Difference between revisions of "Dynkin system generated by"

Revision as of 14:39, 27 July 2015

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

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

1. ↑ ^{1.0 1.1} Measures, Integrals and Martingales - Rene L. Schilling