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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'''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 | + | 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}}]] | + | * 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== | ||
| − | {{:A | + | ==={{M|\sigma}}-algebra {{M|\implies}} both {{M|p}}-system and {{M|d}}-system=== |
| + | It needs to be shown that: | ||
| + | * [[A sigma-algebra is itself a Dynkin system]] | ||
| + | Then it is EVEN more trivial that a sigma-algebra is {{M|\cap}}-closed | ||
| + | ==={{M|p}}-system and {{M|d}}-system {{M|\implies}} a {{m|\sigma}}-algebra=== | ||
| + | <gallery> | ||
| + | File:Panddsystemissigmaalgebra.JPG|Proof done on paper for like the third time | ||
| + | </gallery> | ||
| + | |||
{{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:
- A collection of subsets of X, A, is a σ-algebra if and only if[1][2] it is a d-system (another name for a Dynkin system) and ∩-closed (which is sometimes called a p-system[2]).
Dynkin himself used the p-system/d-system terminology[2] using it we get the much more concise statement below:
Contents
[<hidetoc>]Statement
- A collection of subsets of a set X, say A, is a σ-algebra if and only if is is both a p-system and a d-system[2].
Proof
σ-algebra ⟹ both p-system and d-system
It needs to be shown that:
Then it is EVEN more trivial that a sigma-algebra is ∩-closed
p-system and d-system ⟹ a σ-algebra
Proof done on paper for like the third time
TODO: Page 33 in[1] and like page 3 in[2]
References
- ↑ <cite_references_link_many_accessibility_label> 1.0 1.1 Measures, Integrals and Martingales
- ↑ <cite_references_link_many_accessibility_label> 2.0 2.1 2.2 2.3 2.4 Probability and Stochastics - Erhan Cinlar
