# Borel groupoid

## 0.1 Definitions

• ###### 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.

• 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 2013-03-22 18:23:30 Last modified on 2013-03-22 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