Borel G-space

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

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. 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.

Contributors to this entry (in most recent order):

• bci1

As of this snapshot date, this entry was owned by bci1.