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.

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 $$\Omega$$ be a set, and let $$\Sigma \subseteq \mathcal P(\Omega)$$. Prove that $$\Sigma$$ is a $$\lambda$$-system if and only if
 * 2) $$\emptyset \in \Sigma$$
 * 3) $$A, B \in \Sigma \Rightarrow A \setminus B \in \Sigma$$
 * 4) $$A_1, A_2, \ldots \in \Sigma \wedge A_1 \supseteq A_2 \supseteq A_3 \supseteq \cdots \Rightarrow \bigcap_{n \in \mathbb N} A_n \in \Sigma$$.
 * 5) 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
 * 6) $$\emptyset \in \mathcal F$$
 * 7) $$A, B \in \mathcal F \Rightarrow A \setminus B \in \mathcal F$$
 * 8) $$A_n \in \mathcal F$$ for all $$n \in \mathbb N$$ implies $$\bigcap_{n \in \mathbb N} A_n \in \mathcal F$$.