Dynkin system generated by

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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]

TODO: Be bothered, notes on p75 of my notebook, or page 31 of[1]

See also


  1. 1.0 1.1 Measures, Integrals and Martingales - Rene L. Schilling