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In mathematics, a metric outer measure is an outer measure ''μ'' defined on the subsets of a given metric space (''X'', ''d'') such that : for every pair of positively separated subsets ''A'' and ''B'' of ''X''. ==Construction of metric outer measures== Let ''τ'' : Σ → () 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 : where : 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 as ''δ'' → 0; the two give the same result, since ''μ''''δ''(''E'') increases as ''δ'' decreases.) For the function ''τ'' one can use : where ''s'' is a positive constant; this ''τ'' is defined on the power set of all subsets of ''X''; the associated measure ''μ'' is the ''s''-dimensional Hausdorff measure. More generally, one could use any so-called dimension function. This construction is very important in fractal geometry, since this is how the Hausdorff and packing measures are obtained. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「metric outer measure」の詳細全文を読む スポンサード リンク
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