\begin{definition}[Definition:Monotone Class]
Let $X$ be a set, and let $\powerset X$ be its power set.
Let $\MM \subseteq \powerset X$ be a collection of subsets of $X$.
Then $\MM$ is said to be a '''monotone class (on $X$)''' {{iff}} for every countable, nonempty, index set $I$, it holds that:
:$\ds \family {A_i}_{i \mathop \in I} \in \MM \implies \bigcup_{i \mathop \in I} A_i \in \MM$
:$\ds \family {A_i}_{i \mathop \in I} \in \MM \implies \bigcap_{i \mathop \in I} A_i \in \MM$
that is, {{iff}} $\MM$ is closed under countable unions and countable intersections.
\end{definition}