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Borel groupoid
0.1 Definitions

a. Borel function
Definition 0.1.
A function $f_{B}:(X;\mathcal{B})\to(X;\mathcal{C}$) of Borel spaces is defined to be a Borel function if the inverse image of every Borel set under $f_{B}^{{1}}$ is also a Borel set.

b. Borel groupoid
Definition 0.2.
Let ${\mathbb{G}}$ be a groupoid and ${\mathbb{G}}^{{(2)}}$ a subset of ${\mathbb{G}}\times{\mathbb{G}}$– the set of its composable pairs. A Borel groupoid is defined as a groupoid ${\mathbb{G}}_{B}$ such that ${\mathbb{G}}_{B}^{{(2)}}$ is a Borel set in the product structure on ${\mathbb{G}}_{B}\times{\mathbb{G}}_{B}$, and also with functions $(x,y)\mapsto xy$ from ${\mathbb{G}}_{B}^{{(2)}}$ to ${\mathbb{G}}_{B}$, and $x\mapsto x^{{1}}$ from ${\mathbb{G}}_{B}$ to ${\mathbb{G}}_{B}$ defined such that they are all (measurable) Borel functions (ref. [1]).
0.1.1 Analytic Borel space
${\mathbb{G}}_{B}$ becomes an analytic groupoid if its Borel structure is analytic.
A Borel space $(X;\mathcal{B})$ is called analytic if it is countably separated, and also if it is the image of a Borel function from a standard Borel space.
References
 1 M.R. Buneci. 2006., Groupoid C*Algebras., Surveys in Mathematics and its Applications, Volume 1, p.75 .
Mathematics Subject Classification
54H05 no label found28A05 no label found18B40 no label found28A12 no label found22A22 no label found28C15 no label found Forums
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