Hausdorff measure

To determine the area of a -dimensional area in the -dimensional space (with ) there are various measures in mathematics that are defined for all subsets of the and result in the heuristic area of the "decent" (not degenerate) -dimensional areas. (The "decent" surfaces include, in particular, the submanifolds of the .) ${\displaystyle m}$ ${\displaystyle n}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle m<n}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle m}$ ${\displaystyle \mathbb {R} ^{n}}$

The best known of these measures is the -dimensional Hausdorff measure , named after Felix Hausdorff ; to illustrate the definition, however, the -dimensional spherical dimension should first be explained. ${\displaystyle m}$ ${\displaystyle {\mathcal {H}}^{m}}$ ${\displaystyle m}$ ${\displaystyle {\mathcal {S}}^{m}}$

Definition of the spherical dimension

For a subset of the one considers the sizes ${\displaystyle A}$ ${\displaystyle \mathbb {R} ^{n}}$

${\displaystyle {\mathcal {S}}_{\varepsilon }^{m}(A)=\inf \left\{\sum _{i=1}^{\infty }\alpha (m)\left({\frac {1}{2}}\,\operatorname {diam} (B_{i})\right)^{m}\right|\left.A\subset \bigcup _{i=1}^{\infty }B_{i};\;B_{i}{\text{ Kugel im }}\mathbb {R} ^{n};\;\operatorname {diam} (B_{i})<\varepsilon \right\}}$

for the, infimum over all overlaps of by a countable number -dimensional spheres ... in diameters (Diametern) is formed. Here, the volume of the -dimensional unit sphere (sphere with radius 1) is equivalent to the -dimensional area of ​​the -dimensional unit circle im . The form factor ensures the correct "normalization" of the resulting surface area. The summands are precisely the -dimensional areas of the intersections of the spheres with -dimensional planes in the center running through their center . ${\displaystyle \varepsilon >0}$ ${\displaystyle (B_{i})_{i\in \mathbb {N} }}$ ${\displaystyle A}$ ${\displaystyle n}$ ${\displaystyle B_{1},B_{2},}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \operatorname {diam} (B_{i})<\varepsilon }$ ${\displaystyle \alpha (m)}$ ${\displaystyle m}$ ${\displaystyle \mathbb {R} ^{m}}$ ${\displaystyle m}$ ${\displaystyle m}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \alpha (m)}$ ${\displaystyle \alpha (m)(\operatorname {diam} (B_{i})/2)^{m}}$ ${\displaystyle m}$ ${\displaystyle B_{i}}$ ${\displaystyle m}$ ${\displaystyle \mathbb {R} ^{n}}$

The -dimensional spherical measure of is then, due to the increasing smallness of the spheres, defined by ${\displaystyle m}$ ${\displaystyle A}$

${\displaystyle {\mathcal {S}}^{m}(A)=\lim _{\varepsilon \to 0}{\mathcal {S}}_{\varepsilon }^{m}(A).}$

The refinement of the spherical overlaps by diameters approaching 0 brings about an increasing approach of the -dimensional equatorial surfaces of the spheres to the initial surface . ${\displaystyle m}$ ${\displaystyle A}$

Definition of the Hausdorff measure

One arrives at the definition of the Hausdorff measure if instead of the spheres all subsets of the are allowed for the overlaps. The diameter of is defined by ${\displaystyle {\mathcal {H}}^{m}}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle B\subset \mathbb {R} ^{m}}$

${\displaystyle \operatorname {diam} (B)=\sup \,\left\{|x-y|\,:\,x,y\in B\right\}}$

for and , and you bet accordingly for ${\displaystyle B\neq \emptyset }$ ${\displaystyle \operatorname {diam} (\emptyset )=0}$ ${\displaystyle A\subset \mathbb {R} ^{n}}$

${\displaystyle {\mathcal {H}}_{\varepsilon }^{m}(A)=\inf \left\{\sum _{i=1}^{\infty }\alpha (m)\left({\frac {1}{2}}\operatorname {diam} (B_{i})\right)^{m}\right|\left.A\subset \bigcup _{i=1}^{\infty }B_{i};\;B_{i}\subset \mathbb {R} ^{n};\;\operatorname {diam} (B_{i})<\varepsilon \right\},}$

whereby here the infimum is formed over all coverages of by a countable number of (arbitrary) subsets ... of with . After all, you define ${\displaystyle (B_{i})_{i\in \mathbb {N} }}$ ${\displaystyle A}$ ${\displaystyle B_{1},B_{2},}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \operatorname {diam} (B_{i})<\varepsilon }$

${\displaystyle {\mathcal {H}}^{m}(A)=\lim _{\varepsilon \to 0}{\mathcal {H}}_{\varepsilon }^{m}(A)}$

the metric outer dimension , which is also called the outer Hausdorff dimension. The limitation of the domain to Carathéodory-measurable quantities provides the measure . ${\displaystyle {\mathcal {H}}^{m}}$ ${\displaystyle {\mathcal {H}}^{m}}$

The expressions and are themselves external measures and have different values ​​for certain quantities - the difference does not disappear in some "pathological" cases when the border crosses towards 0 - however, the two measures and the rectifiable (the "decent") dimensional quantities provide same value. In general, the inequality applies ${\displaystyle {\mathcal {S}}_{\varepsilon }^{m}}$ ${\displaystyle {\mathcal {H}}_{\varepsilon }^{m}}$ ${\displaystyle \varepsilon }$ ${\displaystyle {\mathcal {S}}^{m}}$ ${\displaystyle {\mathcal {H}}^{m}}$ ${\displaystyle m}$

${\displaystyle {\mathcal {H}}^{m}\leq {\mathcal {S}}^{m}\leq [2n/(n+1)]^{m/2}{\mathcal {H}}^{m}.}$

Relationship with the area formula

For the explicit calculation of the Hausdorff measure of a parameterized area with an area and an injective differentiable function , the area formula is used: ${\displaystyle A=f(G)}$ ${\displaystyle G\subset \mathbb {R} ^{m}}$ ${\displaystyle f\colon G\to \mathbb {R} ^{n}}$

${\displaystyle {\mathcal {H}}^{m}(A)=\int _{G}{\sqrt {\det(Df\,^{t}Df)}}\,\mathrm {d} {\mathcal {L}}^{m}.}$

It is the generalized Jacobian of , and refers to the dimensional Lebesgue measure (of volume) in the  . ${\displaystyle {\sqrt {\det(Df\,^{t}Df)}}}$ ${\displaystyle f}$ ${\displaystyle {\mathcal {L}}^{m}}$ ${\displaystyle m}$ ${\displaystyle \mathbb {R} ^{m}}$

Generalizations

1. The above definitions of and with are used analogously for “non-integer dimensions” , where denotes the gamma function . The Hausdorff dimension of a subset of is then the (uniquely determined) number with for all and for all . Because of the above inequality, the difference between and in determining the Hausdorff dimension does not matter. In the past few decades, fractals have come under the spotlight in popular science media. Fractals are subsets of the Hausdorff dimension with a broken (“fractal”); In the public, fractals are predominantly perceived as sets which, in addition to their fractal dimension, are also characterized by certain self-similarities . ${\displaystyle m}$ ${\displaystyle {\mathcal {S}}^{m}}$ ${\displaystyle {\mathcal {H}}^{m}}$ ${\displaystyle \alpha (m)=\Gamma (1/2)^{m}/\Gamma (1+m/2)}$ ${\displaystyle \Gamma }$ ${\displaystyle A}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle m}$ ${\displaystyle {\mathcal {H}}^{s}(A)=\infty }$ ${\displaystyle s<m}$ ${\displaystyle {\mathcal {H}}^{s}(A)=0}$ ${\displaystyle s>m}$ ${\displaystyle {\mathcal {H}}^{m}}$ ${\displaystyle {\mathcal {S}}^{m}}$
   ${\displaystyle \mathbb {R} ^{n}}$
2. The definition of the -dimensional Hausdorff measure remains valid without significant changes in any metric space instead of the ; the same applies to the -dimensional spherical measure. (Only the amount function in the definition of the diameter is replaced by the underlying metric , more precisely: becomes off ) ${\displaystyle m}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle m}$ ${\displaystyle d}$ ${\displaystyle |x-y|}$ ${\displaystyle d(x,y).}$

literature

• Herbert Federer : Geometric Measure Theory (=  basic teaching of the mathematical sciences with special consideration of the areas of application . Volume 153 ). Springer, 1969, ISBN 3-540-60656-4 , ISSN  0072-7830 (reprinted there in 1996).