Dynkin system

Definition

$$\newcommand{\bigudot}{ \mathchoice{\mathop{\bigcup\mkern-15mu\cdot\mkern8mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}} }$$ $$\newcommand{\udot}{\cup\mkern-12.5mu\cdot\mkern6.25mu\!}$$ $$\require{AMScd}\newcommand{\d}[1][]{\mathrm{d}^{#1} }$$ 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 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 $\udot_{n=1}^\infty D_n\in\mathcal{D}$

Immediate results

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• $\emptyset\in\mathcal{D}$

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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.

See also

• Dynkin system generated by
• Types of set algebras