\begin{definition}[Definition:Monotone Class Generated by Collection of Subsets]
Let $X$ be a set.
Let $\GG \subseteq \powerset X$ be a collection of subsets of $X$.
Then the '''monotone class generated by $\GG$''', $\map {\mathfrak m} \GG$, is the smallest monotone class on $X$ that contains $\GG$.
That is, $\map {\mathfrak m} \GG$ is subject to:
:$(1): \quad \GG \subseteq \map {\mathfrak m} \GG$
:$(2): \quad \GG \subseteq \MM \implies \map {\mathfrak m} \GG \subseteq \MM$ for any monotone class $\MM$ on $X$
\end{definition}