Difference between revisions of "A collection of subsets is a sigma-algebra iff it is a Dynkin system and closed under finite intersections"

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(Created page with "'''Terminology note:''' *A collection of subsets of {{M|X}}, {{M|\mathcal{A} }}, is a algebra}} ''if and only if''<ref name="MIM">Measures, Integrals...")
 
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{{DISPLAYTITLE:A collection of subsets is a {{sigma|algebra}} {{M|\iff}} it is both a {{M|p}}-system and a {{M|d}}-system}}
 
'''Terminology note:'''
 
'''Terminology note:'''
 
*A collection of subsets of {{M|X}}, {{M|\mathcal{A} }}, is a [[sigma-algebra|{{Sigma|algebra}}]] ''if and only if''<ref name="MIM">Measures, Integrals and Martingales</ref><ref name="PAS">Probability and Stochastics - Erhan Cinlar</ref> it is a {{M|d}}-system (another name for a [[Dynkin system]]) and {{M|\cap}}-closed (which is sometimes called a [[p-system|{{M|p}}-system]]<ref name="PAS"/>).
 
*A collection of subsets of {{M|X}}, {{M|\mathcal{A} }}, is a [[sigma-algebra|{{Sigma|algebra}}]] ''if and only if''<ref name="MIM">Measures, Integrals and Martingales</ref><ref name="PAS">Probability and Stochastics - Erhan Cinlar</ref> it is a {{M|d}}-system (another name for a [[Dynkin system]]) and {{M|\cap}}-closed (which is sometimes called a [[p-system|{{M|p}}-system]]<ref name="PAS"/>).
Dynkin himself used the {{M|p}}-system/{{M|d}}-system terminology<ref name="PAS"/> using it we get the much more concise statement:
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Dynkin himself used the {{M|p}}-system/{{M|d}}-system terminology<ref name="PAS"/> using it we get the much more concise statement below:
 
__TOC__
 
__TOC__
 
==Statement==
 
==Statement==
* A collection of subsets of {{M|X}}, {{M|\mathcal{A} }} is a [[Sigma-algebra|{{sigma|algebra}}]] ''if and only if'' is is both a {{M|p}}-system and a {{m|d}}-system<ref name="PAS"/>.
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* A collection of subsets of a [[set]] {{M|X}}, say {{M|\mathcal{A} }}, is a [[Sigma-algebra|{{sigma|algebra}}]] {{iff}} is is both a [[p-system|{{M|p}}-system]] and a [[d-system|{{m|d}}-system]]<ref name="PAS"/>.
 
==Proof==
 
==Proof==
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==={{M|\sigma}}-algebra {{M|\implies}} both {{M|p}}-system and {{M|d}}-system===
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It needs to be shown that:
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* [[A sigma-algebra is itself a Dynkin system]]
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Then it is EVEN more trivial that a sigma-algebra is {{M|\cap}}-closed
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==={{M|p}}-system and {{M|d}}-system {{M|\implies}} a {{m|\sigma}}-algebra===
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<gallery>
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File:Panddsystemissigmaalgebra.JPG|Proof done on paper for like the third time
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</gallery>
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{{Todo|Page 33 in<ref name="MIM"/> and like page 3 in<ref name="PAS"/>}}
 
{{Todo|Page 33 in<ref name="MIM"/> and like page 3 in<ref name="PAS"/>}}
 
==References==
 
==References==
 
<references/>
 
<references/>
 
{{Theorem Of|Measure Theory}}
 
{{Theorem Of|Measure Theory}}

Latest revision as of 13:21, 17 December 2016

Terminology note:

Dynkin himself used the [ilmath]p[/ilmath]-system/[ilmath]d[/ilmath]-system terminology[2] using it we get the much more concise statement below:

Statement

Proof

[ilmath]\sigma[/ilmath]-algebra [ilmath]\implies[/ilmath] both [ilmath]p[/ilmath]-system and [ilmath]d[/ilmath]-system

It needs to be shown that:

Then it is EVEN more trivial that a sigma-algebra is [ilmath]\cap[/ilmath]-closed

[ilmath]p[/ilmath]-system and [ilmath]d[/ilmath]-system [ilmath]\implies[/ilmath] a [ilmath]\sigma[/ilmath]-algebra



TODO: Page 33 in[1] and like page 3 in[2]


References

  1. 1.0 1.1 Measures, Integrals and Martingales
  2. 2.0 2.1 2.2 2.3 2.4 Probability and Stochastics - Erhan Cinlar