Definition:Dynkin System Generated by Collection of Subsets

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

 * Sigma-Algebra Generated by Collection of Subsets
 * Monotone Class Generated by Collection of Subsets