---- > [!definition] Definition. ([[Dynkin system]]) > > Let $\Omega$ be a nonempty set. A collection of subsets $\mathcal{D} \subset 2^{\Omega}$ is said to be a **Dynkin system** if > 1. $\Omega \in \mathcal{D}$; > 2. *(complements of subsets in supersets)* If $A\subset B$ in $\mathcal{D}$, then $B-A \in D$; > 3. *(countable increasing unions)* if $A_{1} \subset A_{2} \subset \dots$ is an [[monotonic map|increasing]] [[sequence]] of sets in $\mathcal{D}$, then $\bigcup_{n=1}^{\infty}A_{n} \in \mathcal{D}$. > > > ![[class methods in measure theory.canvas|class methods in measure theory]] > [!equivalence] > $\mathcal{D}$ is a Dynkin system (satisfies $(1)$-$(3)$ above) if and only if $\mathcal{D}$ satisfies $(4)$-$(6)$ below: > > 4. $\emptyset \in \mathcal{D}$ > 5. *(complements)* If $A \in \mathcal{D}$, then $\Omega-A \in \mathcal{D}$ > 6. *(countable disjoint unions)* if $A_{1},A_{2},\dots$ is a [[sequence]] of pairwise disjoint sets in $\mathcal{D}$, then $\bigcup_{n=1}^{\infty}A_{n} \in \mathcal{D}$. > > > > [!proof]- Proof. > > > > Suppose $\mathcal{D}$ satisfies $(1)$-$(3)$. Then $(4)$ holds from $(1)$ and $(2)$, because $\Omega \subset \Omega$ and $\emptyset=\Omega - \Omega$. $(5)$ too, letting $B=\Omega$ in $(2)$. If $A_{1},A_{2},\dots$ is a disjoint sequence in of sets $\mathcal{D}$, first note that $A_{1} \sqcup A_{2} \in \mathcal{D}$ since $A_{1}\cap A_{2} = \emptyset$ implies $A_{1} \subset A_{2}^{c}$, so $A_{2}^{c}-A_{1}= (A_{1} \sqcup A_{2})^{c}$ is in $\mathcal{D}$, hence $A_{1} \sqcup A_{2}=\Omega-(A_{1} \sqcup A_{2})^{c}$ is in $\mathcal{D}$. By induction, $A_{1} \sqcup \dots \sqcup A_{n}$ is in $\mathcal{D}$ for all $n \in \mathbb{N}$. Then we have that > > > > $\bigsqcup_{n=1}^{\infty}A_{n}=A_{1} \cup (A_{1} \sqcup A_{2}) \cup (A_{1} \sqcup A_{2} \sqcup A_{3}) \cup \dots$ > > [[countably infinite|which shows]] $(6)$. > > > > > > Conversely, suppose $\mathcal{D}$ satisfies $(4)$-$(6)$. Then obviously $(1)$ holds because $\Omega= \Omega-\emptyset$. If $A \subset B$ in $\mathcal{D}$, then $\begin{align} > > B-A&=B \cap (\Omega- A)= \underbrace{ \Omega - \big(\underbrace{ \underbrace{ (\Omega - B) }_{ \in \mathcal{D} , \ \text{by }(5) } \sqcup A }_{ \in \mathcal{D}, \text{ by }(6) } \big) }_{ \in \mathcal{D}, \ \text{by }(5) } \in \mathcal{D}. > > \end{align}$If $A_{1} \subset A_{2} \subset\dots$ is an increasing union of elements in $\mathcal{D}$, then putting $A_{0}:=\emptyset$ we have that $\bigcup_{n}A_{n}= \bigsqcup_{n} (A_{n}-A_{n-1}) \in \mathcal{D}$, witnessing that $(3)$ holds. > > [!basicproperties] > 1. *(countable decreasing intersections)* If $A_{1} \supset A_{2} \supset \dots$ is a decreasing sequence of sets in $\mathcal{D}$, then $\bigcap_{n=1}^{\infty}A_{n} \in \mathcal{D}$. Thus, any Dynkin system is a [[monotone class]]. > > > [!proof]- Proof. > > [[De Morgan's Laws|Write]] $X-\bigcap_{n=1}^{\infty}=\bigcup_{n=1}^{\infty}(\overbrace{ X - A_{n} }^{ \in \mathcal{D} })$. Since the intersection is assumed decreasing, the union on the RHS is increasing. Hence $X-\bigcap_{n=1}^{\infty}$ is $\mathcal{D}$ (by $(3)$), thus so too is its complement (by $(2)$). > ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```