Borel σ-Algebra
Undergraduate (subsets of $\reals$)
The Borel σ-algebra of $\reals$ is the smallest σ-algebra on $\reals$ that contains every open interval.
Graduate (topological spaces)
Consider a set $X$ with a topology $\mathcal{T}$. The Borel σ-algebra $\mathcal{B}(X)$ of $X$ is the smallest σ-algebra on $X$ that contains every open set:
\[\mathcal{B}(X) = \bigcap \setdef{\mathcal{F} \suchthat \mathcal{F} \text{ is a } \sigma\text{-algebra on } X \text{ and } \mathcal{T} \subseteq \mathcal{F}}.\]Examples
Examples
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The Borel sets of $\reals$
Graduate
The Borel sets of $\reals$ form the Borel σ-algebra of $\reals$, because the Borel σ-algebra of $\reals$ is the smallest σ-algebra that contains every open interval, and every open set of $\reals$ is a countable union of open intervals.
Properties
- The intersection in the definition is itself a σ-algebra that contains every open set, so the smallest such σ-algebra exists.
- The Borel σ-algebra of $\reals^n$ is also the smallest σ-algebra that contains every open box, because every open set of $\reals^n$ is a countable union of open boxes.
- The Borel σ-algebra of $\reals$ is also the smallest σ-algebra that contains every interval of the form $(a, \infty)$.
- The Borel σ-algebra of $\reals$ has the same cardinality $\mathfrak{c}$ as $\reals$.
- The Lebesgue σ-algebra on $\reals^n$ contains the Borel σ-algebra, and the Lebesgue σ-algebra is larger: the Lebesgue σ-algebra also contains every subset of a Borel set of measure zero.
Notes
The sets in the Borel σ-algebra are called Borel sets. The Borel σ-algebra is named after Émile Borel.