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→Carathéodory Construction: ce, clarify notation |
→Carathéodory Construction: having the conditions in a big line hinders readability |
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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| \begin{matrix} C_{i} \in \Sigma, \\ \mathrm{diam} (C_{i}) \leq \delta, \\ \bigcup_{i = 1}^{\infty} C_{i} \supseteq E \end{matrix} \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.)
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