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 $\Bbb A$ be the collection of $\sigma$-algebras containing $\mathcal G$:


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

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

Also, by Set Intersection Preserves Subsets, have $\mathcal G \subseteq \mathcal A$.

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

By construction of $\mathcal A$, and Intersection Subset, $\mathcal A \subseteq \mathcal B$.

Uniqueness
Suppose both $\mathcal{A}_1$ and $\mathcal{A}_2$ are $\sigma$-algebras generated by $\mathcal G$.

Then property $(2)$ for these $\sigma$-algebras implies both $\mathcal{A}_1 \subseteq \mathcal{A}_2$ and $\mathcal{A}_2 \subseteq \mathcal{A}_1$.

Hence, by definition of set equality, $\mathcal{A}_1 = \mathcal{A}_2$.