definition

Fix some $d\geq 0$ and a metric space $X$. Let $S\subset X$ be any subset of $X$. For $\delta >0$ let

$H_{\delta }^{d}(S):=\inf {\Bigl \{}\sum _{i}r_{i}^{d}:{\text{there is a collection of balls with radii }}r_{i}\in (0,\delta ){\text{ which cover }}S{\Bigr \}}.$

Note that $H_{\delta }^{d}(S)$ is monotone decreasing in $\delta$ since the larger $\delta$ is, the more collections of balls are permitted. Thus, the limit $\lim _{\delta \to 0}H_{\delta }^{d}(S)$ exists. Set

$H^{d}(S):=\sup _{\delta >0}H_{\delta }^{d}(S)=\lim _{\delta \to 0}H_{\delta }^{d}(S).$

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.

References

• L. Evans and R. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, 1992
• H. Federer, Geometric Measure Theory, Springer-Verlag, 1969.
• Frank Morgan, Geometric Measure Theory, Academic Press, 1988. Good introductory presentation with lots of illustrations.
• E. Szpilrajn, La dimension et la mesure, Fundamenta Mathematica 28, 1937, pp 81-89.