Definition:Dynkin System Generated by Collection of Subsets

Definition
Let $X$ be a set.

Let $\mathcal G \subseteq \mathcal P \left({X}\right)$ be a collection of subsets of $X$.

Then the Dynkin system generated by $\mathcal G$, $\delta \left({\mathcal G}\right)$, is the smallest Dynkin system on $X$ that contains $\mathcal G$.

That is, $\delta \left({\mathcal G}\right)$ is subject to:


 * $(1):\quad \mathcal G \subseteq \delta \left({\mathcal G}\right)$
 * $(2):\quad \mathcal G \subseteq \mathcal D \implies \delta \left({\mathcal G}\right) \subseteq \mathcal D$ for any Dynkin system $\mathcal D$ on $X$

In fact, $\delta \left({\mathcal G}\right)$ always exists, and is unique, as proved on Existence and Uniqueness of Dynkin System Generated by Collection of Subsets.

Generator
One says that $\mathcal G$ is a generator for $\delta \left({\mathcal G}\right)$.