8.9.E: Problems on Riemann and Stieltjes Integrals
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.
Do Problems \(5-7\) in §5 for R-integrals.
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 \(\left.1 \text { and } 2^{\prime} .\right]\)
Fill in all details in the proof of Theorem \(1,\) Lemmas 3 and \(4,\) and Corollary \(4 .\)
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.
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.
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.]
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).]
(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 ).]
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.
(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 \(\left.\mathcal{P} \text { refines } \mathcal{P}_{\varepsilon} .\right]\)
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.]
(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 \(\left.S \int_{a}^{b} f d \beta .\right]\)