8.10.E: Problems on Generalized Integration

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.]