borel set

Borel set

A borel set is a subset of the real line $\mathbb{R}$ that can be formed through countable unions, countable intersections, and complements of open intervals. The collection of all Borel sets forms the Borel σ-algebra, denoted $\mathcal{B}(\mathbb{R})$.

Examples of Borel sets include:

All open intervals $(a,b)$
All closed intervals $[a,b]$
All singletons $\{x\}$
Any countable union or intersection of these
For instance, the set of rational numbers $\mathbb{Q}$ is a Borel set since it can be written as a countable union of singletons.

Borel sets capture all the "reasonable" subsets of $\mathbb{R}$ that we might want to assign probability to. While more exotic sets exist in $\mathbb{R}$, Borel sets include all sets we encounter in standard probability theory and statistical applications.