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}