View source for Borel sigma-algebra of the real line

{{DISPLAYTITLE:Borel {{sigma|algebra}} of the real line}}
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==Definition==
Let {{M|(\mathbb{R},\mathcal{O})}}<ref group="Note">Traditionally we use {{M|\mathcal{J} }} for the [[topology]] part of a [[topological space]], however later in the article we will introduce {{M|\mathscr{J} }} in several forms, so we avoid {{M|\mathcal{J} }} to avoid confusion.</ref> denote [[the real line]] considered as a [[topological space]]. Recall that the [[Borel sigma-algebra|Borel {{sigma|algebra}}]] is defined to be the [[sigma-algebra generated by|{{sigma|algebra}} generated by]] the [[open sets]] of the [[topology]], recall that {{M|\mathcal{J} }} is the collection of all open sets of the space. Thus:
* {{M|\mathcal{B}(\mathbb{R}):\eq\sigma(\mathcal{O})}}
** where {{M|\sigma(\mathcal{G})}} denotes the [[sigma-algebra generated by|{{sigma|algebra}} generated by]] {{M|\mathcal{G} }}, a collection of sets. 
This is often written just as {{M|\mathcal{B} }}, provided this doesn't lead to ambiguities - this is inline with: {{M|\mathcal{B}^n}}, which we use for the [[Borel sigma-algebra on R^n|Borel {{sigma|algebra}} on {{M|\mathbb{R}^n}}]]
==Other generators==
Let {{M|\mathbb{M} }} denote either [[the real numbers]], {{M|\mathbb{R} }}, or [[the quotient numbers]], {{M|\mathbb{Q} }} (to save us writing the same thing for both {{M|\mathbb{R} }} and {{M|\mathbb{Q} }}, then the following all generate<ref group="Note">This means that if {{M|A}} is any of the families of sets from the list, then:
* {{M|\mathcal{B}(\mathbb{R})\eq\sigma(A)}}.</ref> {{M|\mathcal{B}(\mathbb{R})}}:
# {{M|\{(-\infty,a)\ \vert\ a\in\mathbb{M}\} }}{{rMIAMRLS}}
# {{M|\{(-\infty,b]\ \vert\ b\in\mathbb{M}\} }}{{rMIAMRLS}}
# {{M|\{(c,+\infty)\ \vert\ c\in\mathbb{M}\} }}{{rMIAMRLS}}
# {{M|\{[d,+\infty)\ \vert\ d\in\mathbb{M}\} }}{{rMIAMRLS}}
# {{M|\{(a,b)\ \vert\ a,b\in\mathbb{M}\} }}{{rMIAMRLS}}
# {{M|\{[c,d)\ \vert\ c,d\in\mathbb{M}\} }}{{rMIAMRLS}}
# {{M|\{(p,q]\ \vert\ p,q\in\mathbb{M}\} }}<sup>Suspected:</sup><ref group="Note">I have proved form {{M|6}} before, the order didn't matter there</ref><sup> - almost certain</sup>
# {{Warning|May not be true: }} {{M|\{[u,v]\ \vert\ u,v\in\mathbb{M}\} }}<sup>Suspected:</sup><ref group="Note">I suspect this holds as the open balls basically are open intervals, sort of... anyway "it works" for the [[open balls]], and the [[closed sets]] of {{M|\mathbb{R} }} also generate {{M|\mathcal{B} }} (see: form {{M|9}}) so it might work</ref><sup> - induced from pattern, unsure</sup>
# {{M|\mathcal{C} }}{{rMIAMRLS}} - the [[closed sets]] of {{M|\mathbb{R} }}
# {{M|\mathcal{K} }}{{rMIAMRLS}} - the {{link|compact|topology}} sets of {{M|\mathbb{R} }}
===Proofs===
* '''1, 2, 3 and 4: ''' - ''[[the collection of all open and closed rays based at either rational or real points generate the Borel sigma-algebra on R]]''
* '''5: ''' - ''[[the open rectangles with either rational or real points generate the same sigma-algebra as the Borel sigma-algebra on R^n]]''
* '''6: ''' - ''[[the closed-open rectangles with either rational or real points generate the same sigma-algebra as the Borel sigma-algebra on R^n]]''
* '''7: ''' - {{Warning|Suspected from proof on paper of {{M|6}}}}
* '''8: ''' - {{Warning|'''May not be true!'''}} note to self: the open balls are a basis (even at rational points with rational radiuses - countable basis) of {{M|\mathbb{R} }}, is there like a generator for closed sets?
* '''9: ''' - ''[[the sigma-algebra generated by the closed sets of R^n is the same as the Borel sigma-algebra of R^n]]''
* '''10: ''' - ''[[the sigma-algebra generated by the compact sets of R^n is the same as the Borel sigma-algebra of R^n]]''
{{Requires work|grade=A*|msg=
* Tidy up the proofs section, work on resolving 7 and especially 8, also
* maybe write {{M|\mathcal{C}:\eq\{A\in\mathcal{P}(\mathbb{R})\ \vert\ A\text{ is closed }\} }} or something to give it a more clear definition for 9 and 10 [[User:Alec|Alec]] ([[User talk:Alec|talk]]) 22:15, 26 February 2017 (UTC)}}
==See next==
==See also==
==Notes==
<references group="Note"/>
==References==
<references/>
{{Definition|Measure Theory|Analysis|Functional Analysis}}