Definition:Monotone Class Generated by Collection of Subsets

Definition

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 $X$

One says that $\GG$ is a generator for $\map {\mathfrak m} \GG$.

Existence and Uniqueness of Monotone Class Generated by Collection of Subsets, in which it is proved that $\map {\mathfrak m} \GG$ always exists, and is unique.