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# 8.7.E: Problems on Integration of Complex and Vector-Valued Functions

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## Exercise $$\PageIndex{1}$$

Prove Corollary $$1(\text { iii })-$$ (vii) in §4 componentwise for integrable maps $$f: S \rightarrow E^{n}\left(C^{n}\right) .$$

## Exercise $$\PageIndex{2}$$

Prove Theorems 2 and 3 componentwise for $$E=E^{n}\left(C^{n}\right)$$.

## Exercise $$\PageIndex{2'}$$

Do it for Corollary 3 in §6.

## Exercise $$\PageIndex{3}$$

Prove Theorem 1 with
$\int_{A}|f|<\infty$
replaced by
$\int_{A}\left|f_{k}\right|<\infty, \quad k=1, \ldots, n .$

## Exercise $$\PageIndex{4}$$

Prove that if $$f: S \rightarrow E^{n}\left(C^{n}\right)$$ is integrable on $$A,$$ so is $$|f| .$$ Disprove the converse.

## Exercise $$\PageIndex{5}$$

Disprove Lemma 1 for $$m A=\infty$$.

## Exercise $$\PageIndex{*6}$$

Complete the proof of Lemma 3.

## Exercise $$\PageIndex{*7}$$

Complete the proof of Theorem 3.

## Exercise $$\PageIndex{*8}$$

Do Problem 1 and $$2^{\prime}$$ for $$f: S \rightarrow E$$.

## Exercise $$\PageIndex{*9}$$

Prove formula (1) from definitions of Part II of this section.

## Exercise $$\PageIndex{10}$$

$$\Rightarrow 10$$. 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 $$3 .$$ ]

## Exercise $$\PageIndex{11}$$

Do Problem 11 in §6 for $$f_{n}: S \rightarrow E .$$ Do it componentwise for $$E=$$ $$E^{n}\left(C^{n}\right) .$$

## Exercise $$\PageIndex{12}$$

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
$\left.\left(\exists t \in E^{1}\right) \quad \int_{A}|f-t g|=0 .\right]$

## Exercise $$\PageIndex{13}$$

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}
$$\left.(c f)_{\mathrm{re}}=|c f|(\mathrm{a.e.}), \text { and } c f=|c f|=|f| \text { a.e. on } A .\right]$$

## Exercise $$\PageIndex{14}$$

Do Problem 10 for $$E=C$$ using the method of Problem $$13 .$$

## Exercise $$\PageIndex{15}$$

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 $$\left.E=E^{*}: A=A(f>0) \cup A(f \leq 0) . \text { Use Theorems } 1(\mathrm{h}) \text { and } 2 \text { from } §5 .\right]$$

## Exercise $$\PageIndex{16}$$

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 $$A .$$