Definition:Borel Sigma-Algebra/Metric Space

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Definition

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

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