8.10.E: Problems on Generalized Integration
( \newcommand{\kernel}{\mathrm{null}\,}\)
Fill in the missing details in the proofs of this section. Prove Note 3.
Treat Corollary 1 (ii) as a definition of
∫Afds
for s:M→E and elementary and integrable f, even if E≠En(Cn).
Hence deduce Corollary 1(i)(vi) for this more general case.
Using Lemma 2 in §7, with m=vs,s:M→E, 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.
Define
∫Afds(s:M→E)
also for vsA=∞.
[Hint: Set m=vs in Lemma 3 of §7.]
Prove Theorems 1 to 3 for the general case, s:M→E (see Problem 4 ).
[Hint: Argue as in §7.]
From Problems 2−4, deduce Definition 2 as a theorem in the case E= En(Cn).
Let s,sk:M→E(k=1,2,…) be any set functions. Let A∈M and
MA={X∈M|X⊆A}.
Prove that if
(∀X∈MA)limk→∞skX=sX,
then
limk→∞vskA=vsA,
provided limk→∞vsk exists.
[Hint: Using Problem 2 in Chapter 7, §11, fix a finite disjoint sequence {Xi}⊆MA.
Then
∑i|sXi|=∑ilimk→∞|skXi|=limk→∞∑i|skXi|≤limk→∞vskA.
Infer that
vsA≤limk→∞vskA.
Also,
(∀ε>0)(∃k0)(∀k>k0)∑i|skXi|≤∑i|sXi|+ε≤vsA+ε.
Proceed.]
Let (X,M,m) and (Y,N,n) be two generalized measure spaces (X∈ M,Y∈N) such that mn is defined (Note 1). Set
C={A×B|A∈M,B∈N,vmA<∞,vnB<∞}
and s(A×B)=mA⋅nB for A×B∈C.
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∗.]
Continuing Problem 7, let P be the σ-ring generated by C in X×Y. Prove that (∀D∈P)CD is a Fubini map (as modified).
[Outline: Proceed as in Lemma 5 of §8. For step (ii), use Theorem 2 in §10 twice. ]
Further continuing Problems 7 and 8, define
(∀D∈P)pD=∫X∫YCDdndm.
Show that p is a generalized measure, with p=s on C, and that
(∀D∈P)pD=∫X×YCDdp,
with the following convention: If X×Y∉P, we set
∫X×Yfdp=∫Hfdp
whenever H∈P,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.
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 σ∗α in an interval I⊆E1 (or an LS measure sα=¯σ∗α in I ), there are two monotone functions g↑ and h↑(α=g−h) such that
mg=s+αand mh=s−α,
both satisfying formula ( 3 ) of Chapter 7, §11, inside I.
(ii) If f is sα-integrable on A⊆I, then
∫Afdsα=∫Afdmg−∫Afdmh
for any g↑ and h↑ (finite) such that α=g−h.
[Hints: (i) Define s+α and s−α 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=s−α.
(ii) First let A=(a,b]⊆I, then A∈B. Proceed.]