Definition:Dynkin System Generated by Collection of Subsets

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Definition

Let $X$ be a set.

Let $\GG \subseteq \powerset X$ be a collection of subsets of $X$.


Then the Dynkin system generated by $\GG$, denoted $\map \delta \GG$, is the smallest Dynkin system on $X$ that contains $\GG$.

That is, $\map \delta \GG$ is subject to:

$(1):\quad \GG \subseteq \map \delta \GG$
$(2):\quad \GG \subseteq \DD \implies \map \delta \GG \subseteq \DD$ for any Dynkin system $\DD$ on $X$


In fact, $\map \delta \GG$ always exists, and is unique, as proved on Existence and Uniqueness of Dynkin System Generated by Collection of Subsets.


Generator

One says that $\GG$ is a generator for $\map \delta \GG$.


Also see


Source of Name

This entry was named for Eugene Borisovich Dynkin.


Sources