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8.9.E: Problems on Riemann and Stieltjes Integrals

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Exercise 8.9.E.1

Replacing "M" by "C," and "elementary and integrable" or "elementary and nonnegative" by "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.

Exercise 8.9.E.2

Verify Note 1.

Exercise 8.9.E.2

Do Problems 57 in §5 for R-integrals.

Exercise 8.9.E.3

Do the following for R-integrals.
(i) Prove Theorems 1(a)(g) and 2, both in §5(C-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 (1),(1), and ( 1) of Definition 1 in §5.
[Hint: Use Problems 1 and 2.]

Exercise 8.9.E.4

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

Exercise 8.9.E.5

For f,g:EnEs(Cs), via components, prove the following.
(i) Theorems 13 and
(ii) additivity and linearity of R-integrals.
Do also Problem 13 in §7 for R-integrals.

Exercise 8.9.E.6

Prove that if f:AEs(Cs) is bounded and a.e. continuous on A, then
RA|f||RAf|.
For m= Lebesgue measure, do it assuming R-integrability only.

Exercise 8.9.E.7

Prove that if f,g:AE1 are R-integrable, then
(i) so is f2, and
(ii) so is fg.
[Hints: (i) Use Lemma 1. Let h=|f|K< on A. Verify that
(inf
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.

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