Definition:Sigma-Algebra Generated by Collection of Subsets

Definition

Let $X$ be a set.

Let $\GG \subseteq \powerset X$ be a collection of subsets of $X$.

Definition 1

The $\sigma$-algebra generated by $\GG$, denoted $\map \sigma \GG$, is the smallest $\sigma$-algebra on $X$ that contains $\GG$.

That is, $\map \sigma \GG$ is subject to:

$(1): \quad \GG \subseteq \map \sigma \GG$

$(2): \quad$ for all $\sigma$-algebras $\Sigma$ on $X$: $\GG \subseteq \Sigma \implies \map \sigma \GG \subseteq \Sigma$

Definition 2

The $\sigma$-algebra generated by $\GG$, $\map \sigma \GG$, is the intersection of all $\sigma$-algebras on $X$ that contain $\GG$.

Generator

One says that $\GG$ is a generator for $\map \sigma {\GG}$.

Also, elements $G$ of $\GG$ may be called generators.

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

• Existence and Uniqueness of Sigma-Algebra Generated by Collection of Subsets, where it is shown that $\map \sigma \GG$ always exists, and is unique

• Results about $\sigma$-algebras can be found here.

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