# Set Theory/Systems of sets

In this chapter, we would like to study, for a given set ${\displaystyle \Omega }$, subsets of the power set ${\displaystyle {\mathcal {P}}(\Omega )}$. We consider in particular those subsets of ${\displaystyle {\mathcal {P}}(\Omega )}$ that are closed under certain operations.

Definition (π-system):

Let ${\displaystyle \Omega }$ be a set. A ${\displaystyle \pi }$-system is a collection of sets ${\displaystyle {\mathcal {A}}\subseteq {\mathcal {P}}(\Omega )}$ such that whenever ${\displaystyle A,B\in {\mathcal {A}}}$, then also ${\displaystyle A\cap B\in {\mathcal {A}}}$.

Definition (Dynkin system):

Let ${\displaystyle \Omega }$ be a set. A Dynkin system or ${\displaystyle \lambda }$-system is a collection of sets ${\displaystyle \Sigma \subseteq \Omega }$ such that the following three axioms hold:

1. ${\displaystyle \Omega \in \Sigma }$
2. ${\displaystyle A,B\in \Sigma \Rightarrow A\setminus B\in \Sigma }$
3. ${\displaystyle A_{1},A_{2},\cdots \in \Sigma \wedge A_{1}\subseteq A_{2}\subseteq A_{3}\subseteq \cdots \Rightarrow \bigcup _{n\in \mathbb {N} }A_{n}\in \Sigma }$.

Definition (σ-algebra):

Let ${\displaystyle \Omega }$ be a set. A ${\displaystyle \sigma }$-algebra on ${\displaystyle \Omega }$ is a collection of subsets of ${\displaystyle \Omega }$, say ${\displaystyle {\mathcal {F}}\subseteq {\mathcal {P}}(\Omega )}$, such that the following axioms are satisfied:

1. ${\displaystyle \Omega \in {\mathcal {F}}}$
2. ${\displaystyle A,B\in {\mathcal {F}}\Rightarrow A\setminus B\in {\mathcal {F}}}$
3. ${\displaystyle A_{n}\in {\mathcal {F}}}$ for all ${\displaystyle n\in \mathbb {N} }$ implies ${\displaystyle \bigcup _{n\in \mathbb {N} }A_{n}\in {\mathcal {F}}}$.

Note that being a ${\displaystyle \sigma }$-algebra is a stronger requirement than being a Dynkin system: A ${\displaystyle \sigma }$-algebra is closed under all countable intersections, whereas a Dynkin system is only closed under intersections of countable ascending chains.

Definition (σ-algebra generated by a collection of sets):

Let ${\displaystyle \Omega }$ be a set, and let ${\displaystyle {\mathcal {A}}\subseteq 2^{\Omega }}$. Then we define

${\displaystyle \sigma ({\mathcal {A}}):=\bigcap _{{\mathcal {B}}\supset {\mathcal {A}} \atop {\mathcal {B}}{\text{ is a }}\sigma {\text{-algebra}}}{\mathcal {B}}}$.

Definition (λ-system generated by a collection of sets):

Let ${\displaystyle \Omega }$ be a set, and let ${\displaystyle {\mathcal {A}}\subseteq 2^{\Omega }}$. Then we define

${\displaystyle \lambda ({\mathcal {A}}):=\bigcap _{{\mathcal {B}}\supset {\mathcal {A}} \atop {\mathcal {B}}{\text{ is a }}\lambda {\text{-system}}}{\mathcal {B}}}$.

Theorem (Dynkin's λ-π theorem):

Let ${\displaystyle \Omega }$ be a set, and let ${\displaystyle {\mathcal {A}}}$ be a ${\displaystyle \pi }$-system on ${\displaystyle \Omega }$. Then

${\displaystyle \sigma ({\mathcal {A}})=\lambda ({\mathcal {A}})}$.

Proof: The direction "${\displaystyle \supseteq }$" is clear, so that we only have to prove "${\displaystyle \supseteq }$". To do so, we prove that ${\displaystyle \lambda ({\mathcal {A}})}$ is in fact a ${\displaystyle \sigma }$-algebra that contains ${\displaystyle {\mathcal {A}}}$, using the definition of ${\displaystyle \sigma ({\mathcal {A}})}$ as the intersection of all ${\displaystyle \sigma }$-algebrae that contain ${\displaystyle {\mathcal {A}}}$. ${\displaystyle \Box }$

## Exercises

1. Let ${\displaystyle \Omega }$  be a set, and let ${\displaystyle \Sigma \subseteq {\mathcal {P}}(\Omega )}$ . Prove that ${\displaystyle \Sigma }$  is a ${\displaystyle \lambda }$ -system if and only if
1. ${\displaystyle \emptyset \in \Sigma }$
2. ${\displaystyle A,B\in \Sigma \Rightarrow A\setminus B\in \Sigma }$
3. ${\displaystyle A_{1},A_{2},\ldots \in \Sigma \wedge A_{1}\supseteq A_{2}\supseteq A_{3}\supseteq \cdots \Rightarrow \bigcap _{n\in \mathbb {N} }A_{n}\in \Sigma }$ .
2. Let ${\displaystyle \Omega }$  be a set, and let ${\displaystyle {\mathcal {F}}\subseteq {\mathcal {P}}(\Omega )}$ . Prove that ${\displaystyle {\mathcal {F}}}$  is a ${\displaystyle \sigma }$ -algebra if and only if
1. ${\displaystyle \emptyset \in {\mathcal {F}}}$
2. ${\displaystyle A,B\in {\mathcal {F}}\Rightarrow A\setminus B\in {\mathcal {F}}}$
3. ${\displaystyle A_{n}\in {\mathcal {F}}}$  for all ${\displaystyle n\in \mathbb {N} }$  implies ${\displaystyle \bigcap _{n\in \mathbb {N} }A_{n}\in {\mathcal {F}}}$ .