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 $𝒮$, 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) [[../BorelSpace/|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 [[../BorelGroupoid/|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. [[../TrivialGroupoid/|homomorphisms]], embeddings or [[../IsomorphicObjectsUnderAnIsomorphism/|isomorphisms]] between standard Borel [[../TopologicalGSpace/|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.