# Definition:Borel Sigma-Algebra/Metric Space

## Definition

Let $\struct {S, \norm {\, \cdot \,} }$ be a metric space.

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

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 a topological space.

### Borel Set

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

## Also defined as

Some sources reserve the name **Borel $\sigma$-algebra** for $\map \BB {\R^n, \tau}$, 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 $\map \BB \tau$, $\map \BB S$ or even just $\BB$.

Some authors write $\BB^n$ for $\map \BB {\R^n, \tau}$.

## Also see

- Results about
**Borel $\sigma$-algebras**can be found here.

## Source of Name

This entry was named for Émile Borel.

## Sources

- 1984: Gerald B. Folland:
*Real Analysis: Modern Techniques and their Applications*: $\S 1.2$