Borel groupoid
0.1 Definitions

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

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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 (http://planetmath.org/MeasurableFunctions) (ref. [1]).
0.1.1 Analytic Borel space
${\mathbb{G}}_{B}$ becomes an analytic groupoid (http://planetmath.org/LocallyCompactGroupoids) if its Borel structure is analytic (http://planetmath.org/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., http://www.utgjiu.ro/math/mbuneci/preprint/p0024.pdfGroupoid C*Algebras^{}., Surveys in Mathematics and its Applications, Volume 1, p.75 .
Title  Borel groupoid 
Canonical name  BorelGroupoid 
Date of creation  20130322 18:23:30 
Last modified on  20130322 18:23:30 
Owner  bci1 (20947) 
Last modified by  bci1 (20947) 
Numerical id  39 
Author  bci1 (20947) 
Entry type  Definition 
Classification  msc 54H05 
Classification  msc 28A05 
Classification  msc 18B40 
Classification  msc 28A12 
Classification  msc 22A22 
Classification  msc 28C15 
Synonym  Borel space 
Synonym  measure groupoid 
Related topic  BorelSpace 
Related topic  BorelMeasure 
Related topic  MeasurableFunctions 
Related topic  Groupoid 
Related topic  Groupoids 
Related topic  GroupoidRepresentationsInducedByMeasure 
Related topic  LocallyCompactGroupoids 
Related topic  BorelGSpace 
Related topic  CategoryOfBorelGroupoids 
Defines  Borel function 
Defines  analytic groupoid 
Defines  set of composable pairs 
Defines  $Grp(2)$ 
Defines  analytic (Borel) groupoid 
Defines  analytic Borel space 
Defines  product structure 