Borel groupoid

0.1 Definitions

• a. Borel function

  Definition 0.1.

  A function $f_B:(X;ℬ)\to (X;𝒞$) 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 $𝔾$ be a groupoid and ${𝔾}^{(2)}$ a subset of $𝔾\times 𝔾$– the set of its composable pairs. A Borel groupoid is defined as a groupoid ${𝔾}_B$ such that ${𝔾}_B^{(2)}$ is a Borel set in the product structure on ${𝔾}_B\times {𝔾}_B$, and also with functions $(x,y)\mapsto xy$ from ${𝔾}_B^{(2)}$ to ${𝔾}_B$, and $x\mapsto x^{-1}$ from ${𝔾}_B$ to ${𝔾}_B$ defined such that they are all (measurable) Borel functions (http://planetmath.org/MeasurableFunctions) (ref. [1]).

0.1.1 Analytic Borel space

${𝔾}_B$ becomes an analytic groupoid (http://planetmath.org/LocallyCompactGroupoids) if its Borel structure is analytic (http://planetmath.org/Analytic).

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

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