Processing math: 93%
Skip to main content
Library homepage
 

Text Color

Text Size

 

Margin Size

 

Font Type

Enable Dyslexic Font
Mathematics LibreTexts

7.10.E: Problems on Generalized Measures

( \newcommand{\kernel}{\mathrm{null}\,}\)

Exercise 7.10.E.1

Complete the proofs of Theorems 1,4, and 5.

Exercise 7.10.E.1

Do it also for the lemmas and Corollary 3.

Exercise 7.10.E.2

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:ME is additive on M, a semiring, so is vs.
[Hint: Use Theorem 1 from §4.]

Exercise 7.10.E.3

For any set functions s,t on M, prove that
(i) v|s|=vs, and
(ii) vstavt, provided st is defined and
a=sup{|sX||XM}.

Exercise 7.10.E.4

Given s,t:ME, show that
(i) vs+tvs+vt;
(ii) vks=|k|vs ((\k\) as in Corollary 2); and
(iii) if E=En(Cn) and
s=nk=1sk¯ek,
then
vskvsnk=1vsk.
[Hints: (i) If
AXi (disjoint),
with Ai,XiM, verify that
|(s+t)Xi||sXi|+|tXi|,|(s+t)Xi|vsA+vtA, etc.;
(ii) is analogous.
(iii) Use (ii) and (i), with |¯ek|=1.]

Exercise 7.10.E.5

If g,h, and α=gh on E1, can one define the signed LS measure sα by simply setting sα=mgmh (assuming mh<)?
[Hint: the domains of mg and mh may be different. Give an example. How about taking their intersection?]

Exercise 7.10.E.6

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|,x0,0,x=0,
and

A=n=1(n,n+1n2].]

Exercise 7.10.E.7

Construct complex and vector-valued LS measures sα:MαEn(Cn) in E1.

Exercise 7.10.E.8

Show that if s:MEn(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 sXi0(sXi<0, respectively). Show that
sXi=sU+sU2sup|s|<.]

Exercise 7.10.E.9

For any s:ME and AM, set
s+A=sup{sX|AXM}
and
sA=sup{sX|AXM}.
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(vss)0,s=s+s, and vs=s++s.
[Hints: Use Problem 8. Set
s=12(vs+s).
Then (XM|XA)
2sX=sA+sXs(AX)sA+(|sX|+|s(AX)|)sA+vsA=2sA.
Deduce that s+AsA.
To prove also that sAs+A, let ε>0. By Problems 2 and 8, fix {Xi}M, with
A=ni=1Xi (disjoint)
and
vsAε<ni=1|sXi|.
Show that
2sAε=vsA+sAεsU+sU+sni=1Xi=2sU+
and
2s+A2sU+2sAε.]

Exercise 7.10.E.10

Let
K={compact sets in a topological space (S,G)}
(adopt Theorem 2 in Chapter 4, §7, as a definition). Given
s:ME,M2S,
we call s compact regular (CR) iff
(ε>0)(AM)(FK)(GG)F,GM,FAG, and vsGεvsAvsF+ε.
Prove the following.
(i) If s,t:ME 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(GF).]

Exercise 7.10.E.11

(Aleksandrov.) Show that if s:ME 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 SM 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,AiM;
then
|sAr1i=1sAi|i=rvsAi0
as r, for
\sum_{i=1}^{\infty} v_{s} A_{i}=v_{s} \bigcup_{i=1}^{\infty} A_{i}<\infty.
(Explain!) Now, Theorem 2 of §6 extends v_{s} to a measure on a \sigma-field
\mathcal{M}^{*} \supseteq \mathcal{N} \supseteq \mathcal{M}
(use the minimality of \mathcal{N}). Its restriction to \mathcal{N} is the desired \overline{v}_{s} (unique by Problem 15 in §6).
A similar proof holds for s, too, if s : \mathcal{M} \rightarrow[0, \infty). The case s : \mathcal{M} \rightarrow E^{n}\left(C^{n}\right) results via Theorem 5 and Problem 10(iii) provided S \in \mathcal{M}; for then by Corollary 1, v_{s} S<\infty ensures the finiteness of v_{s}, s^{+}, and s^{-} even on \mathcal{N}.]

Exercise \PageIndex{12}

Do Problem 11 for semirings \mathcal{M}.
[Hint: Use Problem 10(ii).]


7.10.E: Problems on Generalized Measures is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

Support Center

How can we help?