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 $$\grp$$ 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. ).

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.