Editing Borel measure (section)

==Formal definition==
Let <math>X</math> be a [[locally compact]] [[Hausdorff space]], and let <math>\mathfrak{B}(X)</math> be the [[Sigma-algebra#Generated .CF.83-algebra|smallest σ-algebra]] that contains the [[open set]]s of <math>X</math>; this is known as the σ-algebra of [[Borel set]]s. A Borel measure is any measure <math>\mu</math> defined on the σ-algebra of Borel sets.<ref>{{cite book | author=Alan J. Weir | title=General integration and measure | publisher=[[Cambridge University Press]] | year=1974 | isbn=0-521-29715-X | pages=158–184 }}</ref> A few authors require in addition that <math>\mu</math> is [[Locally finite measure|locally finite]], meaning that every point has an open neighborhood with finite measure.  For Hausdorff spaces, this implies that <math>\mu(C)<\infty</math> for every [[compact set]] <math>C</math>; and for locally compact Hausdorff spaces, the two conditions are equivalent. If a Borel measure <math>\mu</math> is both [[inner regular]] and [[Regular_measure#Definition|outer regular]], it is called a [[Borel regular measure|regular Borel measure]]. If <math>\mu</math> is both inner regular, outer regular, and [[Locally finite measure|locally finite]], it is called a [[Radon measure]]. Alternatively, if a regular Borel measure <math>\mu</math> is [[Tightness_of_measures|tight]], it is a Radon measure.

If <math>X</math> is a [[Separable_space|separable]] [[complete metric space]], then every Borel measure <math>\mu</math> on <math>X</math> is a Radon measure.<ref>{{citation| last=Bogachev | first=Vladimir I.| authorlink=Vladimir I. Bogachev| title=Measure Theory | chapter=Measures on topological spaces | publisher=Springer Berlin Heidelberg | publication-place=Berlin, Heidelberg | date=2007 | isbn=978-3-540-34513-8 | doi=10.1007/978-3-540-34514-5_7|page=70}}
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