Definition:Sigma-Algebra Generated by Collection of Subsets

Definition
Let $X$ be a set.

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

Then the $\sigma$-algebra generated by $\mathcal G$, $\sigma \left({\mathcal G}\right)$, is the smallest $\sigma$-algebra on $X$ that contains $\mathcal G$.

That is, $\sigma \left({\mathcal G}\right)$ is subject to:


 * $(1):\quad \mathcal G \subseteq \sigma \left({\mathcal G}\right)$
 * $(2):\quad \mathcal G \subseteq \mathcal A \implies \sigma \left({\mathcal G}\right) \subseteq \mathcal A$ for any $\sigma$-algebra $\mathcal A$ on $X$

In fact, $\sigma \left({\mathcal G}\right)$ always exists, and is unique, as proved on Existence and Uniqueness of Sigma-Algebra Generated by Collection of Subsets.

Generator
One says that $\mathcal G$ is a generator for $\sigma \left({\mathcal G}\right)$.