PlanetPhysics/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

 * {\mathbf a.} Standard Borel space. 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.
 * {\mathbf b.} Borel G-space. 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.
 * {\mathbf c.} Borel morphisms.  homomorphisms, embeddings or isomorphisms between standard Borel G-spaces are called Borel  if they are Borel--measurable.

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.