Borel G-space | Physics Library

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 equipped with a $\sigma$ -algebra $\mathcal{S}$ , such that there exists a Polish topology on with its $\sigma$ -algebra of Borel sets.
b. Borel G-space.
Definition 0.2 Let be a Polish group and a (standard) Borel space. An action of on 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 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.

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