Borel G-space

A (standard) Borel G-space is defined in connection with a standard Borel space which shall bespecified 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 $X$ isdefined to be a Borel action if $a:G\times X\to X$ is a Borel-measurable map or aBorel function (http://planetmath.org/BorelGroupoid).In this case, a standard Borel space $X$ that is acted upon by a Polish group with a Borel actionis called a (standard) Borel G-space.

• c. Borel morphisms

  Definition 0.3.

  Homomorphisms, embeddings or isomorphisms between standard Borel G-spacesare 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 aninvariant Borel set is also a Borel G-space.