Content deleted Content added
→Carathéodory Construction: noly the hausdorff measure is obtained this way. |
The measure I was describing is indeed different to Hausdorff measure and apparently also to the "Spherical measure" page. It is, however, important as it is much easier to construct than the Hausdorff measure. |
||
Line 5:
==Carathéodory Construction==
Let ''τ'' : Σ → [0, +∞] be a set function defined on a class Σ of subsets of ''X'' containing the empty set ∅, such that ''τ''(∅) = 0. One can show that the set function ''μ'' defined by<ref name="Geometric Outer-Measures">http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.517.5903&rep=rep1&type=pdf</ref>
:<math>\mu (E) = \lim_{\delta \to 0} \mu_{\delta} (E),</math>
Line 16:
:<math>\tau(C) = \mathrm{diam} (C)^s,\,</math>
where ''s'' is a positive constant and where ''τ'' is defined on the [[power set]] of all subsets of ''X'' (i.e., <math>\Sigma = 2^X</math>), the associated measure ''μ'' is the ''s''-dimensional [[Hausdorff measure]]. More generally, one could use any so-called [[dimension function]]. If instead ''τ'' is defined only on [[ball (mathematics)| ball]]s of ''X'', the associated measure <math>S^s</math> is
<math>H^s(E) \le S^s(E) \le 2H^s(E)</math>.<ref name="Geometric Outer-Measures"/> {{clarify|reason=Covering by balls gives a different result than covering by general shapes; how this works needs to be explained in detail.}}
This construction is how the Hausdorff and [[packing measure]]s are obtained.{{clarify|reason=This statement is incorrect, the packing measure is NOT obtained in this way. In the above, you are covering the set to be measured, by contrast, the packing measure doe NOT cover, it jams balls into the inside, and only then uses the cartheodory extension theorem to get a measure. Don't confuse the covering used in the extension with the covering of the set itself...}}
| |||