7.5.E: Problems on Premeasures and Related Topics - Mathematics

Exercise (PageIndex{1})

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

Exercise (PageIndex{2})

Describe (m^{*}) on (2^{S}) induced by a premeasure (mu : mathcal{C} ightarrow E^{*}) such that each of the following hold.
(a) (mathcal{C}={S, emptyset}, mu S=1).
(b) (mathcal{C}={S, emptyset, ext {and all singletons}}; mu S=infty, mu{x}=1).
(c) (mathcal{C}) as in (b), with (S) uncountable; (mu S=1,) and (mu X=0) otherwise.
(d) (mathcal{C}={ ext {all proper subsets of } S}; mu X=1) when (emptyset subset X subset S; mu emptyset=0).

Exercise (PageIndex{3})

Show that the premeasures
[v^{prime} : mathcal{C}^{prime} ightarrow[0, infty]]
induce one and the same (Lebesgue) outer measure (m^{*}) in (E^{n},) with (v^{prime}=v) (volume, as in §2):
(a) (mathcal{C}^{prime}={ ext {open intervals}});
(b) (mathcal{C}^{prime}={ ext {half-open intervals}});
(c) (mathcal{C}^{prime}={ ext {closed intervals}});
(d) (mathcal{C}^{prime}=mathcal{C}_{sigma});
(e) (mathcal{C}^{prime}={ ext {open sets}});
(f) (mathcal{C}^{prime}={ ext {half-open cubes}});
[Hints: (a) Let (m^{prime}) be the (v^{prime})-induced outer measure; let (mathcal{C}={ ext {all intervals}}.) As (mathcal{C}^{prime} subseteq mathcal{C}, m^{prime} A geq m^{*} A.) (Why?) Also,
[(forall varepsilon>0)left(existsleft{B_{k} ight} subseteq mathcal{C} ight) quad A subseteq igcup_{k} B_{k} ext { and } sum v B_{k} leq m^{*} A+varepsilon.]
(Why?) By Lemma 1 in §2,
[left(existsleft{C_{k} ight} subseteq mathcal{C}^{prime} ight) quad B_{k} subseteq C_{k} ext { and } v B_{k}+frac{varepsilon}{2^{k}}>v^{prime} C_{k}.]
Deduce that (m^{*} A geq m^{prime} A, m^{*}=m^{prime}). 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 (PageIndex{3'})

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

Exercise (PageIndex{4})

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

Exercise (PageIndex{5})

Show that a set (A subseteq E^{n}) is bounded iff its outer Jordan content is finite.

Exercise (PageIndex{6})

Find a set (A subseteq E^{1}) such that
(i) its Lebesgue outer measure is (0) (left(m^{*} A=0 ight),) while its Jordan outer content (c^{*} A=infty);
(ii) (m^{*} A=0, c^{*} A=1) (see Corollary 6 in §2).

Exercise (PageIndex{7})

Let
[mu_{1}, mu_{2} : mathcal{C} ightarrow[0, infty]]
be two premeasures in (S) and let (m_{1}^{*}) and (m_{2}^{*}) be the outer measures induced by them.
Prove that if (m_{1}^{*}=m_{2}^{*}) on (mathcal{C},) then (m_{1}^{*}=m_{2}^{*}) on all of (2^{S}).

Exercise (PageIndex{8})

With the notation of Definition 3 and Note 6, prove the following.
(i) If (A subseteq B subseteq S) and (m^{*} B=0,) then (m^{*} A=0;) similarly for (c^{*}).
[Hint: Use monotonicity.]
(ii) The set family
[left{X subseteq S | c^{*} A=0 ight}]
is a hereditary set ring, i.e., a ring (mathcal{R}) such that
[(forall B in mathcal{R})(forall A subseteq B) quad A in mathcal{R}.]
(iii) The set family
[left{X subseteq S | m^{*} X=0 ight}]
is a hereditary (sigma)-ring.
(iv) So also is
[mathcal{H}={ ext {those } X subseteq S ext { that have basic coverings}};]
thus (mathcal{H}) is the hereditary (sigma)-ring generated by (mathcal{C}) (see Problem 14 in §3).

Exercise (PageIndex{9})

Continuing Problem 8(iv), prove that if (mu) is (sigma)-finite (Definition 4), so is (m^{*}) when restricted to (mathcal{H}.)
Show, moreover, that if (mathcal{C}) is a semiring, then each (X in mathcal{H}) has a basic covering (left{Y_{n} ight},) with (m^{*} Y_{n}[Hint: Show that
[X subseteq igcup_{n=1}^{infty} igcup_{k=1}^{infty} B_{n k}]
for some sets (B_{n k} in mathcal{C},) with (mu B_{n k}

Exercise (PageIndex{10})

Show that if
[s : mathcal{C} ightarrow E^{*}]
is (sigma)-finite and additive on (mathcal{C},) a semiring, then the (sigma)-ring (mathcal{R}) generated by (mathcal{C}) equals the (sigma)-ring (mathcal{R}^{prime}) generated by
[mathcal{C}^{prime}={X in mathcal{C}| | s X |(cf. Problem 6 in §4).
[Hint: By (sigma)-finiteness,
[(forall X in mathcal{C})left(existsleft{A_{n} ight} subseteq mathcal{C}| | s A_{n} |so
[X=igcup_{n}left(X cap A_{n} ight), quad X cap A_{n} in mathcal{C}^{prime}.]
(Use Lemma 3 in §4.)
Thus ((forall X in mathcal{C}) X) is a countable union of (mathcal{C}^{prime})-sets; so (mathcal{C} subseteq mathcal{R}^{prime}.) Deduce (mathcal{R} subseteq mathcal{R}^{prime}). Proceed.]

Exercise (PageIndex{11})

With all as in Theorem 3, prove that if (A) has basic coverings, then
[left(exists B in mathcal{A}_{delta} ight) quad A subseteq B ext { and } m^{*} A=m^{*} B.]
[Hint: By formula (4),
[(forall n in N)left(exists X_{n} in mathcal{A} | A subseteq X_{n} ight) quad m^{*} A leq m X_{n} leq m^{*} A+frac{1}{n}.]
(Explain!) Set
[B=igcap_{n=1}^{infty} X_{n} in mathcal{A}_{delta}.]
Proceed. For (mathcal{A}_{delta},) see Definition 2(b) in §3.]

Exercise (PageIndex{12})

Let ((S, mathcal{C}, mu)) and (m^{*}) be as in Definition 3. Show that if (mathcal{C}) is a (sigma)-field in (S,) then
[(forall A subseteq S)(exists B in mathcal{C}) quad A subseteq B ext { and } m^{*} A=mu B.]
[Hint: Use Problem 11 and Note 3.]

Exercise (PageIndex{13})

(Rightarrow^{*}) Show that if
[s : mathcal{C} ightarrow E]
is (sigma)-finite and (sigma)-additive on (mathcal{C},) a semiring, then (s) has at most one (sigma)-additive extension to the (sigma)-ring (mathcal{R}) generated by (mathcal{C}.)
(Note that (s) is automatically (sigma)-finite if it is finite, e.g., complex or vector valued.)
[Outline: Let
[s^{prime}, s^{prime prime} : mathcal{R} ightarrow E]
be two (sigma)-additive extensions of (s.) By Problem 10, (mathcal{R}) is also generated by
[mathcal{C}^{prime}={X in mathcal{C}| | s X |Now set
[mathcal{R}^{*}=left{X in mathcal{R} | s^{prime} X=s^{prime prime} X ight}.]
Show that (mathcal{R}^{*}) satisfies properties (i)-(iii) of Theorem 3 in §3, with (mathcal{C}) replaced by (mathcal{C}^{prime};) so (mathcal{R}=mathcal{R}^{*}).]

Exercise (PageIndex{14})

Let (m_{n}^{*}(n=1,2, ldots)) be outer measures in (S) such that
[(forall X subseteq S)(forall n) quad m_{n}^{*} X leq m_{n+1}^{*} X.]
Set
[mu^{*}=lim _{n ightarrow infty} m_{n}^{*}.]
Show that (mu^{*}) is an outer measure in (S) (see Note 5).

Exercise (PageIndex{15})

An outer measure (m^{*}) in a metric space ((S, ho)) is said to have the Carathéodory property (CP) iff
[m^{*}(X cup Y) geq m^{*} X+m^{*} Y]
whenever ( ho(X, Y)>0,) where
[ ho(X, Y)=inf { ho(x, y) | x in X, y in Y}.]
For such (m^{*},) prove that
[m^{*}left(igcup_{k} X_{k} ight)=sum_{k} m^{*} X_{k}]
if (left{X_{k} ight} subseteq 2^{S}) and
[ holeft(X_{i}, X_{k} ight)>0 quad(i eq k).]
[Hint: For finite unions, use the CP, subadditivity, and induction. Deduce that
[(forall n) sum_{k=1}^{n} m^{*} X_{k} leq m^{*} igcup_{k=1}^{infty} X_{k}.]
Let (n ightarrow infty.) Proceed.]

Exercise (PageIndex{16})

Let ((S, mathcal{C}, mu)) and (m^{*}) be as in Definition 3, with ( ho) a metric for (S.) Let (mu_{n}) be the restriction of (mu) to the family (mathcal{C}_{n}) of all (X in mathcal{C}) of diameter
[d X leq frac{1}{n}.]
Let (m_{n}^{*}) be the (mu_{n})-induced outer measure in (S.)
Prove that
(i) (left{m_{n}^{*} ight} uparrow) as in Problem 14;
(ii) the outer measure
[mu^{*}=lim _{n ightarrow infty} m_{n}^{*}]
has the CP (see Problem 15), and
[mu^{*} geq m^{*} ext { on } 2^{S}.]
[Outline: Let ( ho(X, Y)>varepsilon>0(X, Y subseteq S)).
If for some (n, X cup Y) has no basic covering from (mathcal{C}_{n},) then
[mu^{*}(X cup Y) geq m_{n}^{*}(X cup Y)=infty geq mu^{*} X+mu^{*} Y,]
and the CP follows. (Explain!)
Thus assume
[left(forall n>frac{1}{varepsilon} ight)(forall k)left(exists B_{n k} in mathcal{C}_{n} ight) quad X cup Y subseteq igcup_{k=1}^{infty} B_{n k}.]
One can choose the (B_{n k}) so that
[sum_{k=1}^{infty} mu B_{n k} leq m_{n}^{*}(X cup Y)+varepsilon.]
(Why?) As
[d B_{n k} leq frac{1}{n}some (B_{n k}) cover (X) only, others (Y) only. (Why?) Deduce that
[left(forall n>frac{1}{varepsilon} ight) quad m_{n}^{*} X+m_{n}^{*} Y leq sum_{k=1}^{infty} mu_{n} B_{n k} leq m_{n}^{*}(X cup Y)+varepsilon.]
Let (varepsilon ightarrow 0) and then (n ightarrow infty).
Also, (m^{*} leq m_{n}^{*} leq mu^{*}.) (Why?)]

Exercise (PageIndex{17})

Continuing Problem 16, suppose that
((forall varepsilon>0)(forall n, k)(forall B in mathcal{C})left(exists B_{n k} in mathcal{C}_{n} ight))
[B subseteq igcup_{k=1}^{infty} B_{n k} ext { and } mu B+varepsilon geq sum_{k=1}^{infty} mu B_{n k}.]
Show that
[m^{*}=lim _{n ightarrow infty} mu_{n}^{*}=mu^{*},]
so (m^{*}) itself has the CP.
[Hints: It suffices to prove that (m^{*} A geq mu^{*} A) if (m^{*} ANow, given (varepsilon>0, A) has a covering
[left{B_{i} ight} subseteq c]
such that
[m^{*} A+varepsilon geq sum mu B_{i}.]
(Why?) By assumption,
[(forall n) quad B_{i} subseteq igcup_{k=1}^{infty} B_{n k}^{i} in mathcal{C}_{n} ext { and } mu B_{i}+frac{varepsilon}{2^{i}} geq sum_{k=1}^{infty} mu B_{n k}^{i}.]
Deduce that
[m^{*} A+varepsilon>sum mu B_{i} geq sum_{i=1}^{infty}left(sum_{k=1}^{infty} mu B_{n k}^{i}-frac{varepsilon}{2^{i}} ight)=sum_{i, k} mu B_{n k}^{i}-varepsilon geq m_{n}^{*} A-varepsilon.]
Let (varepsilon ightarrow 0;) then (n ightarrow infty).]

Exercise (PageIndex{18})

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