# Definition:Borel Sigma-Algebra

## Contents

## Definition

### Topological Spaces

Let $\left({S, \tau}\right)$ be a topological space

The **Borel** or **topological sigma-algebra** (or **$\sigma$-algebra**) $\mathcal B \left({S, \tau}\right)$ of a topological space $\left({S, \tau}\right)$ is the $\sigma$-algebra generated by $\tau$.

That is, it is the $\sigma$-algebra generated by the set of open sets in $S$.

### Metric Spaces

Let $\left({X,\Vert \cdot \Vert}\right)$ be a metric space.

The **Borel sigma-algebra** (or **$\sigma$-algebra**) on $\left({X,\Vert \cdot \Vert}\right)$ is the $\sigma$-algebra generated by the open sets in $\mathcal P(X)$.

By the definition of a topology induced by a metric, this definition is a particular instance of the definition of a Borel $\sigma$-algebra on topological spaces.

### Borel Set

The elements of $\mathcal B \left({S, \tau}\right)$ are called the **Borel (measurable) sets** of $\left({S, \tau}\right)$.

## Also defined as

Sometimes, the name **Borel sigma-algebra** is reserved for $\mathcal B \left({\R^n, \tau}\right)$, where $\tau$ is the usual (Euclidean) topology.

## Also known as

When the set $S$ or the topology $\tau$ are clear from the context, one may encounter $\mathcal B \left({\tau}\right), \mathcal B \left({S}\right)$ or even just $\mathcal B$.

Also, some authors write $\mathcal{B}^n$ for $\mathcal B \left({\R^n, \tau}\right)$.

## Source of Name

This entry was named for Émile Borel.

## 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): $3.6$ - 2005: René L. Schilling:
*Measures, Integrals and Martingales*... (previous) ... (next): $\S 7$