# Definition:Sigma-Algebra Generated by Collection of Subsets

Jump to navigation
Jump to search

## 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