Difference between revisions of "Dynkin system/Definition 1"

Revision as of 23:14, 2 August 2015 (view source) Alec (Talk | contribs) m (Alec moved page Dynkin system/Definiton 1 to Dynkin system/Definition 1 without leaving a redirect: Another typo) ← Older edit		Latest revision as of 01:52, 19 March 2016 (view source) Alec (Talk | contribs) m (Updated reference to use templates)
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−	<noinclude>{{Extra Maths}}</noinclude>Given a set {{M|X}} and a family of subsets of {{M|X}}, which we shall denote {{M|\mathcal{D}\subseteq\mathcal{P}(X)}} is a ''Dynkin system''<ref name="MIM">Rene L. Schilling - Measures, Integrals and Martingales</ref> if:	+	<noinclude>{{Extra Maths}}</noinclude>Given a set {{M|X}} and a family of subsets of {{M|X}}, which we shall denote {{M|\mathcal{D}\subseteq\mathcal{P}(X)}} is a ''Dynkin system''{{rMIAMRLS}} if:
	* {{M|X\in\mathcal{D} }}		* {{M|X\in\mathcal{D} }}
	* For any {{M|D\in\mathcal{D} }} we have {{M|D^c\in\mathcal{D} }}		* For any {{M|D\in\mathcal{D} }} we have {{M|D^c\in\mathcal{D} }}

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Latest revision as of 01:52, 19 March 2016

$$\newcommand{\bigudot}{ \mathchoice{\mathop{\bigcup\mkern-15mu\cdot\mkern8mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}} }$$ $$\newcommand{\udot}{\cup\mkern-12.5mu\cdot\mkern6.25mu\!}$$ $$\require{AMScd}\newcommand{\d}[1][]{\mathrm{d}^{#1} }$$ Given a set $X$ and a family of subsets of $X$, which we shall denote $\mathcal{D}\subseteq\mathcal{P}(X)$ is a Dynkin system^[1] if:

• $X\in\mathcal{D}$
• For any $D\in\mathcal{D}$ we have $D^c\in\mathcal{D}$
• For any $(D_n)_{n=1}^\infty\subseteq\mathcal{D}$ is a sequence of pairwise disjoint sets we have $\udot_{n=1}^\infty D_n\in\mathcal{D}$

References

1. Jump up ↑ Measures, Integrals and Martingales - René L. Schilling