# Definition:Sigma-Algebra Generated by Collection of Subsets

## Contents

## 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

- Existence and Uniqueness of Sigma-Algebra Generated by Collection of Subsets, where it is shown that $\sigma \left({\mathcal G}\right)$ always exists, and is unique

## Sources

- 1984: Gerald B. Folland:
*Real Analysis: Modern Techniques and their Applications*: $\S 1.2$ - 2005: René L. Schilling:
*Measures, Integrals and Martingales*... (previous) ... (next): $3.2 \ \text{(ii)}$