Borel measure


Definition 1 - Let X be a topological spaceMathworldPlanetmath and be its Borel σ-algebra (http://planetmath.org/BorelSigmaAlgebra). A Borel measure on X is a measureMathworldPlanetmath on the measurable spaceMathworldPlanetmathPlanetmath (X,).

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

Definition 2 - Let X be a topological space and be its Borel σ-algebra. A Borel measure on X is a measure μ on the measurable space (X,) such that μ(K)< for all compact subsets KX. (ref.[1]).

Definition 3 - Let X be a topological space and be the σ-algebra generated by all compact sets of X. A Borel measure on X is a measure μ on the measurable space (X,) such that μ(K)< for all compact subsets KX.

Definition 4 - The restriction (http://planetmath.org/RestrictionOfAFunction) of the Lebesgue measureMathworldPlanetmath to the Borel σ-algebra of n is also sometimes called “the” Borel measure of n.

Remark - Definitions 2 and 3 are technically different. For example, when constructing a Haar measure on a locally compact group one considers the σ-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.pdfGroupoidPlanetmathPlanetmathPlanetmath 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