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→Carathéodory Construction: having the conditions in a big line hinders readability |
→Carathéodory Construction: It's not the spherical measure |
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:<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 is the [[spherical measure]].{{clarify|reason=The spherical measure is just the Borel measure on the sphere, and has nothing to do with what you are describing here. You are just saying that X can be covered by balls, and I think when you do this, you just get the ordinary Hausdorff measure, again. If covering by balls gives a different result than covering by other shapes, then how this works would have to be explained in detail. Anyway, just don't call it the spherical measure cause it's not that.}}
This construction is how the Hausdorff and [[packing measure]]s are obtained.
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