Set Theory/Systems of sets

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

Definition (π-system):

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

Definition (Dynkin system):

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

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

Definition (σ-algebra):

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

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

Note that being a $\sigma$ -algebra is a stronger requirement than being a Dynkin system: A $\sigma$ -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 $\Omega$ be a set, and let ${\mathcal {A}}\subseteq 2^{\Omega }$ . Then we define

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

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

\sigma ({\mathcal {A}})=\lambda ({\mathcal {A}})

.

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

Exercises

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