Borel σ-Algebra and Measure
1 Borel σ-Algebra
To construct a measure for the real numbers \(\mathbb{R}\) we need to first construct a suitable \(\sigma\)-algebra, known as the Borel σ-algebra (of the real numbers). First, we recall the following definition of a Topology, a collection of open sets.
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.
2 Generating Sets of the Borel σ-Algebra
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:
- Open intervals \(\mathcal{E}_{1}:=\{ (a,b):a<b \}\);
- Closed intervals \(\mathcal{E}_{2}:=\{ [a,b]:a<b \}\);
- Half-open intervals \(\mathcal{E}_{3}:=\{ (a,b]:a<b \}\);
- Open rays \(\mathcal{E}_{4}:=\{ (a, \infty):a\in \mathbb{R} \}\);
- 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 \[
\begin{align}
(a,b)&=(-\infty,b)\cap(-\infty,a)=([b,+\infty)^c\cup[a,+\infty)^c)^c\subseteq \sigma(\mathcal{E}_{7}) \\
\implies & \mathcal{E}_{1}\subseteq \sigma(\mathcal{E}_{7})\implies\mathcal{B}_{\mathbb{R}}\subseteq \sigma(\mathcal{E}_{7}).
\end{align}
\]
3 Borel Measure
A measure \(\mu:\mathcal{B}_{\mathbb{R}}\to[0,+\infty]\) are named Borel measures on \(\mathbb{R}\).
We wish to construct the family of Borel measures. Our procedure will be to construct a measure \(\mu\) starting from an increasing, [[right-and-left-continuity|right-continuous]] function \(F\) using the building blocks of left-open,right-closed intervals in \(\mathbb{R}\) i.e. sets of the form \((a,b]\) or \((a,\infty)\) or \(\emptyset\) where \(-\infty \leq a<b<\infty\), referred to as \(h\)-intervals (half-open intervals).
Some preliminary observations, clearly the intersection of two \(h\)-intervals is a \(h\)-interval and the complement of a \(h\)-interval is either a \(h\)-interval or the disjoint union of two \(h\)-intervals. Further, the collection \(\mathcal{A}\) if finite disjoint unions of \(h\)-intervals is an algebra and the \(\sigma\)-algebra generated by \(\mathcal{A}\) is \(\mathcal{B}_{\mathbb{R}}\).
Proposition: Let \(F:\mathbb{R} \to \mathbb{R}\) be increasing and right-continuous. If \((a_{j}, b_{j}]\) for \(j=1,\dots,n\) are disjoint \(h\)-intervals, define \[ \mu_{0}\left( \bigcup_{j=1}^n (a_{j}, b_{j}] \right)=\sum_{j=1}^n (F(b_{j})-F(a_{j})), \] and \(\mu_{0}(\emptyset)=0\). Then \(\mu_{0}\) is a premeasure on the algebra \(\mathcal{A}\).
(\(\star\)) Theorem: For increasing, right continuous function \(F:\mathbb{R} \to \mathbb{R}\) there is a unique Borel measure \(\mu_{F}\) on \(\mathbb{R}\) such that \(\mu_{F}((a,b])=F(b)-F(a)\) for all \(a,b\) known as the Lebesgue-Stieltjes measure. If \(G\) is another such function, we have \(\mu_{F}=\mu_{G}\) if and only if \(F-G\) is constant. Conversely, if \(\mu\) is a Borel measure on \(\mathbb{R}\) that is finite on all bounded Borel sets and we define \[ F(x)=\begin{cases} \mu((0,x]) & \text{if }x>0 \\ 0 & \text{if }x = 0 \\ -\mu((-x,0]) & \text{if }x<0, \end{cases} \] then \(F\) is increasing and right continuous, and \(\mu=\mu_{F}\).
Note that this theorem could equally well be developed by using intervals of the form \([a,b)\) and left continuous function \(F\). Also note that \(\mu\) is a finite Borel measure on \(\mathbb{R}\) then \(\mu=\mu_{F}\) where \(F(x)=\mu((-\infty,x])\) is the cumulative distribution function of \(\mu\).
Furthermore, the theory of outer measures gives, for each increasing and right continuous \(F\), not only the Borel measure \(\mu_{F}\) but a complete measure \(\overline{\mu}_{F}\) whose domain includes \(\mathcal{B}_{\mathbb{R}}\). In fact, \(\overline{\mu}_{F}\) is just the completion of \(\mu_{F}\) and one can show that its domain is always strictly larger than \(\mathcal{B}_{\mathbb{R}}\). This complete measure is often also denoted \(\mu_{F}\) and is called the Lebesgue-Stieltjes measure.
The Lebesgue measure is the Lebesgue-Stieltjes measure \(\lambda=\lambda_{F}\) associated to the function \(F(x)=x\). The domain of \(\lambda\) is the class of Lebesgue measurable sets
UPDATE NOTES FROM END OF FOLLAND CHAPTER 1