Definitions

• a. Borel function
  Definition 0.1   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.
• b. Borel groupoid
  Definition 0.2   Let $\grp$ be a groupoid and $\grp^{(2)}$ a subset of $\grp \times \grp$ - the set of its composable pairs. A Borel groupoid is defined as a groupoid $\grp_B$ such that $\grp_B^{(2)}$ is a Borel set in the product structure on $\grp_B \times \grp_B$ , and also such that the functions $ (x,y) \mapsto xy$ from $\grp_B^{(2)}$ to $\grp_B$ , and $ x \mapsto x^{-1}$ from $\grp_B$ to $\grp_B$ are all (measurable) Borel functions (ref. [1]).

Analytic Borel space

$\grp_B$ 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.

Bibliography

1

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