Borel G-space

A (standard) Borel G-space is defined in connection with a standard Borel space which shall be specified first.

0.1 Basic definitions

  • a. Standard Borel space

    Definition 0.1.

    A standard Borel space is defined as a measurable spaceMathworldPlanetmathPlanetmath, that is, a set X equipped with a σ -algebraPlanetmathPlanetmath 𝒮, such that there exists a Polish topologyMathworldPlanetmath on X with S its σ-algebra of Borel sets.

  • b. Borel G-space

    Definition 0.2.

    Let G be a Polish group and X a (standard) Borel spacePlanetmathPlanetmath. An action a of G on X is defined to be a Borel action if a:G×XX is a Borel-measurable map or a Borel function ( In this case, a standard Borel space X that is acted upon by a Polish group with a Borel action is called a (standard) Borel G-space.

  • Definition 0.3.

    HomomorphismsMathworldPlanetmathPlanetmathPlanetmathPlanetmath, embeddingsMathworldPlanetmath or isomorphismsMathworldPlanetmathPlanetmath between standard Borel G-spaces are called Borel if they are Borel–measurable.

Remark 0.1.

Borel G-spaces have the nice property that the productPlanetmathPlanetmathPlanetmathPlanetmath and sum of a countableMathworldPlanetmath sequencePlanetmathPlanetmath of Borel G-spaces (Xn)nN are also Borel G-spaces. Furthermore, the subspaceMathworldPlanetmathPlanetmath of a Borel G-space determined by an invariantMathworldPlanetmath Borel set is also a Borel G-space.

Title Borel G-space
Canonical name BorelGspace
Date of creation 2013-03-22 18:24:45
Last modified on 2013-03-22 18:24:45
Owner bci1 (20947)
Last modified by bci1 (20947)
Numerical id 14
Author bci1 (20947)
Entry type Definition
Classification msc 22A15
Classification msc 22A25
Classification msc 22A22
Classification msc 54H05
Classification msc 22A05
Classification msc 22A10
Related topic BorelSpace
Related topic BorelMeasure
Related topic BorelGroupoid
Related topic CategoryOfBorelSpaces
Defines Borel action
Defines Borel-measurable map
Defines standard Borel space