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

Source of Name

This entry was named for Eugene Borisovich Dynkin.

Sources

• 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $5.3$