PlanetPhysics/Borel Groupoid

DefinitionsEdit

  • Borel function A function ) of Borel spaces is defined to be a Borel function if the inverse image of every Borel set under 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 from Failed to parse (unknown function "\grp"): {\displaystyle \grp_B^{(2)}} to Failed to parse (unknown function "\grp"): {\displaystyle \grp_B} , and 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. [1]).

Analytic Borel spaceEdit

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

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

All SourcesEdit

[1]

ReferencesEdit

  1. 1.0 1.1 M.R. Buneci. 2006., Groupoid C*-Algebras., Surveys in Mathematics and its Applications , Volume 1, p.75 .