Borel space

Definition 0.1.

A Borel spacePlanetmathPlanetmath (X;(X)) is defined as a set X, together with a Borel σ-algebra ( (X) of subsets of X, called Borel sets. The Borel algebra on X is the smallest σ-algebra containing all open sets (or, equivalently, all closed setsPlanetmathPlanetmath if the topologyMathworldPlanetmath on closed sets is selected).

Remark 0.1.

Borel sets were named after the French mathematician Emile Borel.

Remark 0.2.

A subspaceMathworldPlanetmathPlanetmath of a Borel space (X;(X)) is a subset SX endowed with the relative Borel structure, that is the σ-algebra of all subsets of S of the form SE, where E is a Borel subset of X.

Definition 0.2.

A rigid Borel space (Xr;(Xr)) is defined as a Borel space whose only automorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmath f:XrXr (that is, with f being a bijectionMathworldPlanetmath, and also with f(A)=f-1(A) for any A(Xr)) is the identity function 1(Xr;(Xr)) (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’.


  • 1 M.R. Buneci. 2006., 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., 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 spaceMathworldPlanetmath
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