Definition:Borel Sigma-Algebra/Metric Space
Definition
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.
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 : $\S 1.2$