Definition:Borel Sigma-Algebra
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Definition
Topological Spaces
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 $S$.
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 $\map \BB {S, \tau}$ are called the Borel (measurable) sets of $\struct {S, \tau}$.
Also defined as
Sometimes, the name Borel sigma-algebra is reserved for $\map \BB {\R^n, \tau}$, 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 $\BB$.
Also, some authors write $\BB^n$ for $\map \BB {\R^n, \tau}$.
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): $\S 7$