Difference between revisions of "Dynkin system generated by"
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==Definition== | ==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>: | 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>: | ||
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Definition
Given a set [ilmath]X[/ilmath] and another set [ilmath]\mathcal{G}\subseteq\mathcal{P}(X)[/ilmath] which we shall call the generator then we can define the Dynkin system generated by [ilmath]\mathcal{G} [/ilmath] as[1]:
- The smallest Dynkin system that contains [ilmath]\mathcal{G} [/ilmath]
And we denote this as: [ilmath]\delta(\mathcal{G})[/ilmath]. This is to say that:
- [ilmath]\delta(\mathcal{G})[/ilmath] is the smallest Dynkin system such that [ilmath]\mathcal{G}\subseteq\delta(\mathcal{G})[/ilmath]
(Claim 1) This is the same as:
- [math]\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}[/math]
Proof of claims
Claim 1: [math]\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}[/math] is the smallest Dynkin system containing [ilmath]\mathcal{G} [/ilmath]
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
