Difference between revisions of "Dynkin system"
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| + | '''Note: '''a Dynkin system is also called a "{{M|d}}-system"<ref>Probability and Stochastics - Erhan Cinlar</ref> and the page [[d-system]] just redirects here. | ||
==Definition== | ==Definition== | ||
| − | {{Extra Maths}} | + | ===[[Dynkin system/Definition 1|First Definition]]=== |
| − | + | {{Extra Maths}}{{:Dynkin system/Definition 1}} | |
| − | + | ===[[Dynkin system/Definition 2|Second Definition]]=== | |
| − | + | {{:Dynkin system/Definition 2}} | |
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==Immediate results== | ==Immediate results== | ||
{{Begin Inline Theorem}} | {{Begin Inline Theorem}} | ||
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* [[Dynkin system generated by]] | * [[Dynkin system generated by]] | ||
* [[Types of set algebras]] | * [[Types of set algebras]] | ||
| + | * [[p-system|{{M|p}}-system]] | ||
| + | * [[Conditions for a Dynkin system to be a sigma-algebra|Conditions for a {{M|d}}-system to be a {{sigma|algebra}}]] | ||
| + | ==Notes== | ||
| + | <references group="Note"/> | ||
==References== | ==References== | ||
<references/> | <references/> | ||
{{Definition|Measure Theory}} | {{Definition|Measure Theory}} | ||
Revision as of 23:26, 2 August 2015
Note: a Dynkin system is also called a "[ilmath]d[/ilmath]-system"[1] and the page d-system just redirects here.
Contents
Definition
First Definition
[math]\newcommand{\bigudot}{ \mathchoice{\mathop{\bigcup\mkern-15mu\cdot\mkern8mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}} }[/math][math]\newcommand{\udot}{\cup\mkern-12.5mu\cdot\mkern6.25mu\!}[/math][math]\require{AMScd}\newcommand{\d}[1][]{\mathrm{d}^{#1} }[/math]Given a set [ilmath]X[/ilmath] and a family of subsets of [ilmath]X[/ilmath], which we shall denote [ilmath]\mathcal{D}\subseteq\mathcal{P}(X)[/ilmath] is a Dynkin system[2] if:
- [ilmath]X\in\mathcal{D} [/ilmath]
- For any [ilmath]D\in\mathcal{D} [/ilmath] we have [ilmath]D^c\in\mathcal{D} [/ilmath]
- For any [ilmath](D_n)_{n=1}^\infty\subseteq\mathcal{D}[/ilmath] is a sequence of pairwise disjoint sets we have [ilmath]\udot_{n=1}^\infty D_n\in\mathcal{D}[/ilmath]
Second Definition
Given a set [ilmath]X[/ilmath] and a family of subsets of [ilmath]X[/ilmath] we denote [ilmath]\mathcal{D}\subseteq\mathcal{P}(X)[/ilmath] is a Dynkin system[3] on [ilmath]X[/ilmath] if:
- [ilmath]X\in\mathcal{D} [/ilmath]
- [ilmath]\forall A,B\in\mathcal{D}[B\subseteq A\implies A-B\in\mathcal{D}][/ilmath]
- Given a sequence [ilmath](A_n)_{n=1}^\infty\subseteq\mathcal{D}[/ilmath] that is increasing[Note 1] and has [ilmath]\lim_{n\rightarrow\infty}(A_n)=A[/ilmath] we have [ilmath]A\in\mathcal{D}[/ilmath]
Immediate results
- [ilmath]\emptyset\in\mathcal{D} [/ilmath]
Proof:
- As [ilmath]\mathcal{D} [/ilmath] is closed under complements and [ilmath]X\in\mathcal{D} [/ilmath] by definition, [ilmath]X^c\in\mathcal{D} [/ilmath]
- [ilmath]X^c=\emptyset[/ilmath] so [ilmath]\emptyset\in\mathcal{D} [/ilmath]
This completes the proof.
See also
- Dynkin system generated by
- Types of set algebras
- [ilmath]p[/ilmath]-system
- Conditions for a [ilmath]d[/ilmath]-system to be a [ilmath]\sigma[/ilmath]-algebra
Notes
- ↑ Recall this means [ilmath]A_{n}\subseteq A_{n+1} [/ilmath]
References
- ↑ Probability and Stochastics - Erhan Cinlar
- ↑ Measures, Integrals and Martingales - René L. Schilling
- ↑ Probability and Stochastics - Erhan Cinlar
