Borel groupoid

0.1 Definitions

  • Definition 0.1.

    A function fB:(X;)(X;𝒞) of Borel spaces ( is defined to be a Borel function if the inverse imagePlanetmathPlanetmath of every Borel set under fB-1 is also a Borel set.

  • b. Borel groupoid

    Definition 0.2.

    Let 𝔾 be a groupoidPlanetmathPlanetmathPlanetmath and 𝔾(2) a subset of 𝔾×𝔾– the set of its composable pairs. A Borel groupoid is defined as a groupoid 𝔾B such that 𝔾B(2) is a Borel set in the product structure on 𝔾B×𝔾B, and also with functions (x,y)xy from 𝔾B(2) to 𝔾B, and xx-1 from 𝔾B to 𝔾B defined such that they are all (measurable) Borel functions ( (ref. [1]).

0.1.1 Analytic Borel space

𝔾B becomes an analytic groupoid ( if its Borel structure is analytic (

A Borel space (X;) is called analytic if it is countably separated, and also if it is the image of a Borel function from a standard Borel space.


  • 1 M.R. Buneci. 2006., C*-AlgebrasPlanetmathPlanetmath., 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