Definition:Sigma-Algebra Generated by Collection of Subsets/Definition 1

Definition
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 $X$

Also denoted as
Variations of the letter "$M$" can be seen for the $\sigma$-algebra generated by $\GG$:


 * $\map \MM \GG$
 * $\map {\mathscr M} \GG$

Also see

 * Equivalence of Definitions of Sigma-Algebra Generated by Collection of Subsets