View source for Sigma-algebra

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{{:Sigma-algebra/Infobox}}
: '''Note: ''' A ''Sigma-algebra'' of sets, or {{sigma|algebra}} is very similar to a [[Sigma-ring|{{sigma|ring}}]] of sets. 
:: A [[ring of sets]] is to an [[algebra of sets]] as a [[sigma-ring|{{sigma|ring}}]] is to a ''{{sigma|algebra}}''

__TOC__
==Definition==
{{:Sigma-algebra/Definition}}
===Note on Alternative Definitions=== 
{{Begin Notebox}}
Many books have slightly different definitions of a {{sigma|algebra}}, the definition above is actually equivalent to the longer definitions one might see around.<br/> The two properties above give rise to all the others. 
{{Begin Notebox Content}}
Usually on this project differing definitions are listed, but due to the trivial proofs that they share properties, this was omitted. 

To show this yourself, do the following:
# Show a {{sigma|algebra}} is closed under [[set-subtraction]], {{M|\forall A,B\in\mathcal{A}[A-B\in\mathcal{A}]}}
# Use this to show {{M|\emptyset\in\mathcal{A} }}, notice {{M|1=\emptyset=A-A\in\mathcal{A} }} for any {{M|A\in\mathcal{A} }}
# {{M|X\in\mathcal{A} }} as {{M|\emptyset^C\in\mathcal{A} }}
This resolves most ambiguities. 
{{End Notebox Content}}{{End Notebox}}
==Immediate consequences==
Among other things immediately we see that:
{{Begin Inline Theorem}}
* {{M|\mathcal{A} }} is closed under [[Set subtraction|set subtraction]]
{{Begin Inline Proof}}
:: As {{M|1=A-B=(A^c\cup B)^c}} and a {{sigma|algebra}} is closed under complements and unions, this shows it is closed under set subtraction too
{{Begin Notebox}}
One might argue that we only know {{M|\mathcal{A} }} is closed under ''countable'' union, not ''finite'' (2) union. so we cannot know {{M|A^C\cup B\in\mathcal{A} }}, thus this proof is invalid. It is not invalid.
{{Begin Notebox Content}}
The person who mentioned this argued that we ''require'' [[set-subtraction]] in order to show {{M|\emptyset\in\mathcal{A} }} - which we do not yet know, so we cannot construct the family: 
* {{M|1=\{A_n\}_{n=1}^\infty}} with {{M|1=A_1=A^C}}, {{M|1=A_2=B}}, {{M|1=A_i=\emptyset}} for {{M|i> 2}}
And they are not wrong, however:
* {{M|1=A^C\cup B=A^C\cup B\cup B\cup B\ldots}}
So
* {{M|1=\{A_n\}_{n=1}^\infty}} with {{M|1=A_1=A^C}} and {{M|1=A_i=B}} for {{M|i> 1}} ''is'' something we can construct.
{{End Notebox Content}}{{End Notebox}}
{{End Proof}}{{End Theorem}}
{{Begin Inline Theorem}}
* {{M|\mathcal{A} }} is {{M|\cap}}-closed (furthermore, that {{M|\mathcal{A} }} is in fact {{M|\sigma}}-{{M|\cap}}-closed - that is closed under countable intersections)
{{Begin Inline Proof}}
{{Requires proof|We should do this here}}
: See [[Class of sets closed under set-subtraction properties|Properties of a class of sets closed under set subtraction]]
{{End Proof}}{{End Theorem}}
{{Begin Inline Theorem}}
* {{M|\emptyset\in\mathcal{A} }}
{{Begin Inline Proof}}
:: {{M|\forall A\in\mathcal{A} }} we have {{M|A-A\in\mathcal{A} }} (by closure under set subtraction), as {{M|1=A-A=\emptyset}}, {{M|\emptyset\in\mathcal{A} }}
{{End Proof}}{{End Theorem}}
{{Begin Inline Theorem}}
* {{M|X\in\mathcal{A} }}<ref group="Note">''Measures, Integrals and Martingales'' puts this in the definition of {{sigma|algebras}}</ref>
{{Begin Inline Proof}}
:: As {{M|\emptyset\in\mathcal{A} }} and it is closed under complement we see that {{M|\emptyset^c\in\mathcal{A} }} (by closure under complement) and {{M|1=\emptyset^c=X}} - the claim follows.
{{End Proof}}{{End Theorem}}
{{Begin Inline Theorem}}
* {{M|\mathcal{A} }} is a {{sigma|algebra}} {{M|\implies}} {{M|\mathcal{A} }} is a [[Sigma-ring|{{Sigma|ring}}]]
{{Begin Inline Proof}}
:: To prove this we must check:
::# {{M|\mathcal{A} }} is closed under countable union
::#* True by definition of {{Sigma|algebra}}
::# {{M|\mathcal{A} }} is closed under [[Set subtraction|set subtraction]]
::#* We've already shown this, so this is true too. 
:: This completes the proof.
{{End Proof}}{{End Theorem}}
==Important theorems==
* [[The intersection of sigma-algebras is a sigma-algebra|The intersection of {{sigma|algebras}} is a {{sigma|algebra}}]]

==Common {{Sigma|algebras}}==
'''See also: [[Index of common sigma-algebras|Index of common {{Sigma|algebras}}]]'''
* [[Sigma-algebra generated by|{{Sigma|algebra}} generated by]]
* [[Trace sigma-algebra|Trace {{Sigma|algebra}}]]
* [[Pre-image sigma-algebra|Pre-image {{Sigma|algebra}}]]

==See also==
* [[Types of set algebras]]
* [[Sigma-algebra generated by|{{Sigma|algebra}} generated by]]
* [[Sigma-ring|{{Sigma|ring}}]]
* [[Class of sets closed under set-subtraction properties|Properties of a class of sets closed under set subtraction]]

==Notes==
<references group="Note"/>

==References==
<references/>
{{Measure theory navbox|plain}}
{{Definition|Measure Theory}}
{{Theorem Of|Measure Theory}}


=OLD PAGE=
A '''Sigma-algebra''' of sets, or {{sigma|algebra}} is very similar to a [[Sigma-ring|{{sigma|ring}}]] of sets. 

Like how [[Ring of sets|ring of sets]] and [[Algebra of sets|algebra of sets]] differ, the same applies to [[Sigma-ring|{{sigma|ring}}]] compared to {{sigma|algebra}}

==Definition==
A non empty class of sets {{M|S}} is a {{sigma|algebra}}<ref group="Note">Some books (notably ''Measures, Integrals and Martingales'') give {{M|X\in\mathcal{A} }} as a defining property of {{sigma|algebras}}, however the two listed are sufficient to show this (see the ''immediate consequences'' section)</ref> if<ref>Halmos - Measure Theory - page 28 - Springer - Graduate Texts in Mathematics - 18</ref><ref name="MIAM">Measures, Integrals and Martingales - Rene L. Schilling</ref>
* if <math>A\in S</math> then <math>A^c\in S</math>
* if <math>\{A_n\}_{n=1}^\infty\subset S</math> then <math>\cup^\infty_{n=1}A_n\in S</math>
That is it is closed under [[Complement|complement]] and [[Countable|countable]] [[Union|union]].<br/>
===Immediate consequences===
Among other things immediately we see that:
{{Begin Inline Theorem}}
* {{M|\mathcal{A} }} is closed under [[Set subtraction|set subtraction]]
{{Begin Inline Proof}}
:: As {{M|1=A-B=(A^c\cup B)^c}} and a {{sigma|algebra}} is closed under complements and unions, this shows it is closed under set subtraction too
{{End Proof}}{{End Theorem}}
{{Begin Inline Theorem}}
* {{M|\mathcal{A} }} is {{M|\cap}}-closed (furthermore, that {{M|\mathcal{A} }} is in fact {{M|\sigma}}-{{M|\cap}}-closed - that is closed under countable intersections)
{{Begin Inline Proof}}
: See [[Class of sets closed under set-subtraction properties|Properties of a class of sets closed under set subtraction]]
{{End Proof}}{{End Theorem}}
{{Begin Inline Theorem}}
* {{M|\emptyset\in\mathcal{A} }}
{{Begin Inline Proof}}
:: {{M|\forall A\in\mathcal{A} }} we have {{M|A-A\in\mathcal{A} }} (by closure under set subtraction), as {{M|1=A-A=\emptyset}}, {{M|\emptyset\in\mathcal{A} }}
{{End Proof}}{{End Theorem}}
{{Begin Inline Theorem}}
* {{M|X\in\mathcal{A} }}<ref group="Note">''Measures, Integrals and Martingales'' puts this in the definition of {{sigma|algebras}}</ref>
{{Begin Inline Proof}}
:: As {{M|\emptyset\in\mathcal{A} }} and it is closed under complement we see that {{M|\emptyset^c\in\mathcal{A} }} (by closure under complement) and {{M|1=\emptyset^c=X}} - the claim follows.
{{End Proof}}{{End Theorem}}
{{Begin Inline Theorem}}
* {{M|\mathcal{A} }} is a {{sigma|algebra}} {{M|\implies}} {{M|\mathcal{A} }} is a [[Sigma-ring|{{Sigma|ring}}]]
{{Begin Inline Proof}}
:: To prove this we must check:
::# {{M|\mathcal{A} }} is closed under countable union
::#* True by definition of {{Sigma|algebra}}
::# {{M|\mathcal{A} }} is closed under [[Set subtraction|set subtraction]]
::#* We've already shown this, so this is true too. 
:: This completes the proof.
{{End Proof}}{{End Theorem}}

==Important theorems==
{{Begin Theorem}}
The intersection of {{Sigma|algebras}} is a {{Sigma|algebra}}
{{Begin Proof}}
{{Todo|Proof - see PTACC page 5, also in Halmos AND in that other book}}
{{End Proof}}{{End Theorem}}

==Common {{Sigma|algebras}}==
'''See also: [[Index of common sigma-algebras|Index of common {{Sigma|algebras}}]]'''
* [[Sigma-algebra generated by|{{Sigma|algebra}} generated by]]
* [[Trace sigma-algebra|Trace {{Sigma|algebra}}]]
* [[Pre-image sigma-algebra|Pre-image {{Sigma|algebra}}]]

==See also==
* [[Types of set algebras]]
* [[Sigma-algebra generated by|{{Sigma|algebra}} generated by]]
* [[Sigma-ring|{{Sigma|ring}}]]
* [[Class of sets closed under set-subtraction properties|Properties of a class of sets closed under set subtraction]]

==Notes==
<references group="Note"/>

==References==
<references/>

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