View source for Borel sigma-algebra

'''Note: '''
* Not to be confused with the [[Borel sigma-algebra generated by]] which, for a given [[Topological space|topology]] {{M|(X,\mathcal{O})}} is denoted {{M|1=\mathcal{B}(X,\mathcal{J}):=\sigma(\mathcal{O})}} or just {{M|\mathcal{B}(X)}} if the topology is implicit.
* ''(This page)'' The '''Borel {{sigma|algebra}}''' refers to {{M|\mathcal{B}(\mathbb{R})}}, with it's usual topology (the [[Topology induced by a metric|topology induced by]] the [[Absolute value|absolute value metric]], {{M|\vert\cdot\vert}}).
** Because it is so common we simply denote it {{M|\mathcal{B} }}
** Again, because it is so common, rather than saying a map is {{M|\mathcal{A}/\mathcal{B} }}-measurable, we may just say it is {{M|\mathcal{A} }}-measurable
__TOC__
==Definition==
The ''Borel {{sigma|algebra}}'' is a<ref group="Note">There are of course others, for example {{M|\mathcal{P}(\mathbb{R})}} is always a [[Sigma-algebra|{{sigma|algebra}}]] but is much ''larger'' than the Borel one</ref> [[Sigma-algebra|{{sigma|algebra}}]] on {{M|\mathbb{R} }}<ref name="MIM">Measures, Integrals and Martingales - Rene L. Schilling</ref>. It is [[Sigma-algebra generated by|generated by]] the [[Open set|open sets]] of the [[Metric space|metric space]] {{M|(\mathbb{R},\vert\cdot\vert)}}. We denote it as:
* {{M|1=\mathcal{B}:=\sigma(\mathcal{O})}} where {{M|\mathcal{O} }} denotes the open sets of {{M|(\mathbb{R},\vert\cdot\vert)}}<ref group="Note">Conventionally, {{M|\mathcal{J} }} denotes the open sets, but in measure theory this seems to denote the sets of half-open-half-closed rectangles, and it is too common to ignore</ref>
This is actually a special case of the [[Borel sigma-algebra generated by|Borel {{sigma|algebra}} generated by]], rather than writing {{M|\mathcal{B}(\mathbb{R})}} we simply write {{M|\mathcal{B} }}

The ''Borel {{Sigma|algebra}}'' can also be defined on {{M|\mathbb{R}^n}}, that is done as follows:
* {{M|1=\mathcal{B}^n:=\mathcal{B}(\mathbb{R}^n)}}<ref name="MIM"/> with the usual topology on {{M|\mathbb{R}^n}} (the metric given by the [[Euclidean norm]] will do)
Again, this is a special case of the ''Borel {{sigma|algebra}} generated by'' a topology; this time it is the metric space {{M|(\mathbb{R}^n,\vert\cdot\vert)}}.
==Generators==
There are many generators of {{M|\mathcal{B}^n}} (just use {{M|1=n=1}} for {{M|\mathcal{B} }} itself) - some are listed here. First here are some non-obvious definitions:
* {{MM|1=[ [a,b))\subset\mathbb{R}^n}} means {{M|a}} and {{M|b}} are {{M|n}}-[[Tuple|tuples]] that denote the ''half-open-half-closed'' rectangles:
** {{MM|1=[ [a,b)):=[a_1,b_1)\times[a_2,b_2)\times\cdots\times[a_n,b_n)\subset\mathbb{R}^n}} with the convention of:
*** {{MM|1=[a_i,b_i)=\emptyset}} if {{M|b_i\le a_i}} and of course
*** {{MM|1=[ [a,b))=\emptyset}} if any of the {{M|1=[a_i,b_i)=\emptyset}} - this is trivial to show.
** The notation of {{MM|1=((a,b)):=(a_1,b_1)\times(a_2,b_2)\times\cdots\times(a_n,b_n)}} is similarly defined.
* {{MM|1=\mathscr{J}_\text{rat}^\circ:=\left\{((a,b))\vert\ a,b\in\mathbb{Q}^n\right\} }}
* {{MM|1=\mathscr{J}_\text{rat}:=\left\{[ [a,b))\vert\ a,b\in\mathbb{Q}^n\right\} }}
* {{MM|1=\mathscr{J}^\circ:=\left\{((a,b))\vert\ a,b\in\mathbb{R}^n\right\} }}
* {{MM|1=\mathscr{J}:=\left\{[ [a,b))\vert\ a,b\in\mathbb{R}^n\right\} }}
{| class="wikitable" border="1"
|-
! Claim
! Proof route
! Comment
|-
! {{M|1=\mathcal{B}^n:=\sigma(\mathcal{O})}} - open<ref name="MIM"/>
| Trivial (by definition)
|
|-
! {{M|1=\mathcal{B}^n=\sigma(\mathcal{C})}} - closed<ref name="MIM"/>
| Showing {{M|1=\sigma(\mathcal{C})\subseteq\sigma(\mathcal{O})}} and {{M|1=\sigma(\mathcal{O})\subseteq\sigma(\mathcal{C})}} - see '''Claim 1'''
| This is true for any [[Borel sigma-algebra generated by|Borel {{sigma|algebra}} generated by a topology]]
|-
! {{M|1=\mathcal{B}^n=\sigma(\mathcal{K})}} - compact<ref name="MIM"/>
| {{Todo|There's quite a few steps and theorems required (eg: compact set in Hausdorff space is closed)}}
| Link with generated borel sigma algebra - which requires a Hausdorff metric space I believe
|-
! {{M|1=\mathcal{B}^n=\sigma(\mathcal{O})=\sigma(\mathscr{J}_\text{rat}^\circ)=\sigma(\mathscr{J}^\circ)}}
| '''Claim 2'''
|
|-
!
| {{Todo|Check this and the method from the book - page 67 of my notes}}
|}
Also generated by:
{| class="wikitable" border="1"
|-
| {{M|1=\{(-\infty,a)\vert\ a\in\mathbb{Q}\} }}
| {{M|1=\{(-\infty,a)\vert\ a\in\mathbb{R}\} }}
|-
| {{M|1=\{(-\infty,b]\vert\ b\in\mathbb{Q}\} }}
| {{M|1=\{(-\infty,b]\vert\ b\in\mathbb{R}\} }}
|-
| {{M|1=\{(c,+\infty)\vert\ c\in\mathbb{Q}\} }}
| {{M|1=\{(c,+\infty)\vert\ c\in\mathbb{R}\} }}
|-
| {{M|1=\{[d,+\infty)\vert\ c\in\mathbb{Q}\} }}
| {{M|1=\{[d,+\infty)\vert\ c\in\mathbb{R}\} }}
|}
{{Todo|Integrate these, also find proofs, they're just a remark in<ref name="MIM"/>}}
===Proof of claims===
{{:Borel sigma-algebra generated by/Claim 1|1}}
{{Begin Theorem}}
Claim 2: Todo - even write this
{{Begin Proof}}
{{Todo|This}}
{{End Proof}}{{End Theorem}}
==Notes==
<references group="Note"/>
==References==
<references/>
{{Definition|Measure Theory}}
{{Theorem Of|Measure Theory}}
[[Category:Shorthands]]