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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 $\int_a^b f(x)~d\alpha(x) = \int_a^b f(x) \alpha'(x)~dx .$

FIXME

## Exercises

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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