Monotone class theorem

Let \(\mathcal C\) be a class of subset closed under finite intersections and containing \(\Omega\) (that is, \(\mathcal C\) is a \(\pi\)-system). Let \(\mathcal B\) be the smallest class containing \(\mathcal C\) which is closed under increasing limits and by difference (that is, \(\mathcal B\) is the smallest \(\lambda\) system containing \(\mathcal C\)). Then \(\mathcal B = \sigma(\mathcal C)\)

Dynkin \(\pi-\lambda\) theorem

If \(P\) is a \(\pi\) system and \(D\) is a \(\lambda\) system with \(P\subseteq D\), then \(\sigma(P) \subseteq D\).