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

Theorem
Let $X$ be a set.

Let $\mathcal G \subseteq \mathcal P \left({X}\right)$ be a collection of subsets of $X$.

Then $\sigma \left({\mathcal G}\right)$, the $\sigma$-algebra generated by $\mathcal G$, exists and is unique.

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

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


 * $\Bbb E := \left\{{\Sigma': \mathcal G \subseteq \Sigma', \text{$\Sigma'$ is a $\sigma$-algebra}}\right\}$

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

Also, by Set Intersection Preserves Subsets: General Case, have $\mathcal G \subseteq \Sigma$.

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

By construction of $\Sigma$, and Intersection Subset: General Result, $\Sigma \subseteq \Sigma'$.

Uniqueness
Suppose both $\Sigma_1$ and $\Sigma_2$ are $\sigma$-algebras generated by $\mathcal G$.

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

Hence, by Equality of Sets, $\Sigma_1 = \Sigma_2$.