\begin{definition}[Definition:Sigma-Algebra Generated by Collection of Subsets/Definition 1]
Let $X$ be a set.
Let $\GG \subseteq \powerset X$ be a collection of subsets of $X$.
The '''$\sigma$-algebra generated by $\GG$''', $\map \sigma \GG$, is the smallest $\sigma$-algebra on $X$ that contains $\GG$.
That is, $\map \sigma \GG$ is subject to:
:$(1): \quad \GG \subseteq \map \sigma \GG$
:$(2): \quad \GG \subseteq \Sigma \implies \map \sigma \GG \subseteq \Sigma$ for any $\sigma$-algebra $\Sigma$ on $X$
\end{definition}