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Borel measure
In mathematics, specifically in measure theory, a Borel measure on a topological space is a measure that is defined on all open sets (and thus on all Borel sets).〔D. H. Fremlin, 2000. ''(Measure Theory )''. Torres Fremlin.〕 Some authors require additional restrictions on the measure, as described below.
==Formal definition==
Let ''X'' be a locally compact Hausdorff space, and let be the smallest σ-algebra that contains the open sets of ''X''; this is known as the σ-algebra of Borel sets. A Borel measure is any measure ''μ'' defined on the σ-algebra of Borel sets. Some authors require in addition that ''μ''(''C'') < ∞ for every compact set ''C''. If a Borel measure ''μ'' is both inner regular and outer regular, it is called a regular Borel measure (some authors also require it to be tight). If ''μ'' is both inner regular and locally finite, it is called a Radon measure. Note that a locally finite Borel measure automatically satisfies ''μ''(''C'') < ∞ for every compact set ''C''.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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