Definition:Dynkin System Generated by Collection of Subsets

From ProofWiki
Jump to navigation Jump to search


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 $\mathcal G$, denoted $\delta \left({\mathcal G}\right)$, 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.


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

Also see

Source of Name

This entry was named for Eugene Borisovich Dynkin.