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# 8.10.E: Problems on Generalized Integration

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

Fill in the missing details in the proofs of this section. Prove Note 3.

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

Treat Corollary 1 (ii) as a definition of
$\int_{A} f d s$
for $$s: \mathcal{M} \rightarrow E$$ and elementary and integrable $$f,$$ even if $$E \neq E^{n}\left(C^{n}\right) .$$
Hence deduce Corollary $$1(\mathrm{i})(\mathrm{vi})$$ for this more general case.

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

Using Lemma 2 in §7, with $$m=v_{s}, s: \mathcal{M} \rightarrow E,$$ construct
$\int_{A} f d s$
as in Definition 2 of §7 for the case $$v_{s} A \neq \infty .$$ Show that this agrees with Problem 2 if $$f$$ is elementary and integrable. Then prove linearity for functions with $$v_{s}$$-finite support as in §7.

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

Define
$\int_{A} f d s \quad(s: \mathcal{M} \rightarrow E)$
also for $$v_{s} A=\infty .$$
[Hint: Set $$\left.m=v_{s} \text { in Lemma } 3 \text { of } §7 .\right]$$

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

Prove Theorems 1 to 3 for the general case, $$s: \mathcal{M} \rightarrow E$$ (see Problem 4 ).
[Hint: Argue as in §7.]

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

From Problems $$2-4,$$ deduce Definition 2 as a theorem in the case $$E=$$ $$E^{n}\left(C^{n}\right) .$$

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

Let $$s, s_{k}: \mathcal{M} \rightarrow E(k=1,2, \ldots)$$ be any set functions. Let $$A \in \mathcal{M}$$ and
$\mathcal{M}_{A}=\{X \in \mathcal{M} | X \subseteq A\} .$
Prove that if
$\left(\forall X \in \mathcal{M}_{A}\right) \quad \lim _{k \rightarrow \infty} s_{k} X=s X ,$
then
$\lim _{k \rightarrow \infty} v_{s_{k}} A=v_{s} A ,$
provided $$\lim _{k \rightarrow \infty} v_{s_{k}}$$ exists.
[Hint: Using Problem 2 in Chapter 7, §11, fix a finite disjoint sequence $$\left\{X_{i}\right\} \subseteq \mathcal{M}_{A} .$$
Then
$\sum_{i}\left|s X_{i}\right|=\sum_{i} \lim _{k \rightarrow \infty}\left|s_{k} X_{i}\right|=\lim _{k \rightarrow \infty} \sum_{i}\left|s_{k} X_{i}\right| \leq \lim _{k \rightarrow \infty} v_{s_{k}} A .$
Infer that
$v_{s} A \leq \lim _{k \rightarrow \infty} v_{s k} A .$
Also,
$(\forall \varepsilon>0)\left(\exists k_{0}\right)\left(\forall k>k_{0}\right) \quad \sum_{i}\left|s_{k} X_{i}\right| \leq \sum_{i}\left|s X_{i}\right|+\varepsilon \leq v_{s} A+\varepsilon .$
Proceed.]

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

Let $$(X, \mathcal{M}, m)$$ and $$(Y, \mathcal{N}, n)$$ be two generalized measure spaces $$(X \in$$ $$M, Y \in \mathcal{N}) \text { such that } m n \text { is defined (Note } 1) .$$ Set
$\mathcal{C}=\left\{A \times B | A \in \mathcal{M}, B \in \mathcal{N}, v_{m} A<\infty, v_{n} B<\infty\right\}$
and $$s(A \times B)=m A \cdot n B$$ for $$A \times B \in \mathcal{C}$$.
Define a Fubini map as in §8, Part IV, dropping, however, $$\int_{X \times Y} f d p$$ from Fubini property (c) temporarily.
Then prove Theorem 1 in §8, including formula $$(1),$$ for Fubini maps so modified.
[Hint: For $$\left.\sigma \text { -additivity, use our present Theorem } 2 \text { twice. Omit } \mathcal{P}^{*} .\right]$$

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

Continuing Problem $$7,$$ let $$\mathcal{P}$$ be the $$\sigma$$-ring generated by $$\mathcal{C}$$ in $$X \times Y .$$ Prove that $$(\forall D \in \mathcal{P}) C_{D}$$ is a Fubini map (as modified).
[Outline: Proceed as in Lemma 5 of $$§8 . \text { For step (ii), use Theorem 2 in } §10 \text { twice. }]$$

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

Further continuing Problems 7 and $$8,$$ define
$(\forall D \in \mathcal{P}) \quad p D=\int_{X} \int_{Y} C_{D} d n d m .$
Show that $$p$$ is a generalized measure, with $$p=s$$ on $$\mathcal{C},$$ and that
$(\forall D \in \mathcal{P}) \quad p D=\int_{X \times Y} C_{D} d p ,$
with the following convention: If $$X \times Y \notin \mathcal{P},$$ we set
$\int_{X \times Y} f d p=\int_{H} f d p$
whenever $$H \in \mathcal{P}, f$$ is $$p$$-integrable on $$H,$$ and $$f=0$$ on $$-H .$$
$$\quad$$ Verify that this is unambiguous, i.e.,
$\int_{X \times Y} f d p$
so defined is independent of the choice of $$H$$.
Finally, let $$\overline{p}$$ be the completion of $$p$$ (Chapter $$7,$$ §6, Problem 15 ); let $$\mathcal{P}^{*}$$ be its domain.
Develop the rest of Fubini theory "imitating" Problem 12 in §8.

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

Signed Lebesgue-Stielttjes $$(L S)$$ measures in $$E^{1}$$ are defined as shown in Chapter 7, §11, Part $$V .$$ Using the notation of that section, prove the following:
(i) Given a Borel-Stieltjes measure $$\sigma_{\alpha}^{*}$$ in an interval $$I \subseteq E^{1}$$ (or an LS measure $$s_{\alpha}=\overline{\sigma}^{*}_{\alpha}$$ in $$I$$ ), there are two monotone functions $$g \uparrow$$ and $$h \uparrow(\alpha=g-h)$$ such that
$m_{g}=s_{\alpha}^{+} \text {and } m_{h}=s_{\alpha}^{-} ,$
both satisfying formula ( 3 ) of Chapter 7, §11, inside $$I$$.
(ii) If $$f$$ is $$s_{\alpha}$$-integrable on $$A \subseteq I,$$ then
$\int_{A} f d s_{\alpha}=\int_{A} f d m_{g}-\int_{A} f d m_{h}$
for any $$g \uparrow$$ and $$h \uparrow$$ (finite) such that $$\alpha=g-h$$.
[Hints: (i) Define $$s_{\alpha}^{+}$$ and $$s_{\alpha}^{-}$$ by formula (3) of Chapter $$7,$$ §7. Then arguing as in Theorem 2 in Chapter 7, §9, find $$g \uparrow$$ and $$h \uparrow$$ with $$m_{g}=s_{\alpha}^{+}$$ and $$m_{h}=s_{\alpha}^{-}$$.
(ii) First let $$A=(a, b] \subseteq I,$$ then $$A \in B .$$ Proceed.]