PlanetPhysics/Borel Groupoid

Definitions

• [[../InvariantBorelSet/|Borel function]] A [[../Bijective/|function]] $f_B:(X;ℬ)\to (X;𝒞$) of [[../BorelSpace/|Borel spaces]] is defined to be a Borel function if the inverse image of every [[../InvariantBorelSet/|Borel set]] under $f_B^{-1}$ is also a Borel set.
• Borel groupoid Let Failed to parse (unknown function "\grp"): {\displaystyle \grp} be a [[../QuantumOperatorAlgebra5/|groupoid]] and Failed to parse (unknown function "\grp"): {\displaystyle \grp^{(2)}} a subset of Failed to parse (unknown function "\grp"): {\displaystyle \grp \times \grp} -- the set of its composable pairs. A Borel groupoid is defined as a groupoid Failed to parse (unknown function "\grp"): {\displaystyle \grp_B} such that Failed to parse (unknown function "\grp"): {\displaystyle \grp_B^{(2)}} is a Borel set in the product structure on Failed to parse (unknown function "\grp"): {\displaystyle \grp_B \times \grp_B} , and also such that the functions $(x,y)\mapsto xy$ from Failed to parse (unknown function "\grp"): {\displaystyle \grp_B^{(2)}} to Failed to parse (unknown function "\grp"): {\displaystyle \grp_B} , and $x\mapsto x^{-1}$ from Failed to parse (unknown function "\grp"): {\displaystyle \grp_B} to Failed to parse (unknown function "\grp"): {\displaystyle \grp_B} are all (measurable) Borel functions (ref. ).

Analytic Borel space

Failed to parse (unknown function "\grp"): {\displaystyle \grp_B} becomes an analytic groupoid if its Borel structure is analytic.

A [[../BorelSpace/|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 [[../InvariantBorelSet/|standard Borel space]].