Borel measure

Definition 1 - Let $X$ be a topological space and $\mathcal{B}$ be its Borel $\sigma$-algebra (http://planetmath.org/BorelSigmaAlgebra). A Borel measure on $X$ is a measure on the measurable space $(X,\mathcal{B})$.

In the literature one can find other different definitions of Borel measure, like the following:

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Definition 2 - Let $X$ be a topological space and $\mathcal{B}$ be its Borel $\sigma$-algebra. A Borel measure on $X$ is a measure $\mu$ on the measurable space $(X,\mathcal{B})$ such that $\mu(K)<\infty$ for all compact subsets $K\subset X$. (ref.[1]).

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Definition 3 - Let $X$ be a topological space and $\mathcal{B}$ be the $\sigma$-algebra generated by all compact sets of $X$. A Borel measure on $X$ is a measure $\mu$ on the measurable space $(X,\mathcal{B})$ such that $\mu(K)<\infty$ for all compact subsets $K\subset X$.

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Definition 4 - The restriction (http://planetmath.org/RestrictionOfAFunction) of the Lebesgue measure to the Borel $\sigma$-algebra of $\mathbb{R}^{n}$ is also sometimes called “the” Borel measure of $\mathbb{R}^{n}$.

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Remark - Definitions $2$ and $3$ are technically different. For example, when constructing a Haar measure on a locally compact group one considers the $\sigma$-algebra generated by all compact subsets, instead of all closed (or open) sets.

References

• 1 M.R. Buneci. 2006., http://www.utgjiu.ro/math/mbuneci/preprint/p0024.pdfGroupoid C*-Algebras., Surveys in Mathematics and its Applications, Volume 1: 71–98.
• 2 A. Connes.1979. Sur la théorie noncommutative de l’ integration, Lecture Notes in Math., Springer-Verlag, Berlin, 725: 19-14.
 Title Borel measure Canonical name BorelMeasure Date of creation 2013-03-22 17:34:00 Last modified on 2013-03-22 17:34:00 Owner asteroid (17536) Last modified by asteroid (17536) Numerical id 24 Author asteroid (17536) Entry type Definition Classification msc 60A10 Classification msc 28C15 Classification msc 28A12 Classification msc 28A10 Related topic BorelSigmaAlgebra Related topic RadonMeasure Related topic BorelSpace Related topic Measure Related topic MeasurableSpace Related topic BorelGroupoid Related topic BorelMorphism Related topic BorelGSpace