Dynkin system/Definition 2
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< Dynkin system
Revision as of 23:22, 2 August 2015 by Alec (Talk | contribs) (Created page with "<noinclude>{{Extra Maths}}</noinclude>Given a set {{M|X}} and a family of subsets of {{M|X}} we denote {{M|\mathcal{D}\subseteq\mathcal{P}(X)}} is a ''Dynkin system''<ref name...")
[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] we denote [ilmath]\mathcal{D}\subseteq\mathcal{P}(X)[/ilmath] is a Dynkin system[1] 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]
Notes
- ↑ Recall this means [ilmath]A_{n}\subseteq A_{n+1} [/ilmath]
References
- ↑ Probability and Stochastics - Erhan Cinlar
