Editing Hausdorff dimension (section)

==Formal definition==
{{main| Hausdorff measure}}
The formal definition of the Hausdorff dimension is arrived at by defining first the d-dimensional [[Hausdorff measure]], a fractional-dimension analogue of the [[Lebesgue measure]]. First, an [[outer measure]] is constructed:
Let <math>X</math> be a [[metric space]]. If <math>S\subset X</math> and <math>d\in [0,\infty)</math>,

:<math>H^d_\delta(S)=\inf\left \{\sum_{i=1}^\infty (\operatorname{diam} U_i)^d: \bigcup_{i=1}^\infty U_i\supseteq S, \operatorname{diam} U_i<\delta\right \},</math>

where the [[infimum]] is taken over all countable covers <math>U</math> of <math>S</math>. The Hausdorff d-dimensional outer measure is then defined as <math>\mathcal{H}^d(S)=\lim_{\delta\to 0}H^d_\delta(S)</math>, and the restriction of the mapping to [[non-measurable set| measurable set]]s justifies it as a measure, called the <math>d</math>-dimensional Hausdorff Measure.<ref>{{cite web| last1=Briggs| first1=Jimmy| last2=Tyree|first2=Tim| title=Hausdorff Measure| url=https://sites.math.washington.edu/~farbod/teaching/cornell/math6210pdf/math6210Hausdorff.pdf| date=3 December 2016| access-date=3 February 2022| publisher=University of Washington}}</ref>

===Hausdorff dimension===

The '''Hausdorff dimension''' <math>\dim_{\operatorname{H}}{(X)}</math> of <math>X</math> is defined by
:<math>\dim_{\operatorname{H}}{(X)}:=\inf\{d\ge 0: \mathcal{H}^d(X)=0\}.</math>

This is the same as the [[supremum]] of the set of <math>d\in [0,\infty)</math> such that the <math>d</math>-dimensional Hausdorff measure of <math>X</math> is infinite (except that when this latter set of numbers <math>d</math> is empty the Hausdorff dimension is zero).

===Hausdorff content===
The <math>d</math>-dimensional '''unlimited Hausdorff content''' of <math>S</math> is defined by
:<math>C_H^d(S):= H_\infty^d(S) = \inf\left \{ \sum_{k=1}^\infty (\operatorname{diam} U_k)^d: \bigcup_{k=1}^\infty U_k\supseteq S \right \}</math>

In other words, <math>C_H^d(S)</math> has the construction of the Hausdorff measure where the covering sets are allowed to have arbitrarily large sizes. (Here we use the standard convention that [[infimum|<math>\inf\varnothing=\infty</math>]].)<ref>{{Cite arXiv| last1=Farkas| first1=Abel| last2=Fraser| first2=Jonathan| title=On the equality of Hausdorff measure and Hausdorff content| date=30 July 2015| class=math.MG| eprint=1411.0867}}</ref> The Hausdorff measure and the Hausdorff content can both be used to determine the dimension of a set, but if the measure of the set is non-zero, their actual values may disagree.