Borel sigma algebra

• categories: Measure Theory, Definition

The Borel $\sigma$-algebra on a topological space $X$ (typically $\mathbb{R}$ with its standard topology) is the smallest $\sigma$-algebra that contains all open sets in $X$. Denoted by $\mathcal{B}(X)$, it includes all sets that can be constructed from open sets using countable unions, countable intersections, and complements. Sets in this $\sigma$-algebra are called Borel Sets

Properties:

1. Contains open and closed sets: Since the Borel $\sigma$-algebra is generated by open sets, it contains all open sets and their complements (i.e., all closed sets).
2. Closed under countable operations: If $A_1,A_2,\dots$ are Borel sets, then:
   • The countable union ${\bigcup }_{n=1}^{\infty }{A}_n$ is a Borel set.
   • The countable intersection ${\bigcap }_{n=1}^{\infty }{A}_n$ is a Borel set.
   • The complement $A^c$ of any Borel set is also a Borel set.
3. Contains intervals: In $\mathbb{R}$, all common intervals (open, closed, half-open) such as $(a,b)$, $[a,b]$, $(a,b]$, and $[a,b)$ are Borel sets because they can be expressed as open or closed sets.
4. Generated by a base: For example, in $\mathbb{R}$, the Borel $\sigma$-algebra can be generated by:
   • The collection of all open intervals $(a,b)$.
   • Equivalently, by all closed intervals $[a,b]$.