Definition:Sigma-Algebra Generated by Collection of Subsets

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Definition

Let $X$ be a set.

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


Definition 1

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:

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


Definition 2

Then the $\sigma$-algebra generated by $\mathcal G$, $\sigma \left({\mathcal G}\right)$, is the intersection of all $\sigma$-algebras on $X$ that contain $\mathcal G$.


Generator

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

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


Also denoted as

Variations of the letter "$M$" can be seen for the $\sigma$-algebra generated by $\mathcal G$:

  • $\mathcal M \left({\mathcal G}\right)$
  • $\mathscr M \left({\mathcal G}\right)$

Also see


Sources