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Mathematics LibreTexts

7.5.E: Problems on Premeasures and Related Topics

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Exercise 7.5.E.1

Fill in the missing details in the proofs, notes, and examples of this section.

Exercise 7.5.E.2

Describe m on 2S induced by a premeasure μ:CE such that each of the following hold.
(a) C={S,},μS=1.
(b) C={S,,and all singletons};μS=,μ{x}=1.
(c) C as in (b), with S uncountable; μS=1, and μX=0 otherwise.
(d) C={all proper subsets of S};μX=1 when XS;μ=0.

Exercise 7.5.E.3

Show that the premeasures
v:C[0,]
induce one and the same (Lebesgue) outer measure m in En, with v=v (volume, as in §2):
(a) C={open intervals};
(b) C={half-open intervals};
(c) C={closed intervals};
(d) C=Cσ;
(e) C={open sets};
(f) C={half-open cubes};
[Hints: (a) Let m be the v-induced outer measure; let C={all intervals}. As CC,mAmA. (Why?) Also,
(ε>0)({Bk}C)AkBk and vBkmA+ε.
(Why?) By Lemma 1 in §2,
({Ck}C)BkCk and vBk+ε2k>vCk.
Deduce that mAmA,m=m. Similarly for (b) and (c). For (d), use Corollary 1 and Note 3 in §1. For (e), use Lemma 2 in §2. For (f), use Problem 2 in §2.]

Exercise 7.5.E.3

Do Problem 3(a)-(c), with m replaced by the Jordan outer content c (Note 6).

Exercise 7.5.E.4

Do Problem 3, with v and m replaced by the LS premeasure and outer measure. (Use Problem 7 in §4.)

Exercise 7.5.E.5

Show that a set AEn is bounded iff its outer Jordan content is finite.

Exercise 7.5.E.6

Find a set AE1 such that
(i) its Lebesgue outer measure is 0 (mA=0), while its Jordan outer content cA=;
(ii) mA=0,cA=1 (see Corollary 6 in §2).

Exercise 7.5.E.7

Let
μ1,μ2:C[0,]
be two premeasures in S and let m1 and m2 be the outer measures induced by them.
Prove that if m1=m2 on C, then m1=m2 on all of 2S.

Exercise 7.5.E.8

With the notation of Definition 3 and Note 6, prove the following.
(i) If ABS and mB=0, then mA=0; similarly for c.
[Hint: Use monotonicity.]
(ii) The set family
{XS|cA=0}
is a hereditary set ring, i.e., a ring R such that
(BR)(AB)AR.
(iii) The set family
{XS|mX=0}
is a hereditary σ-ring.
(iv) So also is
H={those XS that have basic coverings};
thus H is the hereditary σ-ring generated by C (see Problem 14 in §3).

Exercise 7.5.E.9

Continuing Problem 8(iv), prove that if μ is σ-finite (Definition 4), so is m when restricted to H.
Show, moreover, that if C is a semiring, then each XH has a basic covering {Yn}, with mYn< and with all Yn disjoint.
[Hint: Show that
Xn=1k=1Bnk
for some sets BnkC, with μBnk<. Then use Note 4 in §5 and Corollary 1 of §1.]

Exercise 7.5.E.10

Show that if
s:CE
is σ-finite and additive on C, a semiring, then the σ-ring R generated by C equals the σ-ring R generated by
C={XC||sX|<}
(cf. Problem 6 in §4).
[Hint: By σ-finiteness,
(XC)({An}C||sAn|<)XnAn;
so
X=n(XAn),XAnC.
(Use Lemma 3 in §4.)
Thus (XC)X is a countable union of C-sets; so CR. Deduce RR. Proceed.]

Exercise 7.5.E.11

With all as in Theorem 3, prove that if A has basic coverings, then
(BAδ)AB and mA=mB.
[Hint: By formula (4),
(nN)(XnA|AXn)mAmXnmA+1n.
(Explain!) Set
B=n=1XnAδ.
Proceed. For Aδ, see Definition 2(b) in §3.]

Exercise 7.5.E.12

Let (S,C,μ) and m be as in Definition 3. Show that if C is a σ-field in S, then
(AS)(BC)AB and mA=μB.
[Hint: Use Problem 11 and Note 3.]

Exercise 7.5.E.13

Show that if
s:CE
is σ-finite and σ-additive on C, a semiring, then s has at most one σ-additive extension to the σ-ring R generated by C.
(Note that s is automatically σ-finite if it is finite, e.g., complex or vector valued.)
[Outline: Let
s,s:RE
be two σ-additive extensions of s. By Problem 10, R is also generated by
C={XC||sX|<}.
Now set
R={XR|sX=sX}.
Show that R satisfies properties (i)-(iii) of Theorem 3 in §3, with C replaced by C; so R=R.]

Exercise 7.5.E.14

Let mn(n=1,2,) be outer measures in S such that
(XS)(n)mnXmn+1X.
Set
μ=limnmn.
Show that μ is an outer measure in S (see Note 5).

Exercise 7.5.E.15

An outer measure m in a metric space (S,ρ) is said to have the Carathéodory property (CP) iff
m(XY)mX+mY
whenever ρ(X,Y)>0, where
ρ(X,Y)=inf{ρ(x,y)|xX,yY}.
For such m, prove that
m(kXk)=kmXk
if {Xk}2S and
ρ(Xi,Xk)>0(ik).
[Hint: For finite unions, use the CP, subadditivity, and induction. Deduce that
(n)nk=1mXkmk=1Xk.
Let n. Proceed.]

Exercise 7.5.E.16

Let (S,C,μ) and m be as in Definition 3, with ρ a metric for S. Let μn be the restriction of μ to the family Cn of all XC of diameter
dX1n.
Let mn be the μn-induced outer measure in S.
Prove that
(i) {mn} as in Problem 14;
(ii) the outer measure
μ=limnmn
has the CP (see Problem 15), and
μm on 2S.
[Outline: Let ρ(X,Y)>ε>0(X,YS).
If for some n,XY has no basic covering from Cn, then
μ(XY)mn(XY)=μX+μY,
and the CP follows. (Explain!)
Thus assume
(n>1ε)(k)(BnkCn)XYk=1Bnk.
One can choose the Bnk so that
k=1μBnkmn(XY)+ε.
(Why?) As
dBnk1n<ε,
some Bnk cover X only, others Y only. (Why?) Deduce that
(n>1ε)mnX+mnYk=1μnBnkmn(XY)+ε.
Let ε0 and then n.
Also, mmnμ. (Why?)]

Exercise 7.5.E.17

Continuing Problem 16, suppose that
(ε>0)(n,k)(BC)(BnkCn)
Bk=1Bnk and μB+εk=1μBnk.
Show that
m=limnμn=μ,
so m itself has the CP.
[Hints: It suffices to prove that mAμA if mA<. (Why?)
Now, given ε>0,A has a covering
{Bi}c
such that
mA+εμBi.
(Why?) By assumption,
(n)Bik=1BinkCn and μBi+ε2ik=1μBink.
Deduce that
mA+ε>μBii=1(k=1μBinkε2i)=i,kμBinkεmnAε.
Let ε0; then n.]

Exercise 7.5.E.18

Using Problem 17, show that the Lebesgue and Lebesgue-Stieltjes outer measures have the CP.


7.5.E: Problems on Premeasures and Related Topics is shared under a CC BY 1.0 license and was authored, remixed, and/or curated by LibreTexts.

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