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): $\S 7$