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 .
随便看	locally integrable function LocallyMaximal locally maximal LocallyNilpotentGroup locally nilpotent group LocallyRingedSpace locally ringed space LocallySimplyConnected locally simply connected LocallyTestable locally testable LocallyTrivialBundle locally trivial bundle locally 𝒫 LocalMartingale local martingale LocalMinimumOfConvexFunctionIsNecessarilyGlobal local minimum of convex function is necessarily global LocalNaganoTheorem local Nagano theorem LocalPropertiesOfProcesses local properties of processes LocalRing local ring Locus

Borel groupoid

0.1 Definitions

Definition 0.1.

Definition 0.2.

0.1.1 Analytic Borel space

References