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: ${\mathcal {E}}_{1}=\{(a,b):a<b\}$,
• b. the closed intervals: ${\mathcal {E}}_{2}=\{[a,b]:a<b\}$,
• c. the half-open intervals: ${\mathcal {E}}_{3}=\{(a,b]:a<b\}$, or ${\mathcal {E}}_{4}=\{[a,b):a<b\}$,
• 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. ${\mathcal {B}}_{\mathbb {R} ^{n}}=\bigotimes _{1}^{n}{\mathcal {B}}_{\mathbb {R} }$