8.10.E: Problems on Generalized Integration
Fill in the missing details in the proofs of this section. Prove Note 3.
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.
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.
Define
\[
\int_{A} f d s \quad(s: \mathcal{M} \rightarrow E)
\]
also for \(v_{s} A=\infty .\)
[Hint: Set \(\left.m=v_{s} \text { in Lemma } 3 \text { of } §7 .\right]\)
Prove Theorems 1 to 3 for the general case, \(s: \mathcal{M} \rightarrow E\) (see Problem 4 ).
[Hint: Argue as in §7.]
From Problems \(2-4,\) deduce Definition 2 as a theorem in the case \(E=\) \(E^{n}\left(C^{n}\right) .\)
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.]
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 \(\left.\sigma \text { -additivity, use our present Theorem } 2 \text { twice. Omit } \mathcal{P}^{*} .\right]\)
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 \(§8 . \text { For step (ii), use Theorem 2 in } §10 \text { twice. }]\)
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.
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.]