Definition:Monotone Class

Definition
Let $X$ be a set, and let $\mathcal P \left({X}\right)$ be its power set.

Let $\mathcal M \subseteq \mathcal P \left({X}\right)$ be a collection of subsets of $X$.

Then $\mathcal M$ is said to be a monotone class (on $X$) iff for every countable, nonempty, index set $I$, it holds that:


 * $\displaystyle \left({A_i}\right)_{i \in I} \in \mathcal M \implies \bigcup_{i \mathop \in I} A_i \in \mathcal M$
 * $\displaystyle \left({A_i}\right)_{i \in I} \in \mathcal M \implies \bigcap_{i \mathop \in I} A_i \in \mathcal M$

that is, iff $\mathcal M$ is closed under countable unions and intersections.

Also defined as
Some sources stipulate only that above closure properties should hold for $I = \N$.

This definition is equivalent, as proved on Equivalence of Definitions of Monotone Class.