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Mathematics LibreTexts

8.10.E: Problems on Generalized Integration

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Exercise 8.10.E.1

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

Exercise 8.10.E.2

Treat Corollary 1 (ii) as a definition of
Afds
for s:ME and elementary and integrable f, even if EEn(Cn).
Hence deduce Corollary 1(i)(vi) for this more general case.

Exercise 8.10.E.3

Using Lemma 2 in §7, with m=vs,s:ME, construct
Afds
as in Definition 2 of §7 for the case vsA. Show that this agrees with Problem 2 if f is elementary and integrable. Then prove linearity for functions with vs-finite support as in §7.

Exercise 8.10.E.4

Define
Afds(s:ME)
also for vsA=.
[Hint: Set m=vs in Lemma 3 of §7.]

Exercise 8.10.E.5

Prove Theorems 1 to 3 for the general case, s:ME (see Problem 4 ).
[Hint: Argue as in §7.]

Exercise 8.10.E.5

From Problems 24, deduce Definition 2 as a theorem in the case E= En(Cn).

Exercise 8.10.E.6

Let s,sk:ME(k=1,2,) be any set functions. Let AM and
MA={XM|XA}.
Prove that if
(XMA)lim
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.]


8.10.E: Problems on Generalized Integration is shared under a CC BY 1.0 license and was authored, remixed, and/or curated by LibreTexts.

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