Existence and Uniqueness of Monotone Class Generated by Collection of Subsets
Theorem
Let $X$ be a set.
Let $\GG \subseteq \powerset X$ be a collection of subsets of $X$.
Then $\map {\mathfrak m} \GG$, the monotone class generated by $\GG$, exists and is unique.
Proof
Existence
By Power Set is Monotone Class, there is at least one monotone class containing $\GG$.
Now let $\Bbb M$ be the collection of monotone classes containing $\GG$:
- $\Bbb M := \set {\mathfrak m': \GG \subseteq \mathfrak m', \mathfrak m' \text{ is a monotone class} }$
By Intersection of Monotone Classes is Monotone Class, $\mathfrak m := \bigcap \Bbb M$ is a monotone class.
Also, by Set Intersection Preserves Subsets:
- $\GG \subseteq \mathfrak m$
Now let $\mathfrak m'$ be a monotone class containing $\GG$.
Then by construction of $\mathfrak m$, and Intersection is Subset: General Result:
- $\mathfrak m \subseteq \mathfrak m'$
$\Box$
Uniqueness
Suppose $\mathfrak m_1$ and $\mathfrak m_2$ are both monotone classes generated by $\GG$.
Then property $(2)$ for these monotone classes implies both $\mathfrak m_1 \subseteq \mathfrak m_2$ and $\mathfrak m_2 \subseteq \mathfrak m_1$.
By definition of set equality:
- $\mathfrak m_1 = \mathfrak m_2$
$\blacksquare$
Sources
- 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $\S 3$: Problem $11 \ \text{(i)}$