Existence and Uniqueness of Dynkin System Generated by Collection of Subsets

Theorem

Let $X$ be a set.

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

Then $\map \delta \GG$, the Dynkin system generated by $\GG$, exists and is unique.

Proof

Existence

By Power Set is Dynkin System, there exists at least one Dynkin system containing $\GG$.

Next, let $\Bbb D$ be the collection of Dynkin systems containing $\GG$:

$\Bbb D := \set {\DD': \GG \subseteq \DD', \text{$\DD'$ is a Dynkin system} }$

By Intersection of Dynkin Systems is Dynkin System, $\DD := \bigcap \Bbb D$ is a Dynkin system.

Also, by Set Intersection Preserves Subsets:

$\GG \subseteq \DD$

Now let $\DD'$ be a Dynkin system containing $\GG$.

By construction of $\DD$, and Intersection is Subset: General Result:

$\DD \subseteq \DD'$

$\Box$

Uniqueness

Suppose both $\DD_1$ and $\DD_2$ are Dynkin systems generated by $\GG$.

Then property $(2)$ for these Dynkin systems implies both $\DD_1 \subseteq \DD_2$ and $\DD_2 \subseteq \DD_1$.

By definition of set equality:

$\DD_1 = \DD_2$

$\blacksquare$

Sources

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