Riemann-Stieltjes integral
FIXME: we’d need to redo a bunch of things from Riemann integral. Perhaps useful, but those are missing below and sort of make this more and more out of scope of the book.
A common useful generalization of the Riemann integral is the Riemann-Stieltjes integral. If we think of the Riemann integral as a sum where all terms are weighted equally, it is natural that we may want to do a weigthed sum. That is, we may wish to give some points “more weight” than to other points. A particular simple example of what we might want to accomplish is an integral which evaluates a function at a point. You may have seen this concept in your calculus class as the delta function.
We will again define this integral using the Darboux approach for simplicity.
Let \(f \colon [a,b] \to \R\) be a bounded function and let \(\alpha \colon [a,b] \to \R\) be a monotone increasing function. Let \(P\) be a partition of \([a,b]\), then define \[\begin{aligned}
& m_i := \inf \{ f(x) : x_{i-1} \leq x \leq x_i \} , \\
& M_i := \sup \{ f(x) : x_{i-1} \leq x \leq x_i \} , \\
& L(P,f,\alpha) :=
\sum_{i=1}^n m_i \bigl( \alpha(x_i) - \alpha(x_{i-1}) \bigr) , \\
& U(P,f,\alpha) :=
\sum_{i=1}^n M_i \bigl( \alpha(x_i) - \alpha(x_{i-1}) \bigr) .\end{aligned}\] We call \(L(P,f,\alpha)\) the and \(U(P,f,\alpha)\) the . Then define \[\begin{aligned}
& \underline{\int_a^b} f~d\alpha := \sup \{ L(P,f,\alpha) : P \text{ a
partition of $[a,b]$} \} , \\
& \overline{\int_a^b} f~d\alpha := \inf \{ U(P,f,\alpha) : P \text{ a
partition of $[a,b]$} \} .\end{aligned}\] And we call \(\underline{\int}\) the and \(\overline{\int}\) the . Finally, if \[\underline{\int_a^b} f~d\alpha = \overline{\int_a^b} f~d\alpha .\] Then we say that \(f\) is with respect to \(\alpha\).
When we need to specify the variable of integration we may write \[\int_a^b f(x) ~d\alpha(x) .\]
When we set \(\alpha(x) := x\) we recover the Riemann integral. The notation \(d\alpha\) suggests derivative, in this case \(\alpha'(x) = 1\) and as we said, the Riemann integral is when all points are weighted equally.
If \(\alpha(x) := x\), then a bounded function \(f \colon [a,b] \to \R\) is Riemann integrable if and only if it is Riemann-Stieltjes integrable with respect to \(\alpha\). In this case \[\int_a^b f = \int_a^b f~d\alpha .\]
Simply plug in \(\alpha(x) = x\) into the definition and note that the definition is now precisely the same as for the Riemann integral.
Suppose that \(f \colon [a,b] \to \R\) is continuous. Given \(c \in (a,b)\), let \[\alpha(x) :=
\begin{cases}
1 & \text{if $x \geq c$,} \\
0 & \text{if $x < c$.}
\end{cases}\] We claim that \(f\) is Riemann-Stieltjes differentiable with respect to \(\alpha\) and that \[\int_a^b f~d\alpha = f(c) .\]
Proof: Given \(\epsilon > 0\) take \(\delta > 0\) such that \(\abs{f(x)-f(c)} < \epsilon\) for all \(x \in [a,b]\) with \(\abs{x-c} < \delta\). Take the partition \(P = \{ a , c-\delta,
c+\delta, b \}\). Then \[\begin{split}
L(P,f,\alpha)
& =
m_1 \bigl( \alpha(c-\delta) - \alpha(a) \bigr)
+
m_2 \bigl( \alpha(c+\delta) - \alpha(c-\delta) \bigr)
+
m_3 \bigl( \alpha(b) - \alpha(c+\delta) \bigr)
\\
& =
m_2 \bigl( 1 - 0 ) = m_2 = \inf \{ f(x) : x \in [c-\delta,c+\delta] \}
\\
& >
f(c) - \epsilon .
\end{split}\] Similarly \(U(P,f,\alpha) < f(c)+\epsilon\). Therefore \[U(P,f,\alpha)-L(P,f,\alpha) < 2 \epsilon .\]
The notion of of integrability really does depend on \(\alpha\). For a very trivial example, it is not difficult to see that if \(\alpha(x) = 0\), then all bounded functions \(f\) on \([a,b]\) are integrable with respect to this \(\alpha\) and \[\int_a^b f~d \alpha = 0.\]
If \(\alpha\) is very nice, we can recover the Riemann-Stieltjes integral using the Riemann integral.
Suppose that \(f \colon [a,b] \to \R\) is Riemann integrable and \(\alpha \colon [a,b] \to \R\) is a continuously differentiable increasing function. Then \(f\) is Riemann-Stieltjes integrable with respect to \(\alpha\) and \[\int_a^b f(x)~d\alpha(x) = \int_a^b f(x) \alpha'(x)~dx .\]
FIXME