Hausdorff measure

In mathematics a Hausdorff measure is a type of outer measure, named for Felix Hausdorff, that assigns a number in to each set in R n {displaystyle mathbb {R} ^{n}} or, more generally, in any metric 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. The one-dimensional Hausdorff measure of a simple curve in R n {displaystyle mathbb {R} ^{n}} is equal to the length of the curve. Likewise, the two dimensional Hausdorff measure of a measurable subset of R 2 {displaystyle mathbb {R} ^{2}} is proportional to the area of the set. Thus, the concept of the Hausdorff measure generalizes counting, length, and area. It also generalizes volume. In fact, there are d-dimensional Hausdorff measures for any d ≥ 0, which is not necessarily an integer. These measures are fundamental in geometric measure theory. They appear naturally in harmonic analysis or potential theory. In mathematics a Hausdorff measure is a type of outer measure, named for Felix Hausdorff, that assigns a number in to each set in R n {displaystyle mathbb {R} ^{n}} or, more generally, in any metric 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. The one-dimensional Hausdorff measure of a simple curve in R n {displaystyle mathbb {R} ^{n}} is equal to the length of the curve. Likewise, the two dimensional Hausdorff measure of a measurable subset of R 2 {displaystyle mathbb {R} ^{2}} is proportional to the area of the set. Thus, the concept of the Hausdorff measure generalizes counting, length, and area. It also generalizes volume. In fact, there are d-dimensional Hausdorff measures for any d ≥ 0, which is not necessarily an integer. These measures are fundamental in geometric measure theory. They appear naturally in harmonic analysis or potential theory. Let ( X , ρ ) {displaystyle (X, ho )} be a metric space. For any subset U ⊂ X {displaystyle Usubset X} , let d i a m U {displaystyle mathrm {diam} ;U} denote its diameter, that is Let S {displaystyle S} be any subset of X , {displaystyle X,} and δ > 0 {displaystyle delta >0} a real number. Define where the infimum is over all countable covers of S {displaystyle S} by sets U i ⊂ X {displaystyle U_{i}subset X} satisfying diam ⁡ U i < δ {displaystyle operatorname {diam} U_{i}<delta } . Note that H δ d ( S ) {displaystyle H_{delta }^{d}(S)} is monotone decreasing in δ {displaystyle delta } since the larger δ {displaystyle delta } is, the more collections of sets are permitted, making the infimum smaller. Thus, lim δ → 0 H δ d ( S ) {displaystyle lim _{delta o 0}H_{delta }^{d}(S)} exists but may be infinite. Let It can be seen that H d ( S ) {displaystyle H^{d}(S)} is an outer measure (more precisely, it is a metric outer measure). By general theory, its restriction to the σ-field of Carathéodory-measurable sets is a measure. It is called the d {displaystyle d} -dimensional Hausdorff measure of S {displaystyle S} . Due to the metric outer measure property, all Borel subsets of X {displaystyle X} are H d {displaystyle H^{d}} measurable. In the above definition the sets in the covering are arbitrary. However, they may be taken to be open or closed, and will yield the same measure, although the approximations H δ d ( S ) {displaystyle H_{delta }^{d}(S)} may be different (Federer 1969, §2.10.2). If X {displaystyle X} is a normed space the sets may be taken to be convex. However, the restriction of the covering families to balls gives a different, yet comparable, measure. Note that if d is a positive integer, the d dimensional Hausdorff measure of R d {displaystyle mathbb {R} ^{d}} is a rescaling of usual d-dimensional Lebesgue measure λ d {displaystyle lambda _{d}} which is normalized so that the Lebesgue measure of the unit cube d is 1. In fact, for any Borel set E, where αd is the volume of the unit d-ball; it can be expressed using Euler's gamma function