Existence and Uniqueness of Sigma-Algebra Generated by Collection of Subsets

Theorem

Let $X$ be a set.

Let $\GG \subseteq \powerset X$ be a collection of subsets of $X$.

Then $\map \sigma \GG$, the $\sigma$-algebra generated by $\GG$, exists and is unique.

Proof

Existence

By Power Set is Sigma-Algebra, there is at least one $\sigma$-algebra containing $\GG$.

Next, let $\Bbb E$ be the collection of $\sigma$-algebras containing $\GG$:

$\Bbb E := \set {\Sigma': \GG \subseteq \Sigma', \text{$\Sigma'$is a$\sigma$-algebra} }$

By Intersection of Sigma-Algebras, $\Sigma := \bigcap \Bbb E$ is a $\sigma$-algebra.

Also, by Set Intersection Preserves Subsets:

$\GG \subseteq \Sigma$

Now let $\Sigma'$ be a $\sigma$-algebra containing $\GG$.

By construction of $\Sigma$, and Intersection is Subset: General Result:

$\Sigma \subseteq \Sigma'$

$\Box$

Uniqueness

Suppose both $\Sigma_1$ and $\Sigma_2$ are $\sigma$-algebras generated by $\GG$.

Then property $(2)$ for these $\sigma$-algebras implies both $\Sigma_1 \subseteq \Sigma_2$ and $\Sigma_2 \subseteq \Sigma_1$.

By definition of set equality:

$\Sigma_1 = \Sigma_2$

$\blacksquare$