Definition
Let be a metric space. For any subset, let denote its diameter, that is
Let be any subset of , and a real number. Define
(The infimum is over all countable covers of by sets satisfying .)
Note that is monotone decreasing in δ since the larger δ is, the more collections of sets are permitted, making the infimum smaller. Thus, the limit exists but may be infinite. Let
It can be seen that 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 -dimensional Hausdorff measure of . Due to the metric outer measure property, all Borel subsets of are 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 may be different (Federer 1969, §2.10.2). If is a normed space the sets may be taken to be convex. However, the restriction of the covering families to balls gives a different measure.
Read more about this topic: Hausdorff Measure
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