Conditions for a generated Dynkin system to be a sigma-algebra

Statement

If [ilmath]\mathcal{G}\subseteq\mathcal{P}(X)[/ilmath] is [ilmath]\cap[/ilmath]-closed then [ilmath]\delta(\mathcal{G})=\sigma(\mathcal{G})[/ilmath] where:

• [ilmath]\delta(\mathcal{G})[/ilmath] denotes the Dynkin system generated by [ilmath]\mathcal{G} [/ilmath] and
• [ilmath]\sigma(\mathcal{G})[/ilmath] denotes the [ilmath]\sigma[/ilmath]-algebra generated by [ilmath]\mathcal{G} [/ilmath]

Proof

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TODO: See page 32 in for help if needed

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See also

• Conditions for a Dynkin system to be a sigma-algebra