7.10.E: Problems on Generalized Measures
( \newcommand{\kernel}{\mathrm{null}\,}\)
Complete the proofs of Theorems 1,4, and 5.
Do it also for the lemmas and Corollary 3.
Verify the following.
(i) In Definition 2, one can equivalently replace "countable {Xi}" by "finite {Xi}."
(ii) If M is a ring, Note 1 holds for finite sequences {Xi}.
(iii) If s:M→E is additive on M, a semiring, so is vs.
[Hint: Use Theorem 1 from §4.]
For any set functions s,t on M, prove that
(i) v|s|=vs, and
(ii) vst≤avt, provided st is defined and
a=sup{|sX||X∈M}.
Given s,t:M→E, show that
(i) vs+t≤vs+vt;
(ii) vks=|k|vs ((\k\) as in Corollary 2); and
(iii) if E=En(Cn) and
s=n∑k=1sk¯ek,
then
vsk≤vs≤n∑k=1vsk.
[Hints: (i) If
A⊇⋃Xi (disjoint),
with Ai,Xi∈M, verify that
|(s+t)Xi|≤|sXi|+|tXi|,∑|(s+t)Xi|≤vsA+vtA, etc.;
(ii) is analogous.
(iii) Use (ii) and (i), with |¯ek|=1.]
If g↑,h↑, and α=g−h on E1, can one define the signed LS measure sα by simply setting sα=mg−mh (assuming mh<∞)?
[Hint: the domains of mg and mh may be different. Give an example. How about taking their intersection?]
Find an LS measure mα such that α is continuous and one-to-one, but mα is not m-finite (m=Lebesgue measure).
[Hint: Take
α(x)={x3|x|,x≠0,0,x=0,
and
A=∞⋃n=1(n,n+1n2].]
Construct complex and vector-valued LS measures sα:M∗α→En(Cn) in E1.
Show that if s:M→En(Cn) is additive and bounded on M, a ring, so is vs.
[Hint: By Problem 4(iii), reduce all to the real case.
Use Problem 2. Given a finite disjoint sequence {Xi}⊆M, let U+(U−) be the union of those Xi for which sXi≥0(sXi<0, respectively). Show that
∑sXi=sU+−sU−≤2sup|s|<∞.]
For any s:M→E∗ and A∈M, set
s+A=sup{sX|A⊇X∈M}
and
s−A=sup{−sX|A⊇X∈M}.
Prove that if s is additive and bounded on M, a ring, so are s+ and s−; furthermore,
s+=12(vs+s)≥0,s−=12(vs−s)≥0,s=s+−s−, and vs=s++s−.
[Hints: Use Problem 8. Set
s′=12(vs+s).
Then (∀X∈M|X⊆A)
2sX=sA+sX−s(A−X)≤sA+(|sX|+|s(A−X)|)≤sA+vsA=2s′A.
Deduce that s+A≤s′A.
To prove also that s′A≤s+A, let ε>0. By Problems 2 and 8, fix {Xi}⊆M, with
A=n⋃i=1Xi (disjoint)
and
vsA−ε<n∑i=1|sXi|.
Show that
2s′A−ε=vsA+sA−ε≤sU+−sU−+sn⋃i=1Xi=2sU+
and
2s+A≥2sU+≥2s′A−ε.]
Let
K={compact sets in a topological space (S,G)}
(adopt Theorem 2 in Chapter 4, §7, as a definition). Given
s:M→E,M⊆2S,
we call s compact regular (CR) iff
(∀ε>0)(∀A∈M)(∃F∈K)(∃G∈G)F,G∈M,F⊆A⊆G, and vsG−ε≤vsA≤vsF+ε.
Prove the following.
(i) If s,t:M→E are CR, so are s±t and ks (k as in Corollary 2).
(ii) If s is additive and CR on M, a semiring, so is its extension to the ring Ms (Theorem 1 in §4 and Theorem 4 of §3).
(iii) If E=En(Cn) and vs<∞ on M, a ring, then s is CR iff its components sk are, or in the case E=E1, iff s+ and s− are (see Problem 9).
[Hint for (iii): Use (i) and Problem 4(iii). Consider vs(G−F).]
(Aleksandrov.) Show that if s:M→E is CR (see Problem 10) and additive on M, a ring in a topological space S, and if vs<∞ on M, then vs and s are σ-additive, and vs has a unique σ-additive extension ¯vs to the σ-ring N generated by M.
The latter holds for s, too, if S∈M and E=En(Cn).
[Proof outline: The σ-additivity of vs results as in Theorem 1 of §2 (first check Lemma 1 in §1 for vs).
For the σ-additivity of s, let
A=∞⋃i=1Ai (disjoint),A,Ai∈M;
then
|sA−r−1∑i=1sAi|≤∞∑i=rvsAi→0
as r→∞, for
∞∑i=1vsAi=vs∞⋃i=1Ai<∞.
(Explain!) Now, Theorem 2 of §6 extends vs to a measure on a σ-field
M∗⊇N⊇M
(use the minimality of N). Its restriction to N is the desired ¯vs (unique by Problem 15 in §6).
A similar proof holds for s, too, if s:M→[0,∞). The case s:M→En(Cn) results via Theorem 5 and Problem 10(iii) provided S∈M; for then by Corollary 1, vsS<∞ ensures the finiteness of vs,s+, and s− even on N.]
Do Problem 11 for semirings M.
[Hint: Use Problem 10(ii).]