# 8.9.E: Problems on Riemann and Stieltjes Integrals

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## Exercise $$\PageIndex{1}$$

Replacing "$$\mathcal{M}$$" by "$$\mathcal{C},$$" and "elementary and integrable" or "elementary and nonnegative" by "$$\mathcal{C}$$-simple," price Corollary 1(ii)(iv)(vii) and Theorems 1(i) and 2(ii), all in §4, and do Problems 5-7 in §4, for R-integrals.

Verify Note 1.

## Exercise $$\PageIndex{2'}$$

Do Problems $$5-7$$ in §5 for R-integrals.

## Exercise $$\PageIndex{3}$$

Do the following for R-integrals.
(i) Prove Theorems $$1(\mathrm{a})-(\mathrm{g})$$ and $$2,$$ both in $$§5(\mathcal{C} \text {-partitions only })$$.
(ii) Prove Theorem 1 and Corollaries 1 and 2, all in §6.
(iii) Show that definition (b) can be replaced by formulas analogous to formulas $$\left(1^{\prime}\right),\left(1^{\prime \prime}\right),$$ and ( 1) of Definition 1 in §5.
[Hint: Use Problems $$\left.1 \text { and } 2^{\prime} .\right]$$

## Exercise $$\PageIndex{4}$$

Fill in all details in the proof of Theorem $$1,$$ Lemmas 3 and $$4,$$ and Corollary $$4 .$$

## Exercise $$\PageIndex{5}$$

For $$f, g: E^{n} \rightarrow E^{s}\left(C^{s}\right),$$ via components, prove the following.
(i) Theorems $$1-3$$ and
(ii) additivity and linearity of R-integrals.
Do also Problem 13 in §7 for R-integrals.

## Exercise $$\PageIndex{6}$$

Prove that if $$f: A \rightarrow E^{s}\left(C^{s}\right)$$ is bounded and a.e. continuous on $$A,$$ then
$R \int_{A}|f| \geq\left|R \int_{A} f\right| .$
For $$m=$$ Lebesgue measure, do it assuming R-integrability only.

## Exercise $$\PageIndex{7}$$

Prove that if $$f, g: A \rightarrow E^{1}$$ are R-integrable, then
(i) so is $$f^{2},$$ and
(ii) so is $$f g$$.
[Hints: (i) Use Lemma 1. Let $$h=|f| \leq K<\infty$$ on A. Verify that
$\left(\inf h\left[A_{i}\right]\right)^{2}=\inf f^{2}\left[A_{i}\right] \text { and }\left(\sup h\left[A_{i}\right]\right)^{2}=\sup f^{2}\left[A_{i}\right] ;$
so
\begin{aligned} \sup f^{2}\left[A_{i}\right]-\inf f^{2}\left[A_{i}\right] &=\left(\sup h\left[A_{i}\right]+\inf h\left[A_{i}\right]\right)\left(\sup h\left[A_{i}\right]-\inf h\left[A_{i}\right]\right) \\ & \leq\left(\sup h\left[A_{i}\right]-\inf h\left[A_{i}\right]\right) 2 K . \end{aligned}
(ii) Use
$f g=\frac{1}{4}\left[(f+g)^{2}-(f-g)^{2}\right] .$
(iii) For $$m=$$ Lebesgue measure, do it using Theorem 3.]

## Exercise $$\PageIndex{8}$$

Prove that if $$m=$$ the volume function $$v$$ (or LS function $$s_{\alpha}$$ for a continuous $$\alpha$$ ), then in formulas ( 1) and $$(2),$$ one may replace $$A_{i}$$ by $$\overline{A}_{i}$$ (closure of $$\left.A_{i}\right) .$$
[Hint: Show that here $$m A=m \overline{A}$$,
$R \int_{A} f=R \int_{\overline{A}} f ,$
and additivity works even if the $$A_{i}$$ have some common "faces" (only their interiors being disjoint).]

## Exercise $$\PageIndex{9}$$

(Riemann sums.) Instead of $$\underline{S}$$ and $$\bar{S}$$, Riemann used sums
$S(f, \mathcal{P})=\sum_{i} f\left(x_{i}\right) d m A_{i} ,$
where $$m=v \text { (see Problem } 8)$$ and $$x_{i}$$ is arbitrarily chosen from $$\overline{A_{i}}$$.
For a bounded $$f,$$ prove that
$r=R \int_{A} f d m$
exists on $$A=[a, b]$$ iff for every $$\varepsilon>0,$$ there is $$\mathcal{P}_{\varepsilon}$$ such that
$|S(f, \mathcal{P})-r|<\varepsilon$
for every refinement
$\mathcal{P}=\left\{A_{i}\right\}$
of $$\mathcal{P}_{\varepsilon}$$ and any choice of $$x_{i} \in \overline{A_{i}}$$.
[Hint: Show that by Problem $$8,$$ this is equivalent to formula ( 3 ).]

## Exercise $$\PageIndex{10}$$

Replacing $$m$$ by the $$\sigma_{\alpha}$$ of Problem 9 of Chapter 7, §4, write $$S(f, \mathcal{P}, \alpha)$$ for $$S(f, \mathcal{P})$$ in Problem $$9,$$ treating Problem 9 as a definition of the Stieltjes integral,
$S \int_{a}^{b} f d \alpha \quad\left(\text { or } S \int_{a}^{b} f d \sigma_{\alpha}\right) .$
Here $$f, \alpha: E^{1} \rightarrow E^{1}$$ (monotone or not; even $$f, \alpha: E^{1} \rightarrow C$$ will do).
Prove that if $$\alpha: E^{1} \rightarrow E^{1}$$ is continuous and $$\alpha \uparrow,$$ then
$S \int_{a}^{b} f d \alpha=R \int_{a}^{b} f d \alpha ,$
the $$R S$$-integral.

## Exercise $$\PageIndex{11}$$

(Integration by parts.) Continuing Problem $$10,$$ prove that
$S \int_{a}^{b} f d \alpha$
exists iff
$S \int_{a}^{b} \alpha d f$
does, and then
$S \int_{a}^{b} f d \alpha+S \int_{a}^{b} \alpha d f=K ,$
where
$K=f(b) \alpha(b)-f(a) \alpha(a) .$
[Hints: Take any $$\mathcal{C}$$-partition $$\mathcal{P}=\left\{A_{i}\right\}$$ of $$[a, b],$$ with
$\overline{A_{i}}=\left[y_{i-1}, y_{i}\right] ,$
say. For any $$x_{i} \in \overline{A}_{i},$$ verify that
$S(f, \mathcal{P}, \alpha)=\sum f\left(x_{i}\right)\left[\alpha\left(y_{i}\right)-\alpha\left(y_{i-1}\right)\right]=\sum f\left(x_{i}\right) \alpha\left(y_{i}\right)-\sum f\left(x_{i}\right) \alpha\left(y_{i-1}\right)$
and
$K=\sum f\left(x_{i}\right) \alpha\left(y_{i}\right)-\sum f\left(x_{i-1}\right) \alpha\left(y_{i-1}\right) .$
Deduce that
$K-S(f, \mathcal{P}, \alpha)=S\left(\alpha, \mathcal{P}^{\prime}, f\right)=\sum \alpha\left(x_{i}\right)\left[f\left(x_{i}\right)-f\left(y_{i}\right)\right]-\sum \alpha\left(x_{i-1}\right)\left[f\left(y_{i}\right)-f\left(x_{i-1}\right)\right] ;$
here $$\mathcal{P}^{\prime}$$ results by combining the partition points $$x_{i}$$ and $$y_{i},$$ so it refines $$\mathcal{P}$$.
Now, if $$S \int_{a}^{b} \alpha d f$$ exists, fix $$\mathcal{P}_{\varepsilon}$$ as in Problem 9 and show that
$\left|K-S(f, \mathcal{P}, \alpha)-S \int_{a}^{b} \alpha d f\right|<\varepsilon$
whenever $$\left.\mathcal{P} \text { refines } \mathcal{P}_{\varepsilon} .\right]$$

## Exercise $$\PageIndex{12}$$

If $$\alpha: E^{1} \rightarrow E^{1}$$ is of class $$C D^{1}$$ on $$[a, b]$$ and if
$S \int_{a}^{b} f d \alpha$
exists (see Problem $$10),$$ it equals
$R \int_{a}^{b} f(x) \alpha^{\prime}(x) d x .$
[Hints: Set $$\phi=f \alpha^{\prime}, \mathcal{P}=\left\{A_{i}\right\}, \overline{A_{i}}=\left[a_{i-1}, a_{i}\right]$$. Then
$S(\phi, \mathcal{P})=\sum f\left(x_{i}\right) \alpha^{\prime}\left(x_{i}\right)\left(a_{i}-a_{i-1}\right), \quad x_{i} \in \overline{A_{i}}$
and (Corollary 3 in Chapter 5, §2)
$S(f, \mathcal{P}, \alpha)=\sum f\left(x_{i}\right)\left[\alpha\left(a_{i}\right)-\alpha\left(a_{i-1}\right)\right]=\sum f\left(x_{i}\right) \alpha^{\prime}\left(q_{i}\right), \quad q_{i} \in A_{i} .$
As $$f$$ is bounded and $$\alpha^{\prime}$$ is uniformly continuous on $$[a, b]$$ (why?), deduce that
\begin{aligned}(\forall \varepsilon>0)\left(\exists \mathcal{P}_{\varepsilon}\right)\left(\forall \mathcal{P}_{\varepsilon}\right)(\forall \mathcal{P}& \text { refining }\left.\mathcal{P}_{\varepsilon}\right) \\ &|S(\phi, \mathcal{P})-S(f, \mathcal{P}, \alpha)|<\frac{1}{2} \varepsilon \text { and }\left|S(f, \mathcal{P}, \alpha)-S \int_{a}^{b} f d \alpha\right|<\frac{1}{2} \varepsilon . \end{aligned}
Proceed. Use Problem 9.]

## Exercise $$\PageIndex{13}$$

(Laws of the mean.) Let $$f, g, \alpha: E^{1} \rightarrow E^{1} ; p \leq f \leq q$$ on $$A=[a, b] ;$$ $$p, q \in E^{1} .$$ Prove the following.
(i) If $$\alpha \uparrow$$ and if
$s \int_{a}^{b} f d \alpha$
exists, then $$(\exists c \in[p, q])$$ such that
$S \int_{a}^{b} f d \alpha=c[\alpha(b)-\alpha(a)] .$
Similarly, if
$R \int_{a}^{b} f d \alpha$
exists, then $$(\exists c \in[p, q])$$ such that
$R \int_{a}^{b} f d \alpha=c[\alpha(b+)-\alpha(a-)] .$
(i') If $$f$$ also has the Darboux property on $$A,$$ then $$c=f\left(x_{0}\right)$$ for some $$x_{0} \in A .$$
(ii) If $$\alpha$$ is continuous, and $$f \uparrow$$ on $$A,$$ then
$S \int_{a}^{b} f d \alpha=[f(b) \alpha(b)-f(a) \alpha(a)]-S \int_{a}^{b} \alpha d f$
exists, and $$(\exists z \in A)$$ such that
\begin{aligned} S \int_{a}^{b} f d \alpha &=f(a) S \int_{a}^{z} d \alpha+f(b) S \int_{z}^{b} d \alpha \\ &=f(a)[\alpha(z)-\alpha(a)]+f(b)[\alpha(b)-\alpha(z)] . \end{aligned}
(ii') If $$g$$ is continuous and $$f \uparrow$$ on $$A,$$ then $$(\exists z \in A)$$ such that
$R \int_{a}^{b} f(x) g(x) d x=p \cdot R \int_{a}^{z} g(x) d x+q \cdot R \int_{z}^{b} g(x) d x .$
If $$f \downarrow,$$ replace $$f$$ by $$-f .$$ (See also Corollary 5 in Chapter $$9,$$ §1.)
[Hints: (i) As $$\alpha \uparrow,$$ we get
$p[\alpha(b)-\alpha(a)] \leq S \int_{a}^{b} f d \alpha \leq q[\alpha(b)-\alpha(a)] .$
(Why?) Now argue as in §6, Theorem 3 and Problem 2.
(ii) Use Problem $$11,$$ and apply (i) to $$\int \alpha d f$$.
(ii') By Theorem 2 of Chapter $$5, \ 10, g$$ has a primitive $$\beta \in C D^{1} .$$ Apply Problem 12 to $$\left.S \int_{a}^{b} f d \beta .\right]$$

8.9.E: Problems on Riemann and Stieltjes Integrals is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.