# PlanetPhysics/Borel Groupoid

### Definitions

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

#### Analytic Borel space

$\displaystyle \grp_B$ becomes an analytic groupoid if its Borel structure is analytic.

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

[1]

## References

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