Dynkin system

Definition

Given a set $X$ and a family of subsets of $X$ , which we shall denote $\mathcal{D}\subseteq\mathcal{P}(X)$ is a Dynkin system^[1] if:

$X\in\mathcal{D}$
For any $D\in\mathcal{D}$ we have $D^c\in\mathcal{D}$
For any $(D_n)_{n=1}^\infty\subseteq\mathcal{D}$ is a sequence of pairwise disjoint sets we have $\udot_{n=1}^\infty D_n\in\mathcal{D}$

Proof:

As

$\mathcal{D}$ is closed under complements and

$X\in\mathcal{D}$ by definition,

$X^c\in\mathcal{D}$

$X^c=\emptyset$ so

$\emptyset\in\mathcal{D}$

This completes the proof.