Definition:Borel Sigma-Algebra/Topological Space
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 $S$.
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
- 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $3.6$
- 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $\S 7$