Set Theory/Systems of sets

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

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.

Exercises

1. Let $\mathrm{\Omega }$ be a set, and let $\mathrm{\Sigma }\subseteq 𝒫(\mathrm{\Omega })$. Prove that $\mathrm{\Sigma }$ is a $\lambda$-system if and only if
   1. $\mathrm{\varnothing }\in \mathrm{\Sigma }$
   2. $A,B\in \mathrm{\Sigma }\Rightarrow A\setminus B\in \mathrm{\Sigma }$
   3. $A_1,A_2,\dots \in \mathrm{\Sigma }\wedge A_1\supseteq A_2\supseteq A_3\supseteq \cdots \Rightarrow \bigcap _{n\in \mathbb{N}}{A}_n\in \mathrm{\Sigma }$.
2. Let $\mathrm{\Omega }$ be a set, and let $ℱ\subseteq 𝒫(\mathrm{\Omega })$. Prove that $ℱ$ is a $\sigma$-algebra if and only if
   1. $\mathrm{\varnothing }\in ℱ$
   2. $A,B\in ℱ\Rightarrow A\setminus B\in ℱ$
   3. $A_n\in ℱ$ for all $n\in \mathbb{N}$ implies $\bigcap _{n\in \mathbb{N}}{A}_n\in ℱ$.