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In mathematics, an outer measure ''μ'' on ''n''-dimensional Euclidean space R''n'' is called a Borel regular measure if the following two conditions hold: * Every Borel set ''B'' ⊆ R''n'' is ''μ''-measurable in the sense of Carathéodory's criterion: for every ''A'' ⊆ R''n'', :: * For every set ''A'' ⊆ R''n'' (which need not be ''μ''-measurable) there exists a Borel set ''B'' ⊆ R''n'' such that ''A'' ⊆ ''B'' and ''μ''(''A'') = ''μ''(''B''). An outer measure satisfying only the first of these two requirements is called a ''Borel measure'', while an outer measure satisfying only the second requirement is called a ''regular measure''. The Lebesgue outer measure on R''n'' is an example of a Borel regular measure. It can be proved that a Borel regular measure, although introduced here as an ''outer'' measure (only countably ''sub''additive), becomes a full measure (countably additive) if restricted to the Borel sets. ==References== * * * 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「Borel regular measure」の詳細全文を読む スポンサード リンク
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