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Chapter 6: The Riemann-Stieltjes integral
The present chapter is based on a definition of the Riemann integral which depends very explicitly on the order structure of the real line. Accordingly, we begin by discussing integration of real-valued functions on intervals. Extensions to complex- and vector-valued functions on intervals follow in later sections. Integration over sets other than intervals is discussed in Chap. 10 (and 11 if possible).
Contents
Contents
1. Definition and existence of the integral
2. Properties of the integral
3. Integration and differentiation
4. Integration of vector-valued functions
5. Rectifiable curves
1. Definition and existence of the integral
\begin{equation}
{\overline\int_a^b f}\, dx=\inf U(P,f),\quad
{\underline\int_a^b f}\, dx=\sup L(P,f).
\end{equation}
where the inf and the sup are taken over all partitions P of [a, b]. The left members of (61) are called the \textbf{upper} and \textbf{lower Riemann integrals} of f over [a, b], respectively.
If the upper and lower integrals are equal, we say that f is \textbf{Riemann-integrable} on [a, b], we write f\in\mathcal{R} (that is, \mathcal{R} denotes the set of Riemann-integrable functions), and we denote the common value of (61) by
\[\int_{a}^{b}f\, dx,\, \text{or by }\int_{a}^{b}f(x)\, dx.\]
This is the \textbf{Riemann integral} of f over [a, b].
Since f is bounded, there exist two numbers, m and M, such that m\leqslant f(x)\leqslant M\, (a\leqslant x\leqslant b). Hence, for every P, m(b-a)\leqslant L(P,f)\leqslant U(P,f)\leqslant M(b-a), so that the numbers L(P,f) and U(P,f) form a bounded set. This shows that the upper and lower integrals are defined for every bounded function f. The question of their equality, and hence the question of the integrability of f, is a more delicate one. Instead of investigating it separately for the Riemann integral, we shall immediately consider a more general situation.
\begin{equation}
{\overline\int_a^b f}\, d\alpha=\inf U(P,f,\alpha),\quad
{\underline\int_a^b f}\, d\alpha=\sup L(P,f,\alpha).
\end{equation}
The inf and sup again being taken over all partitions.
If the left members of (62) are equal, we denote their common value by
\begin{equation}
\int_{a}^{b}f\, d\alpha,\, \text{or by }\int_{a}^{b}f(x)\, d\alpha(x).
\end{equation}
This is the \textbf{Riemann-Stieltjes integral} (or simply the \textbf{Stieltjes integral}) of f with respect to \alpha, over [a, b].
If (63) exists, i.e., if the two of (62) are equal, we say that f is integrable with respect to \alpha, in the Riemann sense, and write f\in\mathcal{R}(\alpha).
By taking \alpha(x) = x, the Riemann integral is seen to be a special case of the Riemann-Stieltjes integral. Let us mention explicitly, however, that in the general case \alpha need not even be continuous.
A few words should be said about the notation. We prefer the former to the latter in (62), since the letter x which appears in the latter adds nothing to the content of the former. It is immaterial which letter we use to represent the so-called "variable of integration." The integral depends on f, \alpha, a and b, but not on the variable of integration, which may as well be omitted.
The role played by the variable of integration is quite analogous to that of the index of summation: The two symbols \sum c_i,\sum c_k mean the same thing, since each means c_1 + c_2 + \dots + c_n.
Of course, no harm is done by inserting the variable of integration, and in many cases it is actually convenient to do so.
We shall now investigate the existence of the integral (63). Without saying so every time, f will be assumed real and bounded, and \alpha monotonically increasing on [a, b]; and, when there can be no misunderstanding, we shall write \int in place of \int_{a}^{b}.
\begin{equation}
U(p,f,\alpha)-L(P,f,\alpha)<\varepsilon.
\end{equation}
This theorem furnishes a convenient criterion for integrability. Before we apply it, we state some closely related facts.
- If (64) holds for some P and some \varepsilon, then (64) holds (with the same \varepsilon) for every refinement of P.
- If (64) holds for P=\{x_0,\dots,x_n\} and if s_i,t_i\in[x_{i-1},x_i], then \sum |f(s_i)-f(t_i)|\Delta\alpha_i<\varepsilon.
- If f\in\mathcal{R}(\alpha) and the hypotheses of (2) hold, then
\[\Big|\sum_{i=1}^{n}f(t_i)\Delta\alpha_i-\int_{a}^{b}f\, d\alpha\Big|<\varepsilon.\]
Note: If f and \alpha have a common point of discontinuity, then f need not be in \mathcal{R}(\alpha).
\textbf{Proof. }Theorem theorem617. Let \varepsilon>0 be given. Put M=\sup f(x), let E be the set of points at which f is discontinuous, Since E is finite and \alpha is continuous at every point of E, we can cover E by finitely many disjoint intervals [u_i,u_j]\subset[a,b] such that the sum of the corresponding differences \alpha(v_j)-\alpha(u_j) is less than \varepsilon. Furthermore, we can place these intervals in such a way that every point of E\cap(a,b) lies in the interior of some [u_j,v_j]. Remove the segments (u_j , v_j) from [a, b]. The remaining set K is compact. Hence f is uniformly continuous on K, and there exists \delta > 0 such that |f(s)-f(t)|<\varepsilon if s\in K, t\in K, |s-t|<\delta.
Now form a partition P=\{x_0,\dots,x_n\} of [a,b], as follows: Each u_j occurs in P. Each v_j occurs in P. No point of any segment (u_j , v_j) occurs in P. If x_{i-1} is not one of the u_j , then \Delta x_i<\delta.
Note that M_i-m_i\leqslant 2M for every i, and that M-i-m_i\leqslant\varepsilon unless x_{i-1} is one of the u_j. Hence, U(P,f,\alpha)-L(P,f,\alpha)\leqslant[\alpha(b)-\alpha(a)]\varepsilon+2M\varepsilon. Since \varepsilon is arbitrary, f\in\mathcal{R}(\alpha).
Theorem theorem618. Choose \varepsilon>0. There exists \delta>0 such that \delta<\varepsilon and |phi(s)-\phi(t)|<\varepsilon if |s-t|\leqslant\delta and s,t\in[m,M]. Since f\in\mathcal{R}(\alpha), there is a partition P=\{x_0,\dots,x_n\} of [a,b] such that U(P,f,\alpha)-L(P,f,\alpha)<\delta^2. Let M_i,m_i be the same meaning of Definition 62, and let M^*_i,m^*_i be the analogous numbers for h. Divide the numbers 1, \dots , n into two classes: i\in A if M_i-m_i<\delta, i\in B if M_i-m_i\geqslant\delta.
For i\in A, our choice of \delta shows that M^*_i-m^*_i\leqslant\varepsilon. For i\in B, M^*_i-m^*_i\leqslant2K, where K=\sup|\phi(t)|, m\leqslant t\leqslant M. We have \delta\sum_{i\in B}\Delta\alpha_i\leqslant\sum_{i\in B}(M_i-m_i)\Delta\alpha_i<\delta^2 so that \sum_{i\in B}\Delta\alpha_i<\delta. It follows that
\begin{align*}
U(P,h,\alpha)-L(P,h,\alpha) & =\sum_{i\in A}(M^*_i-m^*_i)\Delta\alpha_i+\sum_{i\in B}(M^*_i-m^*_i)\Delta\alpha_i \\
& \leqslant\varepsilon[\alpha(b)-\alpha(a)]+2K\delta<\varepsilon[\alpha(b)-\alpha(a)+2K].
\end{align*}
Since \varepsilon is arbitrary, h\in\mathcal{R}(\alpha).
2. Properties of the integral
- If f_1\in\mathcal{R}(\alpha) and f_2\in\mathcal{R}(\alpha) on [a,b], then f_1+f_2\in\mathcal{R}(\alpha), cf\in\mathcal{R}(\alpha) for every constant c, and
\[\int_{a}^{b}(f_1+f_2)\, d\alpha=\int_{a}^{b}f_1\, d\alpha+\int_{a}^{b}f_2\, d\alpha,\quad \int_{a}^{b}cf\, d\alpha=c\int_{a}^{b}f\, d\alpha.\]
- If f_1(x)\leqslant f_2(x) on [a,b], then
\[\int_{a}^{b}f_1\, d\alpha\leqslant\int_{a}^{b}f_2\, d\alpha.\]
- If f\in\mathcal{R}(\alpha) on [a,b] and if a<c<b, then f\in\mathcal{R}(\alpha) on [a,c] and on [c,b], and
\[\int_{a}^{c}f\, d\alpha+\int_{c}^{b}f\, d\alpha=\int_{a}^{b}f\, d\alpha.\]
- If f\in\mathcal{R}(\alpha) on [a,b] and if |f(x)|\leqslant M on [a,b], then
\[\Big|\int_{a}^{b}f\, d\alpha\Big|\leqslant M[\alpha(b)-\alpha(a)].\]
- If f\in\mathcal{R}(\alpha_1) and f\in\mathcal{R}(\alpha_2), then f\in\mathcal{R}(\alpha_1+\alpha_2) and
\[\int_{a}^{b}f\, d(\alpha_1+\alpha_2)=\int_{a}^{b}f\, d\alpha_1+\int_{a}^{b}f\, d\alpha_2;\]
If f\in\mathcal{R}(\alpha) and c is a positive constant, then f\in\mathcal{R}(c\alpha) and
\[\int_{a}^{b}f\, d(c\alpha)=c\int_{a}^{b}f\, d\alpha.\]
- fg\in\mathcal{R}(\alpha);
- |f|\in\mathcal{R}(\alpha) and \displaystyle\Big|\int_{a}^{b}f\, d\alpha\Big|\leqslant\int_{a}^{b}|f|\, d\alpha.
I(x)=\begin{cases}
0,&x\leqslant0;\\
1,&x>0.
\end{cases}
\[\int_{a}^{b}f\, d\alpha=\sum_{n=1}^{\infty}c_nf(s_n).\]
\[\int_{a}^{b}f\, d\alpha=\int_{a}^{b}f(x)\alpha'(x)\, dx.\]
The two preceding theorems illustrate the generality and flexibility which are inherent in the Stieltjes process of integration. If \alpha is a pure step function, the integral reduces to a finite or infinite series. If \alpha has an integrable derivative, the integral reduces to an ordinary Riemann integral. This makes it possible in many cases to study series and integrals simultaneously, rather than separately.
\[\int_{A}^{B}g\, d\beta=\int_{a}^{b}f\, d\alpha.\]
Let us note the following special case: Take \alpha(x) = x. Then \beta = \varphi. Assume \varphi^\prime\in\mathcal{R} on [A, B]. If Theorem theorem625 is applied, we obtain
\[\int_{a}^{b}f(x)\, dx=\int_{A}^{B}f(\varphi(y))\varphi'(y)\, dy.\]
3. Integration and differentiation
We still confine ourselves to real functions in this section. We shall show that integration and differentiation are, in a certain sense, inverse operations.
\[\int_{a}^{b}f(x)\, dx=F(b)-F(a).\]
\[\int_{a}^{b}F(x)g(x)\, dx=F(b)G(b)-F(a)G(a)-\int_{a}^{b}f(x)G(x)\, dx.\]
4. Integration of vector-valued functions
It is clear that parts (1), (3), and (5) of Theorem theorem621 are valid for these vector-valued integrals; we simply apply the earlier results to each coordinate. The same is true of Theorems theorem625, theorem631, and theorem632. To illustrate, we state the analogue of Theorem theorem632.
\begin{equation}
\Big|\int_{a}^{b}\mathbf{f}\, d\alpha\Big|\leqslant\int_{a}^{b}|\mathbf{f}|\, d\alpha.
\end{equation}
\textbf{Proof. }By Theorem theorem622, \mathbf{f}\in\mathcal{R}(\alpha). To prove (65), put \mathbf{y}=(y_1,\dots,y_k), where y_j=\int f_j\, d\alpha. Then we have \mathbf{y}=\int \mathbf{f}\, d\alpha, and
\[|\mathbf{y}|^2=\sum y_i^2=\sum y_j\int f_j\, d\alpha=\int\left(\sum y_jf_j\right)\, d\alpha.\]
By the Schwarz inequality,
\[\sum y_jf_j(t)\leqslant|\mathbf{y}||\mathbf{f}(t)|\; (a\leqslant t\leqslant b)\quad\Longrightarrow\quad |\mathbf{y}|^2\leqslant|\mathbf{y}|\int|\mathbf{f}|\, d\alpha.\]
If \mathbf{y}=\mathbf 0, (65) is trivial. If \mathbf{y}\neq\mathbf 0, division of this by |\mathbf{y}| gives (65).
5. Rectifiable curves
We conclude this chapter with a topic of geometric interest which provides an application of some of the preceding theory. The case k = 2 (i.e., the case of plane curves) is of considerable importance in the study of analytic functions of a complex variable.
We associate to each partition P=\{x_0,\dots,x_n\} of [a,b] and to each curve \gamma on [a,b] the number \Lambda(P,\gamma)=\sum |\gamma(x_i)=\gamma(x_{i-1}|. It seems reasonable to define the length of \gamma as \Lambda(\gamma)=\sup \Lambda(P,\gamma), where the supremum is taken over all partitions of [a,b]. If \Lambda(\gamma)<\infty, we say that \gamma is rectifiable.
In certain cases, \Lambda(\gamma) is given by a Riemann integral. We shall prove this for continuously differentiable curves, i.e., for curves \gamma whose derivative \gamma^\prime is continuous.
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