The Riemann–Stieltjes Integral | Sebastián Fernández Rivera

Sums and definition of the integral

Definition (Riemann–Stieltjes sum)

Let $f, \phi : [a,b] \to \mathbb{R}$ with $f$ bounded. Given a partition $\Gamma = \{a = x_{0} < x_{1} < \dots < x_{n} = b\}$ and intermediate points $\xi_{i} \in [x_{i-1}, x_{i}]$ for $1 \leq i \leq n$, the Riemann–Stieltjes sum of $f$ with respect to $\phi$ is

\[R(f, \Gamma, \phi) = \sum_{i=1}^{n} f(\xi_{i})\,\big[\phi(x_{i}) - \phi(x_{i-1})\big].\]

Definition (Riemann–Stieltjes integral)

We say that $f$ is Riemann–Stieltjes integrable with respect to $\phi$ on $[a,b]$ if there exists $I \in \mathbb{R}$ such that for every $\varepsilon > 0$ there exists $\delta > 0$ with

\[|\Gamma| < \delta \implies |R(f, \Gamma, \phi) - I| < \varepsilon\]

for any choice of intermediate points $\xi_{1}, \dots, \xi_{n}$, where $$

\Gamma

= \max_{i}(x_{i} - x_{i-1})$. In that case we write$I = \int_{a}^{b} f \, d\phi$$.

Exercise (Cauchy criterion for Riemann–Stieltjes integrability)

Prove that $f$ is Riemann–Stieltjes integrable with respect to $\phi$ if and only if for every $\varepsilon > 0$ there exists $\delta > 0$ such that

\[|\Gamma| < \delta \ \text{ and }\ |\Gamma'| < \delta \implies |R(f, \Gamma, \phi) - R(f, \Gamma', \phi)| < \varepsilon\]

for any choices of intermediate points.

Lemma (A common discontinuity prevents integrability)

Let $\phi : [a,b] \to \mathbb{R}$ and let $f : [a,b] \to \mathbb{R}$ be bounded. If there exists $z_{0} \in (a,b)$ at which both $f$ and $\phi$ are discontinuous, then $f$ is not Riemann–Stieltjes integrable with respect to $\phi$.

Proof: We show that the Cauchy criterion fails. Since $f$ and $\phi$ are discontinuous at $z_{0}$, there exists $\varepsilon > 0$ such that, for every $\delta > 0$, there is a point $w_{\delta}$ with $$0 <

w_{\delta} - z_{0}

< \delta$and$

\phi(w_{\delta}) - \phi(z_{0})

\geq \sqrt{\varepsilon}$, and a point$\xi_{\delta}$with$

\xi_{\delta} - z_{0}

w_{\delta} - z_{0}

$and$

f(\xi_{\delta}) - f(z_{0})

\geq \sqrt{\varepsilon}$; in particular$\xi_{\delta}$lies strictly between$z_{0}$and$w_{\delta}$. Take a partition$\Gamma$with$

\Gamma

< \delta$having$z_{0}$and$w_{\delta}$as consecutive nodes$x_{i_{0}-1} = z_{0}$,$x_{i_{0}} = w_{\delta}$, so that$\xi_{\delta} \in (x_{i_{0}-1}, x_{i_{0}})$. Construct two Riemann–Stieltjes sums with the same$\Gamma$and the same intermediate points except on the subinterval$[x_{i_{0}-1}, x_{i_{0}}]$, where one takes$\xi_{i_{0}} = \xi_{\delta}$and the other$\xi_{i_{0}} = z_{0}$$. Then

\[|R(f, \Gamma, \phi) - R'(f, \Gamma, \phi)| = |f(\xi_{\delta}) - f(z_{0})|\,|\phi(x_{i_{0}}) - \phi(x_{i_{0}-1})| = |f(\xi_{\delta}) - f(z_{0})|\,|\phi(w_{\delta}) - \phi(z_{0})| \geq \sqrt{\varepsilon}\cdot\sqrt{\varepsilon} = \varepsilon.\]

Thus, for every $\delta > 0$ there are sums with norm smaller than $\delta$ that differ by at least $\varepsilon$, and the Cauchy criterion fails. The case in which the witnesses $w_{\delta}$ (for $\phi$) and $\xi_{\delta}$ (for $f$) cannot be taken on the same side of $z_{0}$ is handled analogously and is left as an exercise.

Upper and lower sums

Definition (Lower and upper Darboux–Stieltjes sums)

Let $f : [a,b] \to \mathbb{R}$ be bounded and $\Gamma = \{a = x_{0} < \dots < x_{n} = b\}$. For $1 \leq i \leq n$ let

\[m_{i} = \inf_{x_{i-1} \leq \xi \leq x_{i}} f(\xi), \qquad M_{i} = \sup_{x_{i-1} \leq \xi \leq x_{i}} f(\xi).\]

The lower and upper sums with respect to $\phi$ are

\[L(f, \Gamma, \phi) = \sum_{i=1}^{n} m_{i}\,\big[\phi(x_{i}) - \phi(x_{i-1})\big], \qquad U(f, \Gamma, \phi) = \sum_{i=1}^{n} M_{i}\,\big[\phi(x_{i}) - \phi(x_{i-1})\big].\]

Note (Sandwiching of the sums and the Riemann case)

If $\phi$ is increasing, then $\phi(x_{i}) - \phi(x_{i-1}) \geq 0$ and for every choice of intermediate points

\[L(f, \Gamma, \phi) \leq R(f, \Gamma, \phi) \leq U(f, \Gamma, \phi).\]

If $\phi(x) = x$ we recover the usual Riemann sums and the usual Riemann integral.

Lemma (Monotonicity properties of the Darboux–Stieltjes sums)

Let $f : [a,b] \to \mathbb{R}$ be bounded and $\phi : [a,b] \to \mathbb{R}$ increasing.

If $\Gamma_{1} \subseteq \Gamma_{2}$ (refinement), then $L(f, \Gamma_{1}, \phi) \leq L(f, \Gamma_{2}, \phi)$ and $U(f, \Gamma_{2}, \phi) \leq U(f, \Gamma_{1}, \phi)$.
For any partitions $\Gamma_{1}, \Gamma_{2}$, $\ L(f, \Gamma_{1}, \phi) \leq U(f, \Gamma_{2}, \phi)$.

Proof: (1). It suffices to insert a point $y$ into a subinterval $[x_{i-1}, x_{i}]$ of $\Gamma_{1}$. Since

\[\sup_{[x_{i-1}, y]} f, \ \sup_{[y, x_{i}]} f \ \leq\ \sup_{[x_{i-1}, x_{i}]} f,\]

and $\phi(y) - \phi(x_{i-1}) \geq 0$, $\phi(x_{i}) - \phi(y) \geq 0$ with sum $\phi(x_{i}) - \phi(x_{i-1})$, we obtain

\[\big(\textstyle\sup_{[x_{i-1}, y]} f\big)\big(\phi(y) - \phi(x_{i-1})\big) + \big(\textstyle\sup_{[y, x_{i}]} f\big)\big(\phi(x_{i}) - \phi(y)\big) \leq \big(\textstyle\sup_{[x_{i-1}, x_{i}]} f\big)\big(\phi(x_{i}) - \phi(x_{i-1})\big),\]

that is, refinement can only decrease $U$. Iterating over the added points, $U(f, \Gamma_{2}, \phi) \leq U(f, \Gamma_{1}, \phi)$. The argument for $L$ (with infima) is symmetric and gives $L(f, \Gamma_{1}, \phi) \leq L(f, \Gamma_{2}, \phi)$.

(2). Let $\Gamma = \Gamma_{1} \cup \Gamma_{2}$, a common refinement. By (1) and the previous note,

\[L(f, \Gamma_{1}, \phi) \leq L(f, \Gamma, \phi) \leq U(f, \Gamma, \phi) \leq U(f, \Gamma_{2}, \phi).\]

Existence of the integral

Note (Reduction to an increasing integrator)

From the definition one checks that if $\int_{a}^{b} f\,d\phi_{1}$ and $\int_{a}^{b} f\,d\phi_{2}$ exist and $\phi = \phi_{1} - \phi_{2}$, then $\int_{a}^{b} f\,d\phi$ exists and

\[\int_{a}^{b} f\,d\phi = \int_{a}^{b} f\,d\phi_{1} - \int_{a}^{b} f\,d\phi_{2}.\]

Since every function of bounded variation is a difference of two increasing functions (Jordan’s characterisation), the study of integrability with respect to an integrator of bounded variation reduces to the case in which $\phi$ is increasing.

Theorem (Existence of the integral for a continuous integrand and an integrator of bounded variation)

Let $f : [a,b] \to \mathbb{R}$ be continuous and $\phi : [a,b] \to \mathbb{R}$ of bounded variation. Then $\int_{a}^{b} f\,d\phi$ exists and

\[\left| \int_{a}^{b} f\,d\phi \right| \leq \Big(\sup_{[a,b]} |f|\Big)\,\operatorname{Var}(\phi, [a,b]).\]

Proof: By the previous note it suffices to prove existence when $\phi$ is increasing; if $\phi$ is constant every sum vanishes and the integral equals $0$, so we assume $\phi(b) > \phi(a)$. Let $\varepsilon > 0$. Since $f$ is uniformly continuous, there exists $\delta_{1} > 0$ such that

\[|x - y| < \delta_{1} \implies |f(x) - f(y)| < \frac{\varepsilon}{2\big(\phi(b) - \phi(a)\big)}.\]

The upper and lower sums approach each other. If $$

\Gamma

< \delta_{1}$, by continuity there exist$\xi_{i}, \eta_{i} \in [x_{i-1}, x_{i}]$with$f(\xi_{i}) = M_{i}$,$f(\eta_{i}) = m_{i}$; since$

\xi_{i} - \eta_{i}

\leq

\Gamma

< \delta_{1}$$,

\[U(f, \Gamma, \phi) - L(f, \Gamma, \phi) = \sum_{i=1}^{n} (M_{i} - m_{i})\big(\phi(x_{i}) - \phi(x_{i-1})\big) \leq \frac{\varepsilon}{2(\phi(b) - \phi(a))}\big(\phi(b) - \phi(a)\big) = \frac{\varepsilon}{2}.\]

Stability of $U$. If $$

\Gamma

< \delta_{1}$and$

\Gamma’

< \delta_{1}$, using$U - L \leq \varepsilon/2$and$L(f, \cdot, \phi) \leq U(f, \cdot, \phi)$$ between distinct partitions,

\[U(f, \Gamma, \phi) \leq L(f, \Gamma, \phi) + \tfrac{\varepsilon}{2} \leq U(f, \Gamma', \phi) + \tfrac{\varepsilon}{2},\]

and symmetrically, so that $$

U(f, \Gamma, \phi) - U(f, \Gamma’, \phi)

\leq \varepsilon/2$$.

Construction of the limit. Take partitions $\{\Gamma_{k}\}_{k=1}^{\infty}$ with $$

\Gamma_{k}

\to 0$and let$\Gamma_{k}’ = \bigcup_{j=1}^{k} \Gamma_{j}$, so that$\Gamma_{k}’ \subseteq \Gamma_{k+1}’$and$\Gamma_{k} \subseteq \Gamma_{k}’$. By monotonicity (1),${U(f, \Gamma_{k}’, \phi)}_{k}$$ is decreasing and bounded below (by any lower sum), so that

\[U := \inf_{k \geq 1} U(f, \Gamma_{k}', \phi) = \lim_{k \to \infty} U(f, \Gamma_{k}', \phi)\]

exists. Given $\varepsilon$, let $k_{0}$ be such that $0 \leq U(f, \Gamma_{k}', \phi) - U < \varepsilon/2$ for $k \geq k_{0}$, and $k_{1}$ such that $$

\Gamma_{k}

< \delta_{1}$for$k \geq k_{1}$. Since$\Gamma_{k} \subseteq \Gamma_{k}’$and both have norm$< \delta_{1}$, the stability of$U$gives$

U(f, \Gamma_{k}, \phi) - U(f, \Gamma_{k}’, \phi)

\leq \varepsilon/2$, hence for$k \geq \max{k_{0}, k_{1}}$$,

\[|U(f, \Gamma_{k}, \phi) - U| \leq |U(f, \Gamma_{k}, \phi) - U(f, \Gamma_{k}', \phi)| + |U(f, \Gamma_{k}', \phi) - U| < \varepsilon.\]

Independence of the sequence. If $\{\widetilde{\Gamma}_{k}\}$ is another sequence with $$

\widetilde{\Gamma}_{k}

\to 0$, for large$k$both norms are$< \delta_{1}$and the stability of$U$gives$

U(f, \Gamma_{k}, \phi) - U(f, \widetilde{\Gamma}_{k}, \phi)

\leq \varepsilon/2$; hence$U(f, \widetilde{\Gamma}_{k}, \phi) \to U$as well. Thus,$U(f, \Gamma, \phi) \to U$as$

\Gamma

\to 0$$.

Conclusion. It has already been shown that $U(f, \Gamma, \phi) \to U$ as $$

\Gamma

\to 0$. Moreover, for$

\Gamma

< \delta_{1}$we have that$L(f, \Gamma, \phi) \geq U(f, \Gamma, \phi) - \tfrac{\varepsilon}{2}$, so that also$L(f, \Gamma, \phi) \to U$. The sandwiching$L(f, \Gamma, \phi) \leq R(f, \Gamma, \phi) \leq U(f, \Gamma, \phi)$then gives$R(f, \Gamma, \phi) \to U$as$

\Gamma

\to 0$. Hence$\int_{a}^{b} f\,d\phi = U$exists. Finally, for$\phi$of bounded variation and any$\Gamma$$,

\[|R(f, \Gamma, \phi)| \leq \Big(\sup_{[a,b]}|f|\Big)\sum_{i=1}^{n}|\phi(x_{i}) - \phi(x_{i-1})| \leq \Big(\sup_{[a,b]}|f|\Big)\operatorname{Var}(\phi, [a,b]),\]

and passing to the limit we obtain the stated bound.

Properties: linearity, mean value and integration by parts

Theorem (Linearity of the Riemann–Stieltjes integral)

Let $f_{1}, f_{2} : [a,b] \to \mathbb{R}$ be Riemann–Stieltjes integrable with respect to $\phi_{1}, \phi_{2} : [a,b] \to \mathbb{R}$, $c \in \mathbb{R}$ and $a < c' < b$. Then:

$\displaystyle \int_{a}^{b}(c f_{1} + f_{2})\,d\phi_{1} = c\int_{a}^{b} f_{1}\,d\phi_{1} + \int_{a}^{b} f_{2}\,d\phi_{1}$;
$\displaystyle \int_{a}^{b} f_{1}\,d(c\phi_{1}) = c\int_{a}^{b} f_{1}\,d\phi_{1}$;
$\displaystyle \int_{a}^{b} f_{1}\,d(\phi_{1} + \phi_{2}) = \int_{a}^{b} f_{1}\,d\phi_{1} + \int_{a}^{b} f_{1}\,d\phi_{2}$;
$\displaystyle \int_{a}^{b} f_{1}\,d\phi_{1} = \int_{a}^{c'} f_{1}\,d\phi_{1} + \int_{c'}^{b} f_{1}\,d\phi_{1}$.

Proof: Exercise.

Note (Bounds by the supremum and the infimum of the integrand)

If $\phi$ is increasing, from $L(f, \Gamma, \phi) \leq R(f, \Gamma, \phi) \leq U(f, \Gamma, \phi)$, $U(f, \Gamma, \phi) \leq (\phi(b) - \phi(a))\sup_{[a,b]} f$ and $L(f, \Gamma, \phi) \geq (\phi(b) - \phi(a))\inf_{[a,b]} f$ we obtain, passing to the limit,

\[\Big(\inf_{[a,b]} f\Big)\big(\phi(b) - \phi(a)\big) \leq \int_{a}^{b} f\,d\phi \leq \Big(\sup_{[a,b]} f\Big)\big(\phi(b) - \phi(a)\big).\]

Lemma (Mean value theorem for the Riemann–Stieltjes integral)

Let $f : [a,b] \to \mathbb{R}$ be continuous and $\phi : [a,b] \to \mathbb{R}$ increasing, with $f$ Riemann–Stieltjes integrable with respect to $\phi$. Then there exists $\xi \in [a,b]$ such that

\[\int_{a}^{b} f\,d\phi = f(\xi)\big(\phi(b) - \phi(a)\big).\]

Proof: If $\phi(b) = \phi(a)$, the bounds of the previous note give $\int_{a}^{b} f\,d\phi = 0$ and any $\xi$ will do. If $\phi(b) > \phi(a)$, dividing the same bounds by $\phi(b) - \phi(a)$,

\[\inf_{[a,b]} f \ \leq\ \frac{1}{\phi(b) - \phi(a)}\int_{a}^{b} f\,d\phi \ \leq\ \sup_{[a,b]} f.\]

Since $f$ is continuous on the compact set $[a,b]$, it attains its infimum and its supremum, and by the intermediate value theorem it takes every value between them; in particular there exists $\xi \in [a,b]$ at which $f(\xi)$ equals the quotient above, which gives the identity.

Theorem (Integration by parts)

If $\int_{a}^{b} f\,d\phi$ exists, then $\int_{a}^{b} \phi\,df$ exists and

\[\int_{a}^{b} f\,d\phi + \int_{a}^{b} \phi\,df = f(b)\phi(b) - f(a)\phi(a).\]

Proof: Let $\Gamma = \{a = x_{0} < \dots < x_{n} = b\}$ with intermediate points $\xi_{i} \in [x_{i-1}, x_{i}]$. Consider the points $\xi_{0} = a$, $\xi_{n+1} = b$ and the partition $\Gamma' = \{\xi_{0}, \xi_{1}, \dots, \xi_{n+1}\}$, which satisfies $x_{i-1} \leq \xi_{i} \leq x_{i} \leq \xi_{i+1}$, so that each $x_{i}$ is a valid intermediate point of $\Gamma'$. By summation by parts (Abel),

\[R(f, \Gamma, \phi) = \sum_{i=1}^{n} f(\xi_{i})\big(\phi(x_{i}) - \phi(x_{i-1})\big) = -\sum_{i=0}^{n} \phi(x_{i})\big(f(\xi_{i+1}) - f(\xi_{i})\big) + f(b)\phi(b) - f(a)\phi(a),\]

that is, $R(f, \Gamma, \phi) = -R(\phi, \Gamma', f) + f(b)\phi(b) - f(a)\phi(a)$, where $R(\phi, \Gamma', f)$ is a Riemann–Stieltjes sum of $\phi$ with respect to $f$ with partition $\Gamma'$ and intermediate points $x_{i}$. Since $\xi_{i} \in [x_{i-1}, x_{i}]$, each gap satisfies $$\xi_{i+1} - \xi_{i} \leq x_{i+1} - x_{i-1} \leq 2

\Gamma

$, so that$

\Gamma’

\leq 2

\Gamma

\to 0$. Since$\int_{a}^{b} f\,d\phi$exists,$R(f, \Gamma, \phi) \to \int_{a}^{b} f\,d\phi$as$

\Gamma

\to 0$, and the algebraic identity above forces$R(\phi, \Gamma’, f) \to f(b)\phi(b) - f(a)\phi(a) - \int_{a}^{b} f\,d\phi$. Since$

\Gamma’

\to 0$and this limit does not depend on the chosen sequence of partitions,$\int_{a}^{b} \phi\,df$exists and equals$f(b)\phi(b) - f(a)\phi(a) - \int_{a}^{b} f\,d\phi$$, which is the desired identity.

Passing to the limit under the integral sign

Example (An increasing sequence of Riemann-integrable functions with a non-integrable limit)

Let $\{r_{n}\}_{n=1}^{\infty}$ be an enumeration of $\mathbb{Q} \cap [0,1]$ and

\[f_{n} : [0,1] \to \mathbb{R}, \qquad f_{n}(x) = \begin{cases} 1, & x \in \{r_{1}, \dots, r_{n}\},\\ 0, & \text{otherwise.}\end{cases}\]

Then $f_{n} \leq f_{n+1}$ and $f_{n} \to f = \mathbf{1}_{\mathbb{Q} \cap [0,1]}$ pointwise. It is an exercise to check that $\int_{0}^{1} f_{n}(x)\,dx = 0$ for every $n$, whereas $f$ is not Riemann integrable. Thus, the pointwise limit of Riemann-integrable functions need not be one, which motivates a more flexible notion of integral.

Lemma (Passing to the limit under uniform convergence)

Let $\phi : [a,b] \to \mathbb{R}$ be of bounded variation and $\{f_{n}\}_{n=1}^{\infty}$ a sequence of functions Riemann–Stieltjes integrable with respect to $\phi$. If $f_{n} \to f$ uniformly on $[a,b]$ and $f$ is Riemann–Stieltjes integrable with respect to $\phi$, then

\[\lim_{n \to \infty} \int_{a}^{b} f_{n}\,d\phi = \int_{a}^{b} f\,d\phi.\]

Proof: For any function $g$ Riemann–Stieltjes integrable with respect to $\phi$ and every partition $\Gamma$,

\[|R(g, \Gamma, \phi)| \leq \Big(\sup_{[a,b]}|g|\Big)\sum_{i}|\phi(x_{i}) - \phi(x_{i-1})| \leq \Big(\sup_{[a,b]}|g|\Big)\operatorname{Var}(\phi, [a,b]),\]

and passing to the limit, $$\big

\int_{a}^{b} g\,d\phi\big

\leq (\sup_{[a,b]}

)\operatorname{Var}(\phi, [a,b])$. Applying this to$g = f_{n} - f$(integrable, since$f_{n}$and$f$$ are) and using linearity,

\[\left| \int_{a}^{b} f_{n}\,d\phi - \int_{a}^{b} f\,d\phi \right| = \left| \int_{a}^{b} (f_{n} - f)\,d\phi \right| \leq \Big(\sup_{[a,b]}|f_{n} - f|\Big)\operatorname{Var}(\phi, [a,b]).\]

Since $f_{n} \to f$ uniformly, $$\sup_{[a,b]}

f_{n} - f

\to 0$$, and the right-hand side tends to zero.