Definition:Borel Sigma-Algebra
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Contents
Definition
Topological Spaces
Let $\left({S, \tau}\right)$ be a topological space
The Borel or topological sigma-algebra (or $\sigma$-algebra) $\mathcal B \left({S, \tau}\right)$ of a topological space $\left({S, \tau}\right)$ 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 $\left({X,\Vert \cdot \Vert}\right)$ be a metric space.
The Borel sigma-algebra (or $\sigma$-algebra) on $\left({X,\Vert \cdot \Vert}\right)$ is the $\sigma$-algebra generated by the open sets in $\mathcal P(X)$.
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 $\mathcal B \left({S, \tau}\right)$ are called the Borel (measurable) sets of $\left({S, \tau}\right)$.
Also defined as
Sometimes, the name Borel sigma-algebra is reserved for $\mathcal B \left({\R^n, \tau}\right)$, where $\tau$ is the usual (Euclidean) topology.
Also known as
When the set $S$ or the topology $\tau$ are clear from the context, one may encounter $\mathcal B \left({\tau}\right), \mathcal B \left({S}\right)$ or even just $\mathcal B$.
Also, some authors write $\mathcal{B}^n$ for $\mathcal B \left({\R^n, \tau}\right)$.
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): $\S 7$
