PlanetPhysics/Borel Groupoid

Definitions

• Borel function A function $f_{B}:(X;{\mathcal {B}})\to (X;{\mathcal {C}}$) of Borel spaces is defined to be a Borel function if the inverse image of every Borel set under $f_{B}^{-1}$ is also a Borel set.
• Borel groupoid Let Failed to parse (unknown function "\grp"): {\displaystyle \grp} be a 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 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.