Borel groupoid

0.1 Definitions

• Borel function

  Definition 0.1. A function f_B : (X; ℬ) → (X; 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

  Definition 0.2. Let 𝔾 be a groupoid and 𝔾⁽²⁾ a subset of 𝔾 × 𝔾– the set of its composable pairs. A Borel groupoid is defined as a groupoid 𝔾_B such that 𝔾_B⁽²⁾ is a Borel set in the product structure on 𝔾_B × 𝔾_B, and also such that the functions (x,y)xy from 𝔾_B⁽²⁾ to 𝔾 _B, and xx⁻¹ from 𝔾 _B to 𝔾_B are all (measurable) Borel functions (ref. [1]).

0.1.1 Analytic Borel space

𝔾_B becomes an analytic groupoid if its Borel structure is analytic.

A Borel space (X; 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.

References