# Borel sigma-algebra

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

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

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

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

## Product Borel $\sigma$ -algebra

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

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

Corallary. ${\mathcal {B}}_{\mathbb {R} ^{n}}=\bigotimes _{1}^{n}{\mathcal {B}}_{\mathbb {R} }$ 1. Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, Second Edition, §1.2