Borel Sigma-Field

Definition

$\mathcal{R}=\mathcal{B}(X):=\sigma (\mathcal{T})$

Let $(X,\mathcal{T})$ be a Topological Space, where $X$ is the set and $\mathcal{T}$ is the topology on $X$. The Borel sigma-field $\mathcal{B}(X)$ is the smallest Sigma-Field that contains all open sets in $\mathcal{T}$.

Borel Sigma-Field on the Real Number

$\mathcal{R}=\mathcal{B}(\mathbb{R}):=\sigma (\mathcal{U})$ where $\mathcal{U}$ is Usual Topology on $\mathbb{R}$.

Borel Sigma-Field on the Extended Real Number Space

$\bar{\mathcal{R}}=\mathcal{B}(\bar{\mathbb{R}})$ where $\bar{\mathbb{R}}=\mathbb{R}\cup \{-\infty ,\infty \}$.

Facts

Every element of $\mathcal{B}(\mathbb{R})$ is measurable by Lebesgue Measure

• ${\mathcal{A}}_1:=\{A\subset \mathbb{R}:A\text{ is U-open}\}=\mathcal{U}$
• ${\mathcal{A}}_2:=\{(a,b):a,b\in \mathbb{R},a<b\}$
• ${\mathcal{A}}_3:=\{[a,b):a,b\in \mathbb{R},a<b\}$
• ${\mathcal{A}}_4:=\{(a,b]:a,b\in \mathbb{R},a<b\}$
• ${\mathcal{A}}_5:=\{[a,b]:a,b\in \mathbb{R},a<b\}$
• ${\mathcal{A}}_6:=\{(-\infty ,b):a,b\in \mathbb{R},a<b\}$
• ${\mathcal{A}}_7:=\{(-\infty ,b]:a,b\in \mathbb{R},a<b\}$
• ${\mathcal{A}}_8:=\{(a,\infty ):a,b\in \mathbb{R},a<b\}$
• ${\mathcal{A}}_9:=\{[a,\infty ):a,b\in \mathbb{R},a<b\}$

$\mathcal{R}:=\mathcal{B}(\mathbb{R})=\sigma ({\mathcal{A}}_1)=\sigma ({\mathcal{A}}_2)=\cdots =\sigma ({\mathcal{A}}_9)$ The above sets are all the Borel sigma-field.

Proof Every element of $\sigma (\mathcal{A})$ is generated by the elements of $\mathcal{A}$ and the complementary set and countable union by the conditions of Sigma-Field If $\sigma ({\mathcal{A}}_1)$ can be generated by ${\mathcal{A}}_1$ and the operations, and ${\mathcal{A}}_1$ can be generated by ${\mathcal{A}}_2$ and the operations, then ${\mathcal{A}}_2$ also generate $\sigma ({\mathcal{A}}_1)$ Therefore, $\sigma ({\mathcal{A}}_1)=\sigma ({\mathcal{A}}_2)$

An element of Borel sigma-field $B\in \mathcal{B}(X)$ is called Borel set.