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:

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

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:

1. $\Omega \in {\mathcal {F}}$
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 $\bigcup _{n\in \mathbb {N} }A_{n}\in {\mathcal {F}}$.

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

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