8.7.E: Problems on Integration of Complex and Vector-Valued Functions
- Page ID
- 25033
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Prove Corollary \(1(\text { iii })-\) (vii) in §4 componentwise for integrable maps \(f: S \rightarrow E^{n}\left(C^{n}\right) .\)
Prove Theorems 2 and 3 componentwise for \(E=E^{n}\left(C^{n}\right)\).
Do it for Corollary 3 in §6.
Prove Theorem 1 with
\[
\int_{A}|f|<\infty
\]
replaced by
\[
\int_{A}\left|f_{k}\right|<\infty, \quad k=1, \ldots, n .
\]
Prove that if \(f: S \rightarrow E^{n}\left(C^{n}\right)\) is integrable on \(A,\) so is \(|f| .\) Disprove the converse.
Disprove Lemma 1 for \(m A=\infty\).
Complete the proof of Lemma 3.
Complete the proof of Theorem 3.
Do Problem 1 and \(2^{\prime}\) for \(f: S \rightarrow E\).
Prove formula (1) from definitions of Part II of this section.
\(\Rightarrow 10\). Show that
\[
\left|\int_{A} f\right| \leq \int_{A}|f|
\]
for integrable maps \(f: S \rightarrow E .\) See also Problem 14.
[Hint: If \(m A<\infty,\) use Corollary \(1(\text { ii ) of } §4 \text { and Lemma } 1 . \text { If } m A=\infty,\) us imitate" the proof of Lemma \(3 .\) ]
Do Problem 11 in §6 for \(f_{n}: S \rightarrow E .\) Do it componentwise for \(E=\) \(E^{n}\left(C^{n}\right) .\)
Show that if \(f, g: S \rightarrow E^{1}(C)\) are integrable on \(A,\) then
\[
\left|\int_{A} f g\right|^{2} \leq \int_{A}|f|^{2} \cdot \int_{A}|g|^{2} .
\]
In what case does equality hold? Deduce Theorem \(4\left(\mathrm{c}^{\prime}\right)\) in Chapter \(3,\) §§1-3, from this result.
[Hint: Argue as in that theorem. Consider the case
\[
\left.\left(\exists t \in E^{1}\right) \quad \int_{A}|f-t g|=0 .\right]
\]
Show that if \(f: S \rightarrow E^{1}(C)\) is integrable on \(A\) and
\[
\left|\int_{A} f\right|=\int_{A}|f| ,
\]
then
\[
(\exists c \in C) \quad c f=|f| \quad \text { a.e. on } A.
\]
[Hint: Let \(a=\int_{A} f .\) The case \(a=0\) is trivial. If \(a \neq 0,\) let
\[
c=\frac{|a|}{a} ;|c|=1 ; c a=|a| .
\]
Let \(r=(c f)_{\mathrm{re}} .\) Show that \(r \leq|c f|=|f|\),
\[
\begin{aligned}\left|\int_{A} f\right| &=\int_{A} c f=\int_{A} r \leq \int_{A}|f|=\left|\int_{A} f\right| , \\ & \int_{A}|f|=\int_{A} r=\int_{A} (c f)_{\mathrm{re,}} \end{aligned}
\]
\(\left.(c f)_{\mathrm{re}}=|c f|(\mathrm{a.e.}), \text { and } c f=|c f|=|f| \text { a.e. on } A .\right]\)
Do Problem 10 for \(E=C\) using the method of Problem \(13 .\)
Show that if \(f: S \rightarrow E\) is integrable on \(A,\) it is integrable on each \(\mathcal{M}\)-set \(B \subseteq A .\) If, in addition,
\[
\int_{B} f=0
\]
for all such \(B,\) show that \(f=0\) a.e. on \(A .\) Prove it for \(E=E^{n}\) first.
[Hint for \(\left.E=E^{*}: A=A(f>0) \cup A(f \leq 0) . \text { Use Theorems } 1(\mathrm{h}) \text { and } 2 \text { from } §5 .\right]\)
In Problem \(15,\) show that
\[
s=\int f
\]
is a \(\sigma\)-additive set function on
\[
\mathcal{M}_{A}=\{X \in \mathcal{M} | X \subseteq A\} .
\]
(Note \(4 \text { in } §5) ; s\) is called the indefinite integral of \(f\) in \(A .\)