# Definition:Borel Sigma-Algebra

## Definition

### Topological Spaces

Let $\struct {S, \tau}$ be a topological space

The **Borel sigma-algebra** $\map {\mathcal B} {S, \tau}$ of $\struct {S, \tau}$ 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 $\struct {X, \norm {\,\cdot\,} }$ be a metric space.

The **Borel sigma-algebra** (or **$\sigma$-algebra**) on $\struct {X, \norm {\,\cdot\,} }$ is the $\sigma$-algebra generated by the open sets in $\powerset X$.

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 $\map {\mathcal B} {S, \tau}$ are called the **Borel (measurable) sets** of $\struct {S, \tau}$.

## Also defined as

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

## Also known as

The **Borel sigma-algebra** is also found with the name **topological sigma-algebra**, or even just **$\sigma$-algebra**.

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

Also, some authors write $\mathcal B^n$ for $\map {\mathcal B} {\R^n, \tau}$.

## 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): $\S 7$