1 Borel Measure
(\(\star\)) Definition (Borel Measure): A measure \(\mu:\mathcal{B}_{\mathbb{R}}\to[0,+\infty]\) mapping from the Borel set on \(\mathbb{R}\) \(\mathcal{B}_{\mathbb{R}}\) are named Borel measures on \(\mathbb{R}\).
Our aim for this section is to construct the family of Borel measures. Our procedure will be to construct a measure \(\mu\) starting from an increasing, 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