Borel measure
Definition 1 - Let X be a topological space and ℬ be its Borel σ-algebra (http://planetmath.org/BorelSigmaAlgebra). A Borel measure on X is a measure
on the measurable space
(X,ℬ).
In the literature one can find other different definitions of Borel measure, like the following:
Definition 2 - Let X be a topological space and ℬ be its Borel σ-algebra. A Borel measure on X is a measure μ on the measurable space (X,ℬ) such that μ(K)<∞ for all compact subsets K⊂X. (ref.[1]).
Definition 3 - Let X be a topological space and ℬ 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,ℬ) such that μ(K)<∞ for all compact subsets K⊂X.
Definition 4 - The restriction (http://planetmath.org/RestrictionOfAFunction) of the Lebesgue measure to the Borel σ-algebra of ℝn is also sometimes called “the” Borel measure of ℝn.
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.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 |