> [!abstract] > Given a set $\Omega$, $\mathcal{A} \subseteq \mathcal{P}(\Omega)$ is a **π-system** if it is closed to finite intersections. $\mathcal{A}$ is a $\sigma$-algebra $\iff$ $\mathcal{A}$ is both a $\pi$ and a [[λ-Systems|λ-system]]. > [!proof]- > $(\Rightarrow)$ is immediate. > $(\Leftarrow)$ Take any $A \in \mathcal{A}$, then $A^{c}=\Omega - A \in \mathcal{A}$ (by lambda). > Then $A \cup B=\Omega - (A^{c}\cap B^{c}) \in \mathcal{A}$ (by pi and lambda). > Hence $\mathcal{A}$ is stable under finite unions. > Then any countable collection $(A_{i})$ has $\bigcup_{i}A_{i} = \bigcup_{n}\left( \bigcup_{i \le n} A_{i} \right)$where the sets in the parentheses are nested finite unions, hence the whole set is in $\mathcal{A}$.