1 Borel σ-Algebra
To construct a measure for the real numbers \(\mathbb{R}\) we need to first construct a suitable \(\sigma\)-algebra, known informally as the Borel σ-algebra (of the real numbers). First, we recall the following definition of a Topology.
A topology \(\mathcal{T}\) on \(X\) is a collection of subsets of \(X\) that is closed under arbitrary unions and finite intersections. The elements of \(\mathcal{T}\) are open sets. The pair \((X,\mathcal{T})\) is a topological space.
Let \((X,\mathcal{T})\) be a topological space. The Borel σ-algebra of \(X\), \(\mathcal{B}_{X}\) is the \(\sigma\)-algebra generated by \(\mathcal{T}\). Its members are known as Borel sets.
To gain a deeper understanding of what the Borel sets look like we briefly consider the Borel hierarchy. In mathematical logic, the Borel hierarchy is a stratification of the Borel algebra. A countable intersection of open sets is called a \(G_{\delta}\) set; a countable union of closed sets is called an \(F_{\sigma}\) set; a countable union of \(G_{\delta}\) sets is called a \(G_{\delta\sigma}\) set; a countable intersection of \(F_{\sigma}\) sets is called an \(F_{\sigma\delta}\) set; and so forth.
Note: In mathematical etymology the \(\delta\) and \(\sigma\) stand for the German Durchschnitt and Summe, that is, intersection and union.
The Borel \(\sigma\)-algebra of \(\mathbb{R}\), \(\mathcal{B}_{\mathbb{R}}\) is generated by: 1. Open intervals \(\mathcal{E}_{1}:=\{ (a,b):a<b \}\); 2. Closed intervals \(\mathcal{E}_{2}:=\{ [a,b]:a<b \}\); 3. Half-open intervals \(\mathcal{E}_{3}:=\{ (a,b]:a<b \}\); 4. Open rays \(\mathcal{E}_{4}:=\{ (a, \infty):a\in \mathbb{R} \}\); 5. Closed rays \(\mathcal{E}_{5}:=\{ [a,\infty):a\in \mathbb{R} \}\).
PROOF: The Borel \(\sigma\)-algebra \(\mathcal{B}_{\mathbb{R}}:=\sigma(\mathcal{E}_{1})\) is generated from the set of open intervals \[
\mathcal{E}_{1}:=\{ (a,b):a<b \}.
\] (1) Considering arbitrary \((a,b]\in\mathcal{E}_{3}\) we note that since open sets are in \(\mathcal{B}_{\mathbb{R}}\) and \(\sigma\)-algebra are closed under countable intersection we have that \[
(a,b]=\bigcap_{n=1}^\infty \left( a,b+\frac{1}{n} \right)\in\mathcal{B}_{\mathbb{R}}\implies \mathcal{E}_{3}\subseteq\mathcal{B}_{\mathbb{R}\implies}\sigma(\mathcal{E}_{3})\subseteq \mathcal{B}_{\mathbb{R}}.
\] Similarly, considering arbitrary \((a,b)\in\mathcal{E}_{1}\)
\[
(a,b)=\bigcup_{n=1}^\infty \left(a,b-\frac{1}{n}\right]\in \sigma(\mathcal{E}_{3})\implies\mathcal{E}_{1}\subseteq \sigma(\mathcal{E}_{3})\implies\mathcal{B}_{\mathbb{R}}\subseteq \mathcal{E}_{3}.
\] (2) Considering arbitrary \((a,+\infty)\in \mathcal{E}_{5}\) we note that \[
(a,+\infty)=\bigcup_{n=1}^\infty (a,n)\in \mathcal{B}_{\mathbb{R}}\implies\mathcal{E}_{5}\subseteq\mathcal{B}_{\mathbb{R}}\implies \sigma(\mathcal{E}_{5})\subseteq\mathcal{B}_{\mathbb{R}}.
\] Similarly, considering arbitrary \((a,b)\in\mathcal{E}_{1}\) since \(\sigma\)-algebra are closed under countable intersections \[
\begin{align}
(a,b)&=(a,+\infty)\cap[b,+\infty)=(a,+\infty)\cap \bigcap_{n=1}^\infty \left( b-\frac{1}{n},+\infty \right)\in\sigma(\mathcal{E}_{5}) \\
\implies & \mathcal{E}_{1}\subseteq \sigma(\mathcal{E}_{5})\implies \mathcal{B}_{\mathbb{R}}\subseteq \sigma(\mathcal{E}_{5}).
\end{align}
\] (3) Considering arbitrary \([a,+\infty)\in\mathcal{E}_{7}\) we note that \[
\begin{align}
[a,+\infty)&=\bigcap_{n=1}^\infty[a,n)=\bigcap_{n=1}^\infty \bigcap_{m=1}^\infty\left( a-\frac{1}{m}, n \right)\in \mathcal{B}_{\mathbb{R}} \\
\implies & \mathcal{E}_{7}\subseteq \mathcal{B}_{\mathbb{R}}\implies \sigma(\mathcal{E}_{7})\subseteq\mathcal{B}_{\mathbb{R}}.
\end{align}
\] Similarly, considering arbitrary \((a,b)\in \mathcal{E}_{1}\), since \(\sigma\)-algebra are closed under countable unions and complements we have that \[
\begin{align}
(a,b)&=(-\infty,b)\cap(-\infty,a)=([b,+\infty)^c\cup[a,+\infty)^c)^c\subseteq \sigma(\mathcal{E}_{7}) \\
\implies & \mathcal{E}_{1}\subseteq \sigma(\mathcal{E}_{7})\implies\mathcal{B}_{\mathbb{R}}\subseteq \sigma(\mathcal{E}_{7}).
\end{align}
\] \(\square\)