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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| + | {{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 | + | Dynkin himself used the {{M|p}}-system/{{M|d}}-system terminology<ref name="PAS"/> using it we get the much more concise statement. |
__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"/>. | * 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"/>. | ||
==Proof== | ==Proof== | ||
| + | {{:A collection of subsets is a sigma-algebra iff it is a Dynkin system and closed under finite intersections/Proof}} | ||
{{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}} | ||
Revision as of 15:52, 28 August 2015
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.
Contents
[hide]Statement
- A collection of subsets of X, A is a σ-algebra if and only if is is both a p-system and a d-system[2].
Proof
TODO: Page 33 in[1] and like page 3 in[2]
References
- ↑ Jump up to: 1.0 1.1 Measures, Integrals and Martingales
- ↑ Jump up to: 2.0 2.1 2.2 2.3 2.4 Probability and Stochastics - Erhan Cinlar
