8.7.E: Problems on Integration of Complex and Vector-Valued Functions
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 \(A .\)