> [!definition] > > Let $X$ be a [[Set|set]], a family of sets $\cc \subset \pow{X}$ is a **monotone class** if it is closed under countable increasing unions, and countable decreasing intersections. > [!theorem] > > Let $X$ be a [[Set|set]] and $\cm$ be a [[Sigma Algebra|sigma algebra]] over $X$. Then $\cm$ is a monotone class. > [!theorem] > > Let $\cc, \cc'$ be monotone classes, then $\cc \cap \cc'$ is also a monotone class. > [!definition] > > Let $\ce \subset \pow{X}$ be a family of sets. The intersection $\cc(\ce)$ of all monotone classes containing $\ce$ is the smallest monotone class containing $\ce$, known as the monotone class **generated by** $\ce$.