更改

对于自然界中出现的分形，豪斯多夫维数和盒计数维数是一致的。封装尺寸是另一个类似的概念，它给出了许多形状相同的值，但是在所有这些尺寸不同的情况下，有很好的文档说明的例外。

对于自然界中出现的分形，豪斯多夫维数和盒计数维数是一致的。封装尺寸是另一个类似的概念，它给出了许多形状相同的值，但是在所有这些尺寸不同的情况下，有很好的文档说明的例外。

==Formal definitions格式定义==

==Formal definitions形式化定义==

{{unreferenced section|date=March 2015}}

{{unreferenced section|date=March 2015}}

===Hausdorff content豪斯多夫===

===Hausdorff content豪斯多夫集===

Let ''X'' be a [[metric space]]. If ''S'' ⊂ ''X'' and ''d'' ∈ [0, ∞), the ''d''-dimensional '''unlimited Hausdorff content''' of ''S'' is defined by

Let ''X'' be a [[metric space]]. If ''S'' ⊂ ''X'' and ''d'' ∈ [0, ∞), the ''d''-dimensional '''unlimited Hausdorff content''' of ''S'' is defined by

Let X be a metric space. If S ⊂ X and d ∈ [0, ∞), the d-dimensional unlimited Hausdorff content of S is defined by

Let X be a metric space. If S ⊂ X and d ∈ [0, ∞), the d-dimensional unlimited Hausdorff content of S is defined by

设 x 是度量空间。若 s something x 和 d ∈[0，∞) ，则 s 的 d 维无限 豪斯多夫内容定义为

设 x 是度量空间。若 s something x 和 d ∈[0，∞) ，则 s 的 d 维无限 豪斯多夫集定义为

:<math>C_H^d(S):=\inf\Bigl\{\sum_i r_i^d:\text{ there is a cover of } S\text{ by balls with radii }r_i>0\Bigr\}.</math>

:<math>C_H^d(S):=\inf\Bigl\{\sum_i r_i^d:\text{ there is a cover of } S\text{ by balls with radii }r_i>0\Bigr\}.</math>

The Hausdorff outer measure is different from the unbounded Hausdorf content in that rather than considering all possible coverings of S, we see what happens when the sizes of the balls go to zero. This is  for <math>d \geq 0 </math>, we define the d-dimensional Hausdorff outer measure of S as

The Hausdorff outer measure is different from the unbounded Hausdorf content in that rather than considering all possible coverings of S, we see what happens when the sizes of the balls go to zero. This is  for <math>d \geq 0 </math>, we define the d-dimensional Hausdorff outer measure of S as

豪斯多夫外测度不同于无界的豪斯多夫内容，因为我们不考虑 s 的所有可能，我们看到当球的大小为零时会发生什么。这是为了数学 d  geq 0 / math，我们定义了 s 的 d 维豪斯多夫Hausdorff 外测度为

豪斯多夫外测度不同于无界的豪斯多夫，因为我们不考虑 s 的所有可能，我们看到当球的大小为零时会发生什么。这是为了数学 d  geq 0 / math，我们定义了 s 的 d 维豪斯多夫Hausdorff 外测度为

:<math> \mathcal{H}^d(S):=\lim_{r \to 0} \inf\Bigl\{\sum_i r_i^d:\text{ there is a cover of } S\text{ by balls with radii } 0 < r_i < r\Bigr\}.</math>

:<math> \mathcal{H}^d(S):=\lim_{r \to 0} \inf\Bigl\{\sum_i r_i^d:\text{ there is a cover of } S\text{ by balls with radii } 0 < r_i < r\Bigr\}.</math>