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 space, that is, a set $X$ equipped with a $\sigma$ -algebra $𝒮$, such that there exists a Polish topology on $X$ with $S$ its $\sigma$-algebra of Borel sets.

• b. Borel G-space

  Definition 0.2.

  Let $G$ be a Polish group and $X$ a (standard) Borel space. An action $a$ of $G$ on $X$ is defined to be a Borel action if $a:G\times X\to X$ is a Borel-measurable map or a Borel function (http://planetmath.org/BorelGroupoid). 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.

• c. Borel morphisms

  Definition 0.3.

  Homomorphisms, embeddings or isomorphisms 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 product and sum of a countable sequence of Borel G-spaces ${(X_n)}_{n\in N}$ are also Borel G-spaces. Furthermore, the subspace of a Borel G-space determined by an invariant 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