Borel measure: Difference between revisions

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In [[mathematics]], specifically in [[Measure (mathematics)|measure theory]], a Borel measure is defined as follows: let ''X'' 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 ''X''; this is known as the σ-algebra of [[Borel set]]s.  Any measure ''μ'' defined on the σ-algebra of Borel sets is called a '''Borel measure'''. Some authors require in addition that ''μ''(''C'')&nbsp;<&nbsp;∞ for every [[compact set]]&nbsp;''C''. If a Borel measure ''μ'' is both [[inner regular]] and [[outer regular]], it is called a [[Borel regular measure|'''regular Borel measure''']] (some authors also require it to be [[Tightness of measures|tight]]). If ''μ'' is both inner regular and [[Locally finite measure|locally finite]], it is called a [[Radon measure]]. Note that a locally finite Borel measure automatically satisfies ''μ''(''C'')&nbsp;<&nbsp;∞ for every compact set&nbsp;''C''.


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The [[real line]] <math>\mathbb R</math> with its [[Real line#As a topological space|usual topology]] is a locally compact Hausdorff space, hence we can define a Borel measure on it. In this case, <math>\mathfrak{B}(\mathbb R)</math> is the smallest σ-algebra that contains the open intervals of <math>\mathbb R</math>. While there are many Borel measures μ, the choice of Borel measure which assigns <math>\mu([a,b])=b-a</math> for every interval <math>[a,b]</math> is sometimes called "the" Borel measure on <math>\mathbb R</math>. In practice, even "the" Borel measure is not the most useful measure defined on the σ-algebra of Borel sets; indeed, the [[Lebesgue measure]] <math>\lambda</math> is an extension of "the" Borel measure which possesses the crucial property that it is a [[complete measure]] (unlike the Borel measure). To clarify, when one says that the Lebesgue measure <math>\lambda</math> is an extension of the Borel measure <math>\mu</math>, it means that every Borel-measurable set ''E'' is also a Lebesgue-measurable set, and the Borel measure and the Lebesgue measure coincide on the Borel sets (i.e., <math>\lambda(E)=\mu(E)</math> for every Borel measurable set).
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==References==
* {{cite book | author=J. D. Pryce | title=Basic methods of functional analysis | series=Hutchinson University Library | publisher=[[Hutchinson (publisher)|Hutchinson]] | year=1973 | isbn=0-09-113411-0 | page=217 }}
* {{cite book | last=Ransford | first=Thomas | title=Potential theory in the complex plane | series=London Mathematical Society Student Texts | volume=28 | location=Cambridge | publisher=[[Cambridge University Press]] | year=1995 | isbn=0-521-46654-7 | zbl=0828.31001 | pages=209–218}}
* {{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 }}


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In mathematics, specifically in measure theory, a Borel measure is defined as follows: let X be a locally compact Hausdorff space, and let B(X) be the smallest σ-algebra that contains the open sets of X; this is known as the σ-algebra of Borel sets. Any measure μ defined on the σ-algebra of Borel sets is called a Borel measure. 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.

On the real line

The real line with its usual topology is a locally compact Hausdorff space, hence we can define a Borel measure on it. In this case, B() is the smallest σ-algebra that contains the open intervals of . While there are many Borel measures μ, the choice of Borel measure which assigns μ([a,b])=ba for every interval [a,b] is sometimes called "the" Borel measure on . In practice, even "the" Borel measure is not the most useful measure defined on the σ-algebra of Borel sets; indeed, the Lebesgue measure λ is an extension of "the" Borel measure which possesses the crucial property that it is a complete measure (unlike the Borel measure). To clarify, when one says that the Lebesgue measure λ is an extension of the Borel measure μ, it means that every Borel-measurable set E is also a Lebesgue-measurable set, and the Borel measure and the Lebesgue measure coincide on the Borel sets (i.e., λ(E)=μ(E) for every Borel measurable set).

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