# 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 $\mathcal{S}$, 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.

• ###### 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 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