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Borel Gspace
A (standard) Borel Gspace is defined in connection with a standard Borel space which shall be specified first.
0.1 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 Gspace
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 Borelmeasurable 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 Gspace.

c. Borel morphisms
Definition 0.3.
Homomorphisms, embeddings or isomorphisms between standard Borel Gspaces are called Borel if they are Borel–measurable.
Mathematics Subject Classification
22A15 no label found22A25 no label found22A22 no label found54H05 no label found22A05 no label found22A10 no label found Forums
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