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第313行： − 这种不平等可以是严格的。有可能找到两个维数为0的集合，其乘积的维数为1。在相反的方向上，我们知道当 x 和 y 是 r sup n / sup 的 Borel 子集时，x y 的豪斯多夫维数从上面以 x 的豪斯多夫维数加上 y 的填充维数为界。Mattila (1995)讨论了这些事实。+ − −
→Properties of Hausdorff dimension
*刘易斯·弗赖伊·理查森（Lewis Fry Richardson）已经通过豪斯多夫维数去测量了很多海岸线。它的结果涵盖从1.02的南非海岸线到1.25的大英帝国西海岸模型。
*刘易斯·弗赖伊·理查森（Lewis Fry Richardson）已经通过豪斯多夫维数去测量了很多海岸线。它的结果涵盖从1.02的南非海岸线到1.25的大英帝国西海岸模型。
==Properties of Hausdorff dimension==
==Properties of Hausdorff dimension豪斯多夫维数特性==
{{refimprove section|date=March 2015}}
{{refimprove section|date=March 2015}}
=== Hausdorff dimension and Minkowski dimension ===
=== Hausdorff dimension and Minkowski dimension 豪斯多夫维数和闵可夫斯基维度===
The [[Minkowski dimension]] is similar to, and at least as large as, the Hausdorff dimension, and they are equal in many situations. However, the set of [[rational number|rational]] points in [0, 1] has Hausdorff dimension zero and Minkowski dimension one. There are also compact sets for which the Minkowski dimension is strictly larger than the Hausdorff dimension.
The [[Minkowski dimension]] is similar to, and at least as large as, the Hausdorff dimension, and they are equal in many situations. However, the set of [[rational number|rational]] points in [0, 1] has Hausdorff dimension zero and Minkowski dimension one. There are also compact sets for which the Minkowski dimension is strictly larger than the Hausdorff dimension.
The Minkowski dimension is similar to, and at least as large as, the Hausdorff dimension, and they are equal in many situations. However, the set of rational points in [0, 1] has Hausdorff dimension zero and Minkowski dimension one. There are also compact sets for which the Minkowski dimension is strictly larger than the Hausdorff dimension.
The Minkowski dimension is similar to, and at least as large as, the Hausdorff dimension, and they are equal in many situations. However, the set of rational points in [0, 1] has Hausdorff dimension zero and Minkowski dimension one. There are also compact sets for which the Minkowski dimension is strictly larger than the Hausdorff dimension.
闵可夫斯基维度与豪斯多夫维数相似，至少和它一样大，而且在许多情况下是相等的。然而，[0,1]中有理点集的豪斯多夫维数为0，Minkowski 维数为1。还有一些紧集的 Minkowski 维数严格大于豪斯多夫维数。
闵可夫斯基维度与豪斯多夫维数相似，至少和它一样大，而且在许多情况下是相等的。然而，[0,1]中有理点集的豪斯多夫维数为0，Minkowski 维数为1。还有一些紧集的 Minkowski 维数绝对大于豪斯多夫维数。
=== Hausdorff dimensions and Frostman measures ===
=== Hausdorff dimensions and Frostman measures 豪斯多夫维度和弗洛斯曼测度===
If there is a [[measure (mathematics)|measure]] μ defined on [[Borel measure|Borel]] subsets of a metric space ''X'' such that ''μ''(''X'') > 0 and ''μ''(''B''(''x'', ''r'')) ≤ ''r<sup>s</sup>'' holds for some constant ''s'' > 0 and for every ball ''B''(''x'', ''r'') in ''X'', then dim<sub>Haus</sub>(''X'') ≥ ''s''. A partial converse is provided by [[Frostman's lemma]].{{citation needed|date=March 2015}}<ref>This Wikipedia article also discusses further useful characterizations of the Hausdorff dimension.{{clarify|date=March 2015}}</ref>
If there is a [[measure (mathematics)|measure]] μ defined on [[Borel measure|Borel]] subsets of a metric space ''X'' such that ''μ''(''X'') > 0 and ''μ''(''B''(''x'', ''r'')) ≤ ''r<sup>s</sup>'' holds for some constant ''s'' > 0 and for every ball ''B''(''x'', ''r'') in ''X'', then dim<sub>Haus</sub>(''X'') ≥ ''s''. A partial converse is provided by [[Frostman's lemma]].{{citation needed|date=March 2015}}<ref>This Wikipedia article also discusses further useful characterizations of the Hausdorff dimension.{{clarify|date=March 2015}}</ref>
If there is a measure μ defined on Borel subsets of a metric space X such that μ(X) > 0 and μ(B(x, r)) ≤ r<sup>s</sup> holds for some constant s > 0 and for every ball B(x, r) in X, then dim<sub>Haus</sub>(X) ≥ s. A partial converse is provided by Frostman's lemma.
If there is a measure μ defined on Borel subsets of a metric space X such that μ(X) > 0 and μ(B(x, r)) ≤ r<sup>s</sup> holds for some constant s > 0 and for every ball B(x, r) in X, then dim<sub>Haus</sub>(X) ≥ s. A partial converse is provided by Frostman's lemma.
如果在度量空间 x 的 Borel 子集上定义一个测度，使得(x)0和(b (x，r))≤ rsup s / sup 对于某个常数 s0和 x 中的每个球 b (x，r)成立，则 dim sub Haus / sub (x)≥ s。 部分逆向由霜人引理提供。
如果在度量空间 x 的 Borel 子集上定义一个测度，使得(x)0和(b (x，r))≤ rsup s / sup 对于某个常数 s0和 x 中的每个球 b (x，r)成立，则 dim sub Haus / sub (x)≥ s。 部分逆向由弗洛斯曼引理提供。
If X and Y are non-empty metric spaces, then the Hausdorff dimension of their product satisfies
If X and Y are non-empty metric spaces, then the Hausdorff dimension of their product satisfies
如果 x 和 y 是非空度量空间，那么它们产品的豪斯多夫维数满足
如果 x 和 y 是非空度量空间，那么它们乘积的豪斯多夫维数满足
This inequality can be strict. It is possible to find two sets of dimension 0 whose product has dimension 1. In the opposite direction, it is known that when X and Y are Borel subsets of R<sup>n</sup>, the Hausdorff dimension of X × Y is bounded from above by the Hausdorff dimension of X plus the upper packing dimension of Y. These facts are discussed in Mattila (1995).
This inequality can be strict. It is possible to find two sets of dimension 0 whose product has dimension 1. In the opposite direction, it is known that when X and Y are Borel subsets of R<sup>n</sup>, the Hausdorff dimension of X × Y is bounded from above by the Hausdorff dimension of X plus the upper packing dimension of Y. These facts are discussed in Mattila (1995).
这种不平等可以是绝对的。有可能找到两个维数为0的集合，其乘积的维数为1。在相反的方向上，我们知道当 x 和 y 是 r sup n / sup 的 Borel 子集时，x y 的豪斯多夫维数从上面以 x 的豪斯多夫维数加上 y 的填充维数为界。Mattila (1995)曾就这些情况进行了讨论。
==Self-similar sets==
==Self-similar sets==