View source for Dynkin system generated by

{{Refactor notice|grade=A|msg=Working on it!
* Add: [[the intersection of an arbitrary family of Dynkin systems is itself a Dynkin system]] to the mix}}
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==Definition==
Given a set {{M|X}} and another set {{M|\mathcal{G}\subseteq\mathcal{P}(X)}} which we shall call the ''generator'' then we can define ''the [[Dynkin system]] generated by {{M|\mathcal{G} }}'' as<ref name="MIM">Measures, Integrals and Martingales - Rene L. Schilling</ref>:
* The smallest Dynkin system that contains {{M|\mathcal{G} }}
And we denote this as: {{M|\delta(\mathcal{G})}}. This is to say that:
* {{M|\delta(\mathcal{G})}} is the smallest Dynkin system such that {{M|\mathcal{G}\subseteq\delta(\mathcal{G})}}
(Claim 1) This is the same as:
* {{MM|1=\delta(\mathcal{G}):=\bigcap_{\begin{array}{c}\mathcal{D}\text{ is a Dynkin system}\\ \text{and }\mathcal{G}\subseteq\mathcal{D}\end{array} }\mathcal{D} }}
==Proof of claims==
{{Begin Theorem}}
Claim 1: {{MM|1=\delta(\mathcal{G}):=\bigcap_{\begin{array}{c}\mathcal{D}\text{ is a Dynkin system}\\ \text{and }\mathcal{G}\subseteq\mathcal{D}\end{array} }\mathcal{D} }} is the smallest Dynkin system containing {{M|\mathcal{G} }}
{{Begin Proof}}
{{Todo|Be bothered, notes on p75 of my notebook, or page 31 of<ref name="MIM"/>}}
{{End Proof}}{{End Theorem}}
==See also==
* [[Types of set algebras]]
* [[Sigma-algebra]]
* [[Conditions for a Dynkin system to be a sigma-algebra]]
* [[Conditions for a generated Dynkin system to be a sigma-algebra]]

==References==
<references/>

{{Definition|Measure Theory}}
{{Theorem Of|Measure Theory}}