# Borel sigma-algebra

If X is any metric space, or more generally any topological space, the ${\displaystyle \sigma }$-algebra generated by the family of open sets in X (or, equivalently, by the family of closed sets in X) is called the Borel ${\displaystyle \sigma }$-algebra on X and is denoted by ${\displaystyle {\mathcal {B}}_{X}}$. Its members are called Borel sets. ${\displaystyle {\mathcal {B}}_{X}}$ thus includes open sets, closed sets, countable intersections of open sets, countable unions of closed sets, and so forth. [1]

The Borel ${\displaystyle \sigma }$-algebra on ${\displaystyle \mathbb {R} }$ is denoted by ${\displaystyle {\mathcal {B}}_{\mathbb {R} }}$.

## Generating ${\displaystyle {\mathcal {B}}_{\mathbb {R} }}$

${\displaystyle {\mathcal {B}}_{\mathbb {R} }}$ is generated by each of the following:

• a. the open intervals: ${\displaystyle {\mathcal {E}}_{1}=\{(a,b):a,
• b. the closed intervals: ${\displaystyle {\mathcal {E}}_{2}=\{[a,b]:a,
• c. the half-open intervals: ${\displaystyle {\mathcal {E}}_{3}=\{(a,b]:a, or ${\displaystyle {\mathcal {E}}_{4}=\{[a,b):a,
• d. the open rays: ${\displaystyle {\mathcal {E}}_{5}=\{(a,\infty ):a\in \mathbb {R} \}}$ or ${\displaystyle {\mathcal {E}}_{6}=\{(\infty ,a):a\in \mathbb {R} \}}$,
• e. the closed rays: ${\displaystyle {\mathcal {E}}_{7}=\{[a,\infty ):a\in \mathbb {R} \}}$ or ${\displaystyle {\mathcal {E}}_{8}=\{(\infty ,a]:a\in \mathbb {R} \}}$,

## Product Borel ${\displaystyle \sigma }$-algebra

Let ${\displaystyle X_{1},\dots ,X_{n}}$ be metric spaces and let ${\displaystyle X=\prod _{1}^{n}X_{j}}$, equipped with the produc metric. Then ${\displaystyle \bigotimes _{1}^{n}{\mathcal {B}}_{X_{j}}\subset {\mathcal {B}}_{X}}$. If the ${\displaystyle X_{j}}$'s are separable, then ${\displaystyle \bigotimes _{1}^{n}{\mathcal {B}}_{X_{j}}={\mathcal {B}}_{X}}$.

Proof. By proposition in product ${\displaystyle \sigma }$-algebra, ${\displaystyle \bigotimes _{1}^{n}{\mathcal {B}}_{X_{j}}}$ is generated by the sets ${\displaystyle \pi _{j}^{-1}(U_{j}),1\leq j\leq n}$, where ${\displaystyle U_{j}}$ is open in ${\displaystyle X_{j}}$. Since these sets are open in ${\displaystyle X}$, ${\displaystyle \bigotimes _{1}^{n}{\mathcal {B}}_{X_{j}}\subset {\mathcal {B}}_{X}}$. Suppose ${\displaystyle C_{j}}$ is a countable dense set in ${\displaystyle X_{j}}$, and let ${\displaystyle {\mathcal {E}}_{j}}$ be the collection of balls in ${\displaystyle X_{j}}$ with rational radius and center in ${\displaystyle C_{j}}$. Then every open set in ${\displaystyle X_{j}}$ is a union of members of ${\displaystyle {\mathcal {E}}_{j}}$, which is countable. Moreover, the set of points in X whose jth coordinate is in ${\displaystyle C_{j}}$ for all j is a countable dense subset of X, and the balls of radius r in X are merely products of balls of radius r in the ${\displaystyle X_{j}}$'s. It follows that ${\displaystyle {\mathcal {B}}_{X_{j}}}$ is generated by ${\displaystyle {\mathcal {E}}_{j}}$ and ${\displaystyle {\mathcal {B}}_{X}}$ is generated by ${\displaystyle \{\prod _{1}^{n}E_{j}:E_{j}\in {\mathcal {E}}_{j}\}}$. Therefore, ${\displaystyle \bigotimes _{1}^{n}{\mathcal {B}}_{X_{j}}={\mathcal {B}}_{X}}$.

Corallary. ${\displaystyle {\mathcal {B}}_{\mathbb {R} ^{n}}=\bigotimes _{1}^{n}{\mathcal {B}}_{\mathbb {R} }}$

1. Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, Second Edition, §1.2