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:
$$
Definition 2 - Let $X$ be a topological space and $\mathcal{B}$ be its Borel $\sigma $-algebra. A Borel measure on $X$ is a measure $\mu $ on the measurable space $(X,\mathcal{B})$ such that $$ for all compact subsets $K\subset X$. (ref.[1]).
$$
Definition 3 - Let $X$ be a topological space and $\mathcal{B}$ be the $\sigma $-algebra generated by all compact sets of $X$. A Borel measure on $X$ is a measure $\mu $ on the measurable space $(X,\mathcal{B})$ such that $$ for all compact subsets $K\subset X$.
$$
Definition 4 - The restriction (http://planetmath.org/RestrictionOfAFunction) of the Lebesgue measure^{} to the Borel $\sigma $-algebra of ${\mathbb{R}}^{n}$ is also sometimes called “the” Borel measure of ${\mathbb{R}}^{n}$.
$$
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 |