Dynkin_System
Definition
A family $\mathcal{D} \subseteq \mathcal{P}\left. (X) \right.$ is a Dynkin system if
$X \in \mathcal{D}$
$D \in \mathcal{D} \Longrightarrow D^{c} \in \mathcal{D}$
$\left. \left( D_{n} \right) \right._{n \in \mathbb{N}} \subseteq \mathcal{D}\mathrm{\text{ pairwise disjoint }} \Longrightarrow \bigsqcup(n \in \mathbb{N})D_{n} \in \mathcal{D}$
where $\sqcup$ represents the union of disjoint sets.
We have that $\varnothing \in \mathcal{D}$ and that finite disjoint unions are again in $\mathcal{D}:D,E \in \mathcal{D} \Longrightarrow D \cap E = \varnothing \Longrightarrow D \sqcup E \in D$. Of course, every $\sigma$-algebra is a Dynkin system, but the converse is, in general, wrong (See Measures,_Integrals_and_Martingales – Exercises 5.1 and 5.2).
Theorems
Generators
Let $\mathcal{G} \subseteq \mathcal{P}\left. (X) \right.$. Then there is a smallest Dynkin system $\delta\left. \left( \mathcal{G} \right) \right.$ containing $\mathcal{G}$. $\delta\left. \left( \mathcal{G} \right) \right.$ is called the Dynkin system generated by $\mathcal{G}$. Moreover, $G \subseteq \delta\left. \left( \mathcal{G} \right) \right. \subseteq \sigma\left. \left( \mathcal{G} \right) \right.$.
Proof: The proof that $\delta\left. \left( \mathcal{G} \right) \right.$ exists parallels the proof of Theorem 3.4(ii). As in the case of $\sigma$-algebras, $\delta\left. \left( \mathcal{D} \right) \right. = \mathcal{D}$ if $\mathcal{D}$ is a Dynkin system (by minimality) and so $\delta\left. \left( \sigma\left. \left( \mathcal{G} \right) \right. \right) \right. = \sigma\left. \left( \mathcal{G} \right) \right.$. Hence $\mathcal{G} \subseteq \sigma\left. \left( \mathcal{G} \right) \right.$ implies that $\delta\left. \left( \mathcal{G} \right) \right. \subseteq \delta\left. \left( \sigma\left. \left( \mathcal{G} \right) \right. \right) \right. = \sigma\left. \left( \mathcal{G} \right) \right.$.