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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 integral1. 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 \[\int_a^b f(x)~d\alpha(x) = \int_a^b f(x) \alpha'(x)~dx .\]



Directly from the definition of the Riemann-Stieltjes integral prove that if \(\alpha(x) = px\) for some \(p \geq 0\), then If \(f\) is Riemann integrable, then it is Riemann-Stieltjes integrable with respect to \(\alpha\) and \(p \int_a^b f = \int_a^b f~d\alpha\).

Let \(\alpha \colon [a,b] \to \R\) and \(\beta \colon [a,b] \to \R\) be increasing functions and suppose that \(\alpha(x) = \beta(x) + C\) for some constant \(C\). If \(f \colon [a,b] \to \R\) is integrable with respect to \(\alpha\), show that it is integrable with respect to \(\beta\) and \(\int_a^b f~d\alpha = \int_a^b f~d\beta\).

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