Difference between revisions of "Dynkin system/Definition 1"
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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'' | + | <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} }} | ||
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}
