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

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  • {\mathbf a.} Standard Borel space. A standard Borel space is defined as a measurable space , that is, a set   equipped with a   -algebra  , such that there exists a Polish topology on   with   its  -algebra of Borel sets.
  • {\mathbf b.} Borel G-space. Let   be a Polish group and   a (standard) Borel space. An action   of   on   is defined to be a Borel action if   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 .
  • {\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   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.