Generated Monotone Class
Definition
Suppose $X$ is a set and $\mathcal{C} $ is a set of subsets of $X$. The generated monotone class of $\mathcal{C} $ is the smallest monotone class containing the $\mathcal{C} $. We can show that such monotone class exists and is unique. This is the content of the following proposition.
Suppose $X$ is a set and define the set
$\mathcal{G} \in \powerset{X}$ by
\[ \mathcal{G} = \bigcap \Set{\mathcal{C} \subset \powerset{X}}{\mathcal{C} \text{ is a monotone class}} \]
is a monotone class.1- Future editions will include a treatment. ↩︎