A monotone class is a family (i.e. class) \(M\) of sets that is closed under countable monotone unions and under countable monotone intersections. Explicitly, this means that \(M\) has the following properties: 1. \(A_{1}, A_{2}, \dots \in M\) and \(A_1\subseteq A_{2} \subseteq \dots\) then \(\bigcup_{i=1}^\infty A_{i}\in M\); 2. \(B_{1}, B_{2}, \dots \in M\) and \(B_1\supseteq B_{2} \subseteq \dots\) then \(\bigcap_{i=1}^\infty B_{i}\in M\).
The key result for monotone classes is the Monotone Class Theorem which linked monotone classes and [[sigma-algebra|sigma algebra]].