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