# 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 $\displaystyle \grp$ 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 $(x,y)\mapsto xy$ from $\displaystyle \grp_B^{(2)}$ to $\displaystyle \grp_B$ , and $x\mapsto x^{-1}$ from $\displaystyle \grp_B$ to $\displaystyle \grp_B$ are all (measurable) Borel functions (ref. ).

#### Analytic Borel space

$\displaystyle \grp_B$ 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.