View source for Borel sigma-algebra generated by

==Definition==
The ''Borel {{sigma|algebra}}'' is the [[Sigma-algebra generated by|{{sigma|algebra}} generated by]] the [[Open set|open sets]] of a [[Topological space|topological space]], that is<ref name="MIM">Measures, Integrals and Martingales - Rene L. Schilling</ref>: (where {{M|(X,\mathcal{O})}}<ref group="Note">Note the letter {{M|\mathcal{O} }} for the open sets of the topology, conventionally {{M|\mathcal{J} }} is used, however in measure theory this notation is often used to denote the set of half-open-half-closed rectangles in {{M|\mathbb{R}^n}} - a totally separate thing</ref> is any [[Topological space|topology]])
* {{M|1=\mathcal{B}(X,\mathcal{O}):=\sigma(\mathcal{O})}} - if the topology on {{M|X}} is obvious, we may simply write: {{M|\mathcal{B}(X)}}<ref name="MIM"/>
==Generators==
For a topological space {{M|(X,\mathcal{O})}} the following can be shown:
{| class="wikitable" border="1"
|-
! Claim
! Proof route
! Comment
|-
! {{M|1=\mathcal{B}(X):=\sigma(\mathcal{O})}}
| Trivial (by definition)
| 
|-
! {{M|1=\mathcal{B}(X)=\sigma(\mathcal{C})}} - the closed sets
| {{M|1=\mathcal{B}(X):=\sigma(\mathcal{O})=\sigma(\mathcal{C})}} - see '''claim 1''' below
|}
===Proof of claims===
{{:Borel sigma-algebra generated by/Claim 1}}

==See also==
* [[Borel sigma-algebra|Borel {{sigma|algebra}}]] - a special case, where {{M|1=\mathcal{B}:=\mathcal{B}(\mathbb{R},\vert\cdot\vert)}} and {{M|1=\mathcal{B}^n:=\mathcal{B}(\mathbb{R}^n,\vert\cdot\vert)}}
==Notes==
<references group="Note"/>
==References==
<references/>

{{Definition|Measure Theory}}
{{Theorem Of|Measure Theory}}