Borel Set
Graduate (topological spaces)
Consider a set $X$ with a topology $\mathcal{T}$. A set $B \subseteq X$ is a Borel set if $B$ is in the Borel σ-algebra of $X$, which is the smallest σ-algebra on $X$ that contains every open set.
Examples
Examples
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$[0, 1)$, as a subset of $\reals$
High school
The interval $[0, 1)$ is a Borel set, because it is the intersection of the open set $(-1, 1)$ and the closed set $[0, \infty)$.
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$\mathbb{Q}$, as a subset of $\reals$
Undergraduate
$\mathbb{Q}$ is a Borel set, because $\mathbb{Q}$ is the union of countably many one-point sets $\setdef{q}$, each of which is closed.
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The Cantor set
Undergraduate
The Cantor set is a Borel set, because it is closed.
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A “fat” Cantor set
Graduate
A fat Cantor set is a Borel set, because it is closed: it is the intersection of the finite unions of closed intervals that remain after each step of its construction.
Properties
- Every open set and every closed set is a Borel set.
- Every countable subset of $\reals$ is a Borel set, because it is a countable union of one-point sets, which are closed.
- Every Borel set of $\reals^n$ is Lebesgue measurable.
- Some Lebesgue-measurable sets are not Borel sets: every subset of the Cantor set is Lebesgue measurable, because the Cantor set has measure zero, but some subsets of the Cantor set are not Borel sets.
- The collection of Borel sets of $\reals$ has the same cardinality $\mathfrak{c}$ as $\reals$, but $\reals$ has $2^{\mathfrak{c}}$ subsets, so most subsets of $\reals$ are not Borel sets.
- The preimage of every Borel set under a continuous function is a Borel set, so every continuous function is Borel measurable.
Notes
Borel sets are named after Émile Borel.