$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

# 7.5.E: Problems on Premeasures and Related Topics

$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

## 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 $$\mu : \mathcal{C} \rightarrow E^{*}$$ such that each of the following hold.
(a) $$\mathcal{C}=\{S, \emptyset\}, \mu S=1$$.
(b) $$\mathcal{C}=\{S, \emptyset, \text {and all singletons}\}; \mu S=\infty, \mu\{x\}=1$$.
(c) $$\mathcal{C}$$ as in (b), with $$S$$ uncountable; $$\mu S=1,$$ and $$\mu X=0$$ otherwise.
(d) $$\mathcal{C}=\{\text {all proper subsets of } S\}; \mu X=1$$ when $$\emptyset \subset X \subset S; \mu \emptyset=0$$.

## Exercise $$\PageIndex{3}$$

Show that the premeasures
$v^{\prime} : \mathcal{C}^{\prime} \rightarrow[0, \infty]$
induce one and the same (Lebesgue) outer measure $$m^{*}$$ in $$E^{n},$$ with $$v^{\prime}=v$$ (volume, as in §2):
(a) $$\mathcal{C}^{\prime}=\{\text {open intervals}\}$$;
(b) $$\mathcal{C}^{\prime}=\{\text {half-open intervals}\}$$;
(c) $$\mathcal{C}^{\prime}=\{\text {closed intervals}\}$$;
(d) $$\mathcal{C}^{\prime}=\mathcal{C}_{\sigma}$$;
(e) $$\mathcal{C}^{\prime}=\{\text {open sets}\}$$;
(f) $$\mathcal{C}^{\prime}=\{\text {half-open cubes}\}$$;
[Hints: (a) Let $$m^{\prime}$$ be the $$v^{\prime}$$-induced outer measure; let $$\mathcal{C}=\{\text {all intervals}\}.$$ As $$\mathcal{C}^{\prime} \subseteq \mathcal{C}, m^{\prime} A \geq m^{*} A.$$ (Why?) Also,
$(\forall \varepsilon>0)\left(\exists\left\{B_{k}\right\} \subseteq \mathcal{C}\right) \quad A \subseteq \bigcup_{k} B_{k} \text { and } \sum v B_{k} \leq m^{*} A+\varepsilon.$
(Why?) By Lemma 1 in §2,
$\left(\exists\left\{C_{k}\right\} \subseteq \mathcal{C}^{\prime}\right) \quad B_{k} \subseteq C_{k} \text { and } v B_{k}+\frac{\varepsilon}{2^{k}}>v^{\prime} C_{k}.$
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 $$0$$ $$\left(m^{*} A=0\right),$$ while its Jordan outer content $$c^{*} A=\infty$$;
(ii) $$m^{*} A=0, c^{*} A=1$$ (see Corollary 6 in §2).

## Exercise $$\PageIndex{7}$$

Let
$\mu_{1}, \mu_{2} : \mathcal{C} \rightarrow[0, \infty]$
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
$\left\{X \subseteq S | c^{*} A=0\right\}$
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
$\left\{X \subseteq S | m^{*} X=0\right\}$
is a hereditary $$\sigma$$-ring.
(iv) So also is
$\mathcal{H}=\{\text {those } X \subseteq S \text { that have basic coverings}\};$
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 $$\left\{Y_{n}\right\},$$ with $$m^{*} Y_{n}<\infty$$ and with all $$Y_{n}$$ disjoint.
[Hint: Show that
$X \subseteq \bigcup_{n=1}^{\infty} \bigcup_{k=1}^{\infty} B_{n k}$
for some sets $$B_{n k} \in \mathcal{C},$$ with $$\mu B_{n k}<\infty.$$ Then use Note 4 in §5 and Corollary 1 of §1.]

## Exercise $$\PageIndex{10}$$

Show that if
$s : \mathcal{C} \rightarrow E^{*}$
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
$\mathcal{C}^{\prime}=\{X \in \mathcal{C}| | s X |<\infty\}$
(cf. Problem 6 in §4).
[Hint: By $$\sigma$$-finiteness,
$(\forall X \in \mathcal{C})\left(\exists\left\{A_{n}\right\} \subseteq \mathcal{C}| | s A_{n} |<\infty\right) \quad X \subseteq \bigcup_{n} A_{n};$
so
$X=\bigcup_{n}\left(X \cap A_{n}\right), \quad X \cap A_{n} \in \mathcal{C}^{\prime}.$
(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
$\left(\exists B \in \mathcal{A}_{\delta}\right) \quad A \subseteq B \text { and } m^{*} A=m^{*} B.$
[Hint: By formula (4),
$(\forall n \in N)\left(\exists X_{n} \in \mathcal{A} | A \subseteq X_{n}\right) \quad m^{*} A \leq m X_{n} \leq m^{*} A+\frac{1}{n}.$
(Explain!) Set
$B=\bigcap_{n=1}^{\infty} X_{n} \in \mathcal{A}_{\delta}.$
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
$(\forall A \subseteq S)(\exists B \in \mathcal{C}) \quad A \subseteq B \text { and } m^{*} A=\mu B.$
[Hint: Use Problem 11 and Note 3.]

## Exercise $$\PageIndex{13}$$

$$\Rightarrow^{*}$$ Show that if
$s : \mathcal{C} \rightarrow E$
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
$s^{\prime}, s^{\prime \prime} : \mathcal{R} \rightarrow E$
be two $$\sigma$$-additive extensions of $$s.$$ By Problem 10, $$\mathcal{R}$$ is also generated by
$\mathcal{C}^{\prime}=\{X \in \mathcal{C}| | s X |<\infty\}.$
Now set
$\mathcal{R}^{*}=\left\{X \in \mathcal{R} | s^{\prime} X=s^{\prime \prime} X\right\}.$
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
$\mu^{*}=\lim _{n \rightarrow \infty} m_{n}^{*}.$
Show that $$\mu^{*}$$ is an outer measure in $$S$$ (see Note 5).

## Exercise $$\PageIndex{15}$$

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

## Exercise $$\PageIndex{16}$$

Let $$(S, \mathcal{C}, \mu)$$ and $$m^{*}$$ be as in Definition 3, with $$\rho$$ 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) $$\left\{m_{n}^{*}\right\} \uparrow$$ as in Problem 14;
(ii) the outer measure
$\mu^{*}=\lim _{n \rightarrow \infty} m_{n}^{*}$
has the CP (see Problem 15), and
$\mu^{*} \geq m^{*} \text { on } 2^{S}.$
[Outline: Let $$\rho(X, Y)>\varepsilon>0(X, Y \subseteq S)$$.
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
$\left(\forall n>\frac{1}{\varepsilon}\right)(\forall k)\left(\exists B_{n k} \in \mathcal{C}_{n}\right) \quad X \cup Y \subseteq \bigcup_{k=1}^{\infty} B_{n k}.$
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
$d B_{n k} \leq \frac{1}{n}<\varepsilon,$
some $$B_{n k}$$ cover $$X$$ only, others $$Y$$ only. (Why?) Deduce that
$\left(\forall n>\frac{1}{\varepsilon}\right) \quad m_{n}^{*} X+m_{n}^{*} Y \leq \sum_{k=1}^{\infty} \mu_{n} B_{n k} \leq m_{n}^{*}(X \cup Y)+\varepsilon.$
Let $$\varepsilon \rightarrow 0$$ and then $$n \rightarrow \infty$$.
Also, $$m^{*} \leq m_{n}^{*} \leq \mu^{*}.$$ (Why?)]

## Exercise $$\PageIndex{17}$$

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

## Exercise $$\PageIndex{18}$$

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