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Mathematics LibreTexts

8.9.E: Problems on Riemann and Stieltjes Integrals

  • Page ID
    32380
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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.

    Exercise \(\PageIndex{2}\)

    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 \(\left.S \int_{a}^{b} f d \beta .\right]\)

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