In mathematics a Hausdorff measure assigns a number in ${\displaystyle [0,\infty ]}$ 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 ${\displaystyle \infty }$ if the space is infinite. The one dimensional Hausdorff measure of a metric space which is an imbedded path in ${\displaystyle \mathbb {R} ^{n}}$ is proportional to the length of the path. Likewise, the two dimensional Hausdorff measure of a subset of ${\displaystyle \mathbb {R} ^{2}}$ 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 ${\displaystyle d}$-dimensional Hausdorff measures for any ${\displaystyle d\geq 0}$ which is not necessarily an integer. These measures are useful for studying the size of fractals.

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

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

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

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

This is the ${\displaystyle d}$-dimensional Hausdorff measure of ${\displaystyle S}$.

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

• 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.