Inmathematics,Hausdorff measure is a generalization of the traditional notions ofarea andvolume to non-integer dimensions, specificallyfractals and theirHausdorff dimensions. It is a type ofouter measure, named forFelix Hausdorff, that assigns a number in [0,∞] to each set in${\displaystyle {\mathbb{R}}^n}$ or, more generally, in anymetric space.

The zero-dimensional Hausdorff measure is the number of points in the set (if the set is finite) or ∞ if the set is infinite. Likewise, the one-dimensional Hausdorff measure of asimple curve in${\displaystyle {\mathbb{R}}^n}$ is equal to the length of the curve, and the two-dimensional Hausdorff measure of aLebesgue-measurable subset of${\displaystyle {\mathbb{R}}^2}$ is proportional to the area of the set. Thus, the concept of the Hausdorff measure generalizes theLebesgue measure and its notions of counting, length, and area. It also generalizes volume. In fact, there ared-dimensional Hausdorff measures for anyd ≥ 0, which is not necessarily an integer. These measures are fundamental ingeometric measure theory. They appear naturally inharmonic analysis orpotential theory.

Let${\displaystyle (X,\rho )}$ be ametric space. For any subset${\displaystyle U\subset X}$, let${\displaystyle \mathrm{diam}U}$ denote its diameter, that is

${\displaystyle \mathrm{diam}U:=sup\{\rho (x,y):x,y\in U\},\mathrm{diam}\mathrm{\varnothing }:=0.}$

Let${\displaystyle S}$ be any subset of${\displaystyle X,}$ and${\displaystyle \delta >0}$ a real number. Define

where the infimum is over all countable covers of${\displaystyle S}$ by sets${\displaystyle U_i\subset X}$ satisfying.

Note that${\displaystyle H_{\delta }^d(S)}$ is monotone nonincreasing in${\displaystyle \delta }$ since the larger${\displaystyle \delta }$ is, the more collections of sets are permitted, making the infimum not larger. Thus,${\displaystyle \underset{\delta \to 0}{lim}{H}_{\delta }^d(S)}$ exists but may be infinite. Let

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

It can be seen that${\displaystyle H^d(S)}$ is anouter measure (more precisely, it is ametric outer measure). ByCarathéodory's extension theorem, its restriction to the σ-field ofCarathéodory-measurable sets is a measure. It is called the${\displaystyle d}$-dimensional Hausdorff measure of${\displaystyle S}$. Due to themetric outer measure property, allBorel subsets of${\displaystyle X}$ are${\displaystyle H^d}$ measurable.

In the above definition the sets in the covering are arbitrary. However, we can require the covering sets to be open or closed, or innormed spaces even convex, that will yield the same${\displaystyle H_{\delta }^d(S)}$ numbers, hence the same measure. In${\displaystyle {\mathbb{R}}^n}$ restricting the covering sets to be balls may change the measures but does not change the dimension of the measured sets.

Note that ifd is a positive integer, thed-dimensional Hausdorff measure of${\displaystyle {\mathbb{R}}^d}$ is a rescaling of the usuald-dimensionalLebesgue measure${\displaystyle {\lambda }_d}$, which is normalized so that the Lebesgue measure of the unit cube [0,1]^d is 1. In fact, for any Borel setE,

${\displaystyle {\lambda }_d(E)=2^{-d}{\alpha }_d{H}^d(E),}$

where${\displaystyle 2^{-d}}$ scales diameter to radius; while${\displaystyle {\alpha }_d}$ is thevolume of the unitd-ball withradius one, which can be expressed usingEuler's gamma function

${\displaystyle {\alpha }_d=\frac{\mathrm{\Gamma }{\left(\frac{1}{2}\right)}^d}{\mathrm{\Gamma }\left(\frac{d}{2}+1\right)}=\frac{{\pi }^{d/2}}{\mathrm{\Gamma }\left(\frac{d}{2}+1\right)}.}$

This is

${\displaystyle {\lambda }_d(E)={\beta }_d{H}^d(E)}$,

where${\displaystyle {\beta }_d=2^{-d}{\alpha }_d}$ is the volume of thed-ball withdiameter one.

Some authors (e.g. Evans & Gariepy (2015), chapters 2,3) adopt a definition of Hausdorff measure slightly different from the one chosen here, the difference being that the value${\displaystyle H^d(E)}$ defined above is multiplied by the factor${\displaystyle {\beta }_d}$, so that the scaled Hausdorffd-dimensional measure coincides exactly with Lebesgue measure in the case of Euclidean space. In this article we adopt the notation forscaled Hausdorff measure:

${\displaystyle {\overline{H}}^d(E)={\beta }_d{H}^d(E)}$

Further examples of the agreement of this scaled measure with Lebesgue measure include:

• For${\displaystyle B_r^d}$, thed-ball with radius${\displaystyle r}$ the agreement is direct. The volume is:${\displaystyle {\overline{H}}^d(B_r^d)={\lambda }_d(B_r^d)={\alpha }_d{r}^d.}$
• For${\displaystyle S_r^{d-1}}$, the(d-1)-sphere (surface of${\displaystyle B_r^d}$) the agreement is more indirect.The area is:${\displaystyle {\overline{H}}^{d-1}(S_r^{d-1})=\frac{d}{dr}{\lambda }_d(B_r^d)=d{\alpha }_d{r}^{d-1}.}$ Note however that${\displaystyle {\lambda }_d(S_r^{d-1})=0}$, while${\displaystyle {\lambda }_{d-1}(S_r^{d-1})}$ is not defined.
• More generally, for a positive integer, let${\displaystyle {\mathcal{M}}^m}$ be asmoothm-dimensional manifold embedded in${\displaystyle {\mathbb{R}}^d}$. For a measureable subset,${\displaystyle E\subseteq {\mathcal{M}}^m}$, Lebesgue measure is not directly applicable, because${\displaystyle {\lambda }_d(E)=0}$, while${\displaystyle {\lambda }_m(E)}$ is not defined. But (informally stated), if we make${\displaystyle E}$ small enough, so that it is indistinguishable from a subset in the localtangent space, which is anm-dimensional linear subspace of${\displaystyle {\mathbb{R}}^d}$, we can apply${\displaystyle {\lambda }_m}$ in the tangent space, where it will closely approximate${\displaystyle {\overline{H}}^m(E)}$. An example is where local area on Earth's surface can be approximated by applying${\displaystyle {\lambda }_2}$ to a locallyisometric chart.

It turns out that${\displaystyle H^d(S)}$ may have a finite, nonzero value for at most one${\displaystyle d}$. That is, the Hausdorff Measure is zero for any value above a certain dimension and infinity below a certain dimension, analogous to the idea that the area of a line is zero and the length of a 2D shape is in some sense infinity. This leads to one of several possible equivalent definitions of the Hausdorff dimension:

${\displaystyle {\dim }_{\mathrm{H}\mathrm{a}\mathrm{u}\mathrm{s}}(S)=inf\{d\ge 0:H^d(S)=0\}=sup\{d\ge 0:H^d(S)=\mathrm{\infty }\},}$

where we take${\displaystyle inf\mathrm{\varnothing }=+\mathrm{\infty }}$and${\displaystyle sup\mathrm{\varnothing }=0}$.

Note that it is not guaranteed that the Hausdorff measure must be finite and nonzero for somed, and indeed the measure at the Hausdorff dimension may still be zero; in this case, the Hausdorff dimension still acts as a change point between measures of zero and infinity.

Ingeometric measure theory and related fields, theMinkowski content is often used to measure the size of a subset of a metric measure space. For suitable domains in Euclidean space, the two notions of size coincide, up to overall normalizations depending on conventions. More precisely, a subset of${\displaystyle {\mathbb{R}}^n}$ is said to be${\displaystyle m}$-rectifiable if it is the image of abounded set in${\displaystyle {\mathbb{R}}^m}$ under aLipschitz function. If, then the${\displaystyle m}$-dimensional Minkowski content of a closed${\displaystyle m}$-rectifiable subset of${\displaystyle {\mathbb{R}}^n}$ is equal to${\displaystyle 2^{-m}{\alpha }_m}$ times the${\displaystyle m}$-dimensional Hausdorff measure (Federer 1969, Theorem 3.2.39, pp 275).

Infractal geometry, some fractals with Hausdorff dimension${\displaystyle d}$ have zero or infinite${\displaystyle d}$-dimensional Hausdorff measure. For example,almost surely the image of planarBrownian motion has Hausdorff dimension 2 and its two-dimensional Hausdorff measure is zero. In order to "measure" the "size" of such sets, the following variation on the notion of the Hausdorff measure can be considered:

In the definition of the measure${\displaystyle (\mathrm{diam}{U}_i{)}^d}$ is replaced with${\displaystyle \varphi (\mathrm{diam}{U}_i),}$ where${\displaystyle \varphi }$ is any monotone increasing function satisfying${\displaystyle \varphi (0)=0.}$

This is the Hausdorff measure of${\displaystyle S}$ withgauge function${\displaystyle \varphi ,}$ or${\displaystyle \varphi }$-Hausdorff measure. A${\displaystyle d}$-dimensional set${\displaystyle S}$ may satisfy${\displaystyle H^d(S)=0,}$ but${\displaystyle H^{\varphi }(S)\in (0,\mathrm{\infty })}$ with an appropriate${\displaystyle \varphi .}$ Examples of gauge functions include

${\displaystyle \varphi (t)=t^2\log \log \frac{1}{t}\text{or}\varphi (t)=t^2\log \frac{1}{t}\log \log \log \frac{1}{t}.}$

The former gives almost surely positive and${\displaystyle \sigma }$-finite measure to the Brownian path in${\displaystyle {\mathbb{R}}^n}$ when${\displaystyle n>2}$, and the latter when${\displaystyle n=2}$.

• Measure theory

• Evans, Lawrence C.; Gariepy, Ronald F. (2015),Measure Theory and Fine Properties of Functions (Revised Edition), CRC Press.
• Federer, Herbert (1969),Geometric Measure Theory, Springer-Verlag,ISBN 3-540-60656-4.
• Hausdorff, Felix (1918),"Dimension und äusseres Mass"(PDF),Mathematische Annalen,79 (1–2):157–179,doi:10.1007/BF01457179,S2CID 122001234.
• Morgan, Frank (1988),Geometric Measure Theory, Academic Press.
• Rogers, C. A. (1998),Hausdorff measures, Cambridge Mathematical Library (3rd ed.),Cambridge University Press,ISBN 0-521-62491-6
• Szpilrajn, E (1937),"La dimension et la mesure"(PDF),Fundamenta Mathematicae,28:81–89,doi:10.4064/fm-28-1-81-89.

• Hausdorff dimension atEncyclopedia of Mathematics
• Hausdorff measure atEncyclopedia of Mathematics

• Fractals
• Measures (measure theory)
• Metric geometry
• Dimension theory

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