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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)limkskX=sX,
then
limkvskA=vsA,
provided limkvsk exists.
[Hint: Using Problem 2 in Chapter 7, §11, fix a finite disjoint sequence {Xi}MA.
Then
i|sXi|=ilimk|skXi|=limki|skXi|limkvskA.
Infer that
vsAlimkvskA.
Also,
(ε>0)(k0)(k>k0)i|skXi|i|sXi|+εvsA+ε.
Proceed.]

Exercise 8.10.E.7

Let (X,M,m) and (Y,N,n) be two generalized measure spaces (X M,YN) such that mn is defined (Note 1). Set
C={A×B|AM,BN,vmA<,vnB<}
and s(A×B)=mAnB for A×BC.
Define a Fubini map as in §8, Part IV, dropping, however, X×Yfdp from Fubini property (c) temporarily.
Then prove Theorem 1 in §8, including formula (1), for Fubini maps so modified.
[Hint: For σ -additivity, use our present Theorem 2 twice. Omit P.]

Exercise 8.10.E.8

Continuing Problem 7, let P be the σ-ring generated by C in X×Y. Prove that (DP)CD is a Fubini map (as modified).
[Outline: Proceed as in Lemma 5 of §8. For step (ii), use Theorem 2 in §10 twice. ]

Exercise 8.10.E.9

Further continuing Problems 7 and 8, define
(DP)pD=XYCDdndm.
Show that p is a generalized measure, with p=s on C, and that
(DP)pD=X×YCDdp,
with the following convention: If X×YP, we set
X×Yfdp=Hfdp
whenever HP,f is p-integrable on H, and f=0 on H.
Verify that this is unambiguous, i.e.,
X×Yfdp
so defined is independent of the choice of H.
Finally, let ¯p be the completion of p (Chapter 7, §6, Problem 15 ); let P be its domain.
Develop the rest of Fubini theory "imitating" Problem 12 in §8.

Exercise 8.10.E.10

Signed Lebesgue-Stielttjes (LS) measures in E1 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 σα in an interval IE1 (or an LS measure sα=¯σα in I ), there are two monotone functions g and h(α=gh) such that
mg=s+αand mh=sα,
both satisfying formula ( 3 ) of Chapter 7, §11, inside I.
(ii) If f is sα-integrable on AI, then
Afdsα=AfdmgAfdmh
for any g and h (finite) such that α=gh.
[Hints: (i) Define s+α and sα by formula (3) of Chapter 7, §7. Then arguing as in Theorem 2 in Chapter 7, §9, find g and h with mg=s+α and mh=sα.
(ii) First let A=(a,b]I, then AB. 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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