Riemann-Stieltjes Integration

Section 4.4 Riemann-Stieltjes Integration

Subsection 4.4.1 Riemann integral

Recall that the Riemann integral can be defined as a limit of Riemann sums, as follows. Let \(f\) be a function on an interval \([a,b]\) and let \(P = \{x_0,x_1,\dotsc,x_n\}\) be a partition of \([a,b]\text{,}\) so that

\begin{equation} a = x_0 \lt x_1 \lt \dotsb \lt x_n = b\text{.}\tag{4.4.1} \end{equation}

The norm of \(P\) is \(\lvert P \rvert = \max\{x_i-x_{i-1}\}\text{.}\) Let \(T = \{t_1,\dotsc,t_n\}\) be a tag of the partition \(P\text{,}\) meaning that for each \(i\text{,}\) \(x_{i-1} \leq t_i \leq x_i\text{.}\) The pair \((P,T)\) is called a tagged partition. The Riemann sum for this data is

\begin{equation} S(P,T,f) = \sum_{i=1}^n f(t_i)(x_i - x_{i-1})\text{.}\tag{4.4.2} \end{equation}

The Riemann integral \(\int_a^b f dt = L\text{,}\) and \(f\) is Riemann integrable on the interval \([a,b]\text{,}\) if the limit of Riemann sums converges to \(L\text{,}\) in the sense that for every \(\epsilon \gt 0\) there exists \(\delta \gt 0\) such that for every tagged partition \((P,T)\) of norm less than \(\delta\text{,}\) the Riemann sum \(S(P,T,f)\) is within \(\delta\) of \(L\text{.}\)

The Riemann integral enjoys various properties, for example:

The Riemann integral is linear: \(\int_a^b (pf+qg) dt = p \int_a^b f dt + q \int_a^b g dt \text{.}\)
Every continuous function is Riemann integrable.
If \(f\) and \(g\) are continuously differentiable on \([a,b]\text{,}\) then integration by parts is valid:
\begin{equation*} \int_a^b f(t)g'(t) dt = \left. f(t)g(t)\right|_a^b - \int_a^b f'(t)g(t) dt\text{.} \end{equation*}

Riemann integration also has some limitations. For example, if \(f\) is unbounded on \([a,b]\) then \(f\) is not Riemann integrable (the Riemann sums do not converge). (It is possible to assign values to improper integrals via a limiting process.)

Subsection 4.4.2 Riemann-Stieltjes integration

Let \(f\) and \(g\) be functions on an interval \([a,b]\text{.}\) We do not assume they are monotone, positive, differentiable, or even continuous. Let \((P,T)\) be a tagged partition of the interval as before. To this data we assign the Riemann-Stieltjes sum

\begin{equation} S(P,T,f,g) = \sum_{i=1}^n f(t_i)(g(x_i)-g(x_{i-1}))\text{.}\tag{4.4.3} \end{equation}

If these approach a value as before (for partitions of sufficiently small norm, the sum should be arbitrarily close to a value \(L\)) then this limit is the Riemann-Stieltjes integral \(\int_a^b f dg\text{.}\) If this exists we say \(f\) is Riemann(-Stieltjes) integrable with respect to \(g\) on \([a,b]\text{.}\) We write \(f \in R(g)\text{,}\) the set of functions integrable with respect to \(g\text{.}\)

Riemann-Stieltjes integration enjoys some similar properties to those of Riemann integration.

Theorem 4.4.1.

Assume all functions are defined and bounded on \([a,b]\text{.}\)

Riemann-Stieltjes integration is linear as a function of \(f\text{:}\) If \(f_1,f_2 \in R(g)\) on \([a,b]\) and \(c_1,c_2\) are constants, then \(c_1 f_1 + c_2 f_2 \in R(g)\) on \([a,b]\text{,}\) and
\begin{equation} \int_a^b (c_1 f_1 + c_2 f_2) dg = c_1 \int_a^b f_1 dg + c_2 \int_a^b f_2 dg\text{.}\tag{4.4.4} \end{equation}
Likewise, Riemann-Stieltjes integration is linear as a function of \(g\text{:}\) if \(f \in R(g_1)\) and \(f \in R(g_2)\) on \([a,b]\) and \(g = c_1 g_1 + c_2 g_2\) where \(c_1,c_2\) are constants, then \(f \in R(g)\) and
\begin{equation} \int_a^b f dg = \int_a^b f d(c_1 g_1 + c_2 g_2) = c_1 \int_a^b f dg_1 + c_2 \int_a^b f dg_2\text{.}\tag{4.4.5} \end{equation}
If \(f\) is continuous on \([a,b]\) and \(g\) is of bounded variation on \([a,b]\text{,}\) then the Riemann-Stieltjes integral \(\int_a^b f dg\) exists. In particular, the class of functions of bounded variation includes all monotone functions, and all difference of monotone functions. Thus if \(g\) is monotone (increasing or decreasing), or \(g = g_1 - g_2\) where each of \(g_1,g_2\) is monotone, and \(f\) is continuous, then the integral exists.
The Riemann-Stieltjes integral satisfies integration by parts: If \(\int_a^b f dg\) exists, then \(\int_a^b g df\) exists, and
\begin{equation} \int_a^b g df = \left. fg \right|_a^b - \int_a^b f dg\text{.}\tag{4.4.6} \end{equation}
If \(f\) is Riemann integrable on \([a,b]\) and \(g'\) is continuous on \([a,b]\text{,}\) then
\begin{equation} \int_a^b f dg = \int_a^b fg' dx\tag{4.4.7} \end{equation}
(here the right hand side is a Riemann, not Riemann-Stieltjes, integral).

A function \(g\) has bounded variation on an interval \([a,b]\) if it satisfies the following condition. First, for a partition \(P = \{x_0,\dotsc,x_n\}\) of \([a,b]\text{,}\) the variation of \(g\) with respect to \(P\) is the sum

\begin{equation} V(P,g) = \sum_{i=1}^n |g(x_i)-g(x_{i-1})|\tag{4.4.8} \end{equation}

(note the absolute value signs). The variation of \(g\) on \([a,b]\) is the supremum of the variation over all partitions of the interval. We say \(g\) has bounded variation if this supremum is finite. (It is an easy exercise to show that monotone functions have bounded variation; and not too hard to show that sums and differences of functions of bounded variation have again bounded variation. However, not every continuous function has bounded variation!)

Subsection 4.4.3 Step functions

A function \(g\) on \([a,b]\) is called a step function if there is a partition \(a = x_0 \lt x_1 \lt \dotsb \lt x_n = b\) such that \(g\) is constant on each interval \((x_{i-1},x_i)\text{.}\) (In the terminology of Lebesgue integration, these are simple functions, with the restriction that the level sets are intervals, as opposed to more general measurable sets.) The jump of \(g\) at each point \(x\) is defined as the difference between the one-sided limits at \(x\text{,}\) except if \(x\) is one of the endpoints \(a,b\text{,}\) in which case the jump is the difference between the one-sided limit and the value at that point.

Every step function has bounded variation. (The variation is equal to the sum of the absolute values of the jumps.)

Every step function is a difference of monotone increasing functions. Here is a simple construction which almost achieves this: Let \(g_1(x)\) be the sum of all the positive jumps of \(g\) between \(a\) and \(x\text{,}\) and let \(g_2\) be the negative of the sum of all the negative jumps of \(g\) between \(a\) and \(x\text{.}\) Then \(g_1\) and \(g_2\) are monotone increasing, and \(g(x) = g(a) + g_1(x) - g_2(x)\) for all \(x\) which are not jump points of \(g\text{;}\) that is, for all \(x\) other than the points \(x_i\) in the partition. This simple construction does not give fine control over the values at the jump points. It should be possible to get precisely the step function \(g\) at all \(x\) by using one-sided limits more carefully.

The reason we are interested in step functions is that Riemann-Stieltjes integration with respect to a step function is equivalent to a sum. Of course our main interest is in evaluating sums by considering them as integrals (with respect to step functions).

Theorem 4.4.2.

Let \(g\) be a step function on \([a,b]\) with respect to a partition \(a = x_1 \lt x_1 \lt \dotsb \lt x_n = b\text{.}\) For each \(k\) let \(g_k\) be the jump of \(g\) at \(x_k\text{.}\) Let \(f\) be defined on \([a,b]\text{.}\) Assume that for each \(k\text{,}\) it is not the case that both \(f\) and \(g\) are discontinuous from the left at \(x_k\text{,}\) or both discontinuous from the right at \(x_k\text{.}\) (It is allowed for \(f\) to be discontinuous from one side, and \(g\) from the other side; or for \(g\) to be discontinuous from both sides, if \(f\) is continuous; or vice versa.) Then \(\int_a^b f dg\) exists and

\begin{equation} \int_a^b f dg = \sum_{k=1}^n f(x_k) g_k\text{.}\tag{4.4.9} \end{equation}

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