Definition:Borel Sigma-Algebra

Definition

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

The Borel sigma-algebra $\map \BB {S, \tau}$ 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 $S$.

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

The Borel sigma-algebra (or $\sigma$-algebra) on $\struct {S, d}$ 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.

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

Some sources reserve the name Borel $\sigma$-algebra for $\map \BB {\R^n, \tau}$, where $\tau$ is the usual (Euclidean) topology.

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 $\map \BB {\R^n, \tau}$.

This entry was named for Émile Borel.