Borel space
Definition 0.1.
A Borel space^{} $(X;\mathcal{B}(X))$ is defined as a set $X$, together with a Borel $\sigma $-algebra (http://planetmath.org/SigmaAlgebra) $\mathcal{B}(X)$ of subsets of $X$, called Borel sets. The Borel algebra on $X$ is the smallest $\sigma $-algebra containing all open sets (or, equivalently, all closed sets^{} if the topology^{} on closed sets is selected).
Remark 0.1.
Borel sets were named after the French mathematician Emile Borel.
Remark 0.2.
A subspace^{} of a Borel space $(X;\mathcal{B}(X))$ is a subset $S\subset X$ endowed with the relative Borel structure, that is the $\sigma $-algebra of all subsets of $S$ of the form $S\bigcap E$, where $E$ is a Borel subset of $X$.
Definition 0.2.
A rigid Borel space $({X}_{r};\mathcal{B}({X}_{r}))$ is defined as a Borel space whose only automorphism^{} $f:{X}_{r}\to {X}_{r}$ (that is, with $f$ being a bijection^{}, and also with $f(A)={f}^{-1}(A)$ for any $A\in \mathcal{B}({X}_{r})$) is the identity function ${1}_{({X}_{r};\mathcal{B}({X}_{r}))}$ (ref.[2]).
Remark 0.3.
R. M. Shortt and J. Van Mill provided the first construction of a rigid Borel space on a ‘set of large cardinality’.
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 B. Aniszczyk. 1991. A rigid Borel space., Proceed. AMS., 113 (4):1013-1015., http://www.jstor.org/pss/2048777available online.
- 3 A. Connes.1979. Sur la théorie noncommutative de l’ integration, Lecture Notes in Math., Springer-Verlag, Berlin, 725: 19-14.
Title | Borel space |
Canonical name | BorelSpace |
Date of creation | 2013-03-22 18:23:02 |
Last modified on | 2013-03-22 18:23:02 |
Owner | bci1 (20947) |
Last modified by | bci1 (20947) |
Numerical id | 22 |
Author | bci1 (20947) |
Entry type | Definition |
Classification | msc 60A10 |
Classification | msc 28C15 |
Classification | msc 28A12 |
Classification | msc 54H05 |
Classification | msc 28A05 |
Synonym | measurable space^{} |
Related topic | BorelSet |
Related topic | SigmaAlgebra |
Related topic | MeasurableSpace |
Related topic | BorelMeasure |
Related topic | BorelGroupoid |
Related topic | BorelMorphism |
Defines | rigid Borel space |
Defines | Borel subset space |