%%% This file is part of PlanetPhysics snapshot of 2011-09-01
%%% Primary Title: Borel G-space
%%% Primary Category Code: 02.
%%% Filename: BorelGSpace.tex
%%% Version: 5
%%% Owner: bci1
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\begin{document}
A (standard) Borel G-space is defined in connection with a standard Borel space which needs to be specified first.
\subsection{Basic definitions}
\begin{itemize}
\item {\bf a.} Standard Borel space.
\begin{definition}
A \emph{standard Borel space} is defined as a {\em 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.
\end{definition}
\item {\bf b.} Borel G-space.
\begin{definition}
Let $G$ be a Polish group and $X$ a (standard) \htmladdnormallink{Borel space}{http://planetphysics.us/encyclopedia/BorelSpace.html}. An action $a$ of $G$ on $X$ is
defined to be a \emph{Borel action} if $a: G \times X \to X$ is a Borel-measurable map or a
\htmladdnormallink{Borel function}{http://planetphysics.us/encyclopedia/BorelGroupoid.html}.
In this case, a standard Borel space $X$ that is acted upon by a Polish group with a Borel action
is called a \emph{(standard) Borel G-space}.
\end{definition}
\item {\bf c.} Borel morphisms.
\begin{definition}
\htmladdnormallink{homomorphisms}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}, embeddings or \htmladdnormallink{isomorphisms}{http://planetphysics.us/encyclopedia/IsomorphicObjectsUnderAnIsomorphism.html} between standard Borel \htmladdnormallink{G-spaces}{http://planetphysics.us/encyclopedia/TopologicalGSpace.html} are called \emph{Borel} if they are Borel--measurable.
\end{definition}
\end{itemize}
\begin{remark}
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
{\em invariant} Borel set is also a Borel G-space.
\end{remark}
\end{document}
