Difference between revisions of "Dynkin system/Definition 1"

Revision as of 23:10, 2 August 2015 (view source) Alec (Talk | contribs) (Created page with "<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 syst...")	Revision as of 23:12, 2 August 2015 (view source) Alec (Talk | contribs) m (Alec moved page Dynkin System/Definiton 1 to Dynkin system/Definiton 1 without leaving a redirect: Typo) Newer edit →
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Revision as of 23:12, 2 August 2015

$$\newcommand{\bigudot}{ \mathchoice{\mathop{\bigcup\mkern-15mu\cdot\mkern8mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}} }$$ $$\newcommand{\udot}{\cup\mkern-12.5mu\cdot\mkern6.25mu\!}$$ $$\require{AMScd}\newcommand{\d}[1][]{\mathrm{d}^{#1} }$$ 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 ↑ Rene L. Schilling - Measures, Integrals and Martingales