Dynkin system generated by

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• Add: the intersection of an arbitrary family of Dynkin systems is itself a Dynkin system to the mix

Definition

Given a set $X$ and another set $\mathcal{G}\subseteq\mathcal{P}(X)$ which we shall call the generator then we can define the Dynkin system generated by $\mathcal{G}$ as:

• The smallest Dynkin system that contains $\mathcal{G}$

And we denote this as: $\delta(\mathcal{G})$. This is to say that:

• $\delta(\mathcal{G})$ is the smallest Dynkin system such that $\mathcal{G}\subseteq\delta(\mathcal{G})$

(Claim 1) This is the same as:

• $$\delta(\mathcal{G}):=\bigcap_{\begin{array}{c}\mathcal{D}\text{ is a Dynkin system}\\ \text{and }\mathcal{G}\subseteq\mathcal{D}\end{array} }\mathcal{D}$$

Proof of claims

[Expand]

Claim 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 $\mathcal{G}$

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

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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