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starting to include Catheodory construction: It's *super* messy and all over the pace right now, but I'm just trying to get the required information in before I start formatting it, along with still figuring out whether the construction should become the primary focus. It's definitely a work in progress |
→Carathéodory Construction: ce, clarify notation |
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:<math>\mu (E) = \lim_{\delta \to 0} \mu_{\delta} (E),</math>
where
:<math>\mu_{\delta} (E) = \inf \left\{ \left. \sum_{i = 1}^{\infty} \tau (C_{i}) \right| C_{i} \in \Sigma, \mathrm{diam} (C_{i}) \leq \delta, \bigcup_{i = 1}^{\infty} C_{i} \supseteq E \right\},</math>
is not only an outer measure, but in fact a [[metric outer measure]] as well. (Some authors prefer to take a [[supremum]] over ''δ'' > 0 rather than a [[Limit of a function|limit]] as ''δ'' → 0; the two give the same result, since ''μ''<sub>''δ''</sub>(''E'') increases as ''δ'' decreases.)
The function and ___domain of ''τ'' may determine the specific measure obtained. For instance, if we give
:<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]].
This construction is how the Hausdorff and [[packing measure]]s are obtained.
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