Borel groupoid

0.1 Definitions

•
a. Borel function
Definition 0.1.
A function $f_{B}:(X;\\mathcal{B})\\to(X;\\mathcal{C}$ ) ofBorel spaces (http://planetmath.org/BorelSpace) 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 ${\\mathbb{G}}$ be a groupoid and ${\\mathbb{G}}^{(2)}$ a subset of ${\\mathbb{G}}\\times{\\mathbb{G}}$ – the set of its composable pairs.A Borel groupoid is defined as a groupoid ${\\mathbb{G}}_{B}$ such that ${\\mathbb{G}}_{B}^{(2)}$ is a Borel set in the product structure on ${\\mathbb{G}}_{B}\\times{\\mathbb{G}}_{B}$ , and also with functions $(x,y)\\mapsto xy$ from ${\\mathbb{G}}_{B}^{(2)}$ to ${\\mathbb{G}}_{B}$ , and $x\\mapsto x^{-1}$ from ${\\mathbb{G}}_{B}$ to ${\\mathbb{G}}_{B}$ defined such that they are all(measurable) Borel functions (http://planetmath.org/MeasurableFunctions) (ref. [1]).

0.1.1 Analytic Borel space

${\\mathbb{G}}_{B}$ becomes an analytic groupoid (http://planetmath.org/LocallyCompactGroupoids) if its Borel structure isanalytic (http://planetmath.org/Analytic).

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

References

1 M.R. Buneci. 2006.,http://www.utgjiu.ro/math/mbuneci/preprint/p0024.pdfGroupoid C*-Algebras.,Surveys in Mathematics and its Applications, Volume 1, p.75 .

单词	BorelGroupoid
释义	Borel groupoid 0.1 Definitions • a. Borel function Definition 0.1. A function $f_{B}:(X;\\mathcal{B})\\to(X;\\mathcal{C}$ ) ofBorel spaces (http://planetmath.org/BorelSpace) 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 ${\\mathbb{G}}$ be a groupoid and ${\\mathbb{G}}^{(2)}$ a subset of ${\\mathbb{G}}\\times{\\mathbb{G}}$ – the set of its composable pairs.A Borel groupoid is defined as a groupoid ${\\mathbb{G}}_{B}$ such that ${\\mathbb{G}}_{B}^{(2)}$ is a Borel set in the product structure on ${\\mathbb{G}}_{B}\\times{\\mathbb{G}}_{B}$ , and also with functions $(x,y)\\mapsto xy$ from ${\\mathbb{G}}_{B}^{(2)}$ to ${\\mathbb{G}}_{B}$ , and $x\\mapsto x^{-1}$ from ${\\mathbb{G}}_{B}$ to ${\\mathbb{G}}_{B}$ defined such that they are all(measurable) Borel functions (http://planetmath.org/MeasurableFunctions) (ref. [1]). 0.1.1 Analytic Borel space ${\\mathbb{G}}_{B}$ becomes an analytic groupoid (http://planetmath.org/LocallyCompactGroupoids) if its Borel structure isanalytic (http://planetmath.org/Analytic). A Borel space $(X;\\mathcal{B})$ is called analytic if it iscountably separated, and also if it is the image of a Borel function from a standardBorel space. References 1 M.R. Buneci. 2006.,http://www.utgjiu.ro/math/mbuneci/preprint/p0024.pdfGroupoid C*-Algebras.,Surveys in Mathematics and its Applications, Volume 1, p.75 .
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Borel groupoid

0.1 Definitions

Definition 0.1.

Definition 0.2.

0.1.1 Analytic Borel space

References