7.10.E: Problems on Generalized Measures

Exercise $\PageIndex{1}$

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

Exercise $\PageIndex{1'}$

Do it also for the lemmas and Corollary 3.

Exercise $\PageIndex{2}$

Verify the following.
(i) In Definition 2, one can equivalently replace "countable $\left\{X_{i}\right\}$" by "finite $\left\{X_{i}\right\}$."
(ii) If $\mathcal{M}$ is a ring, Note 1 holds for finite sequences $\left\{X_{i}\right\}$.
(iii) If $s : \mathcal{M} \rightarrow E$ is additive on $\mathcal{M},$ a semiring, so is $v_{s}$.
[Hint: Use Theorem 1 from §4.]

Exercise $\PageIndex{3}$

For any set functions $s, t$ on $\mathcal{M},$ prove that
(i) $\quad v_{|s|}=v_{s},$ and
(ii) $v_{s t} \leq a v_{t},$ provided $s t$ is defined and
$$a=\sup \{|s X| | X \in \mathcal{M}\}.$$

Exercise $\PageIndex{4}$

Given $s, t : \mathcal{M} \rightarrow E,$ show that
(i) $v_{s+t} \leq v_{s}+v_{t}$;
(ii) $v_{k s}=|k| v_{s}$ ((\k\) as in Corollary 2); and
(iii) if $E=E^{n}\left(C^{n}\right)$ and
$$s=\sum_{k=1}^{n} s_{k} \overline{e}_{k},$$
then
$$v_{s_{k}} \leq v_{s} \leq \sum_{k=1}^{n} v_{s k}.$$
[Hints: (i) If
$$A \supseteq \bigcup X_{i} \text { (disjoint),}$$
with $A_{i}, X_{i} \in \mathcal{M},$ verify that
$$\begin{array}{c}{\left|(s+t) X_{i}\right| \leq\left|s X_{i}\right|+\left|t X_{i}\right|,} \\ {\sum\left|(s+t) X_{i}\right| \leq v_{s} A+v_{t} A, \text { etc.;}}\end{array}$$
(ii) is analogous.
(iii) Use (ii) and (i), with $\left|\overline{e}_{k}\right|=1$.]

Exercise $\PageIndex{5}$

If $g \uparrow, h \uparrow,$ and $\alpha=g-h$ on $E^{1}$, can one define the signed LS measure $s_{\alpha}$ by simply setting $s_{\alpha}=m_{g}-m_{h}$ (assuming $m_{h}<\infty$)?
[Hint: the domains of $m_{g}$ and $m_{h}$ may be different. Give an example. How about taking their intersection?]

Exercise $\PageIndex{6}$

Find an LS measure $m_{\alpha}$ such that $\alpha$ is continuous and one-to-one, but $m_{\alpha}$ is not $m$-finite ( $m=$Lebesgue measure).
[Hint: Take
$$\alpha(x)=\left\{\begin{array}{ll}{\frac{x^{3}}{|x|},} & {x \neq 0,} \\ {0,} & {x=0,}\end{array}\right.$$
and

$$\left.A=\bigcup_{n=1}^{\infty}\left(n, n+\frac{1}{n^{2}}\right] .\right]$$

Exercise $\PageIndex{7}$

Construct complex and vector-valued LS measures $s_{\alpha}: \mathcal{M}_{\alpha}^{*} \rightarrow E^{n}\left(C^{n}\right)$ in $E^{1}.$

Exercise $\PageIndex{8}$

Show that if $s : \mathcal{M} \rightarrow E^{n}\left(C^{n}\right)$ is additive and bounded on $\mathcal{M},$ a ring, so is $v_{s}$.
[Hint: By Problem 4(iii), reduce all to the real case.
Use Problem 2. Given a finite disjoint sequence $\left\{X_{i}\right\} \subseteq \mathcal{M},$ let $U^{+}\left(U^{-}\right)$ be the union of those $X_{i}$ for which $s X_{i} \geq 0 (s X_{i}<0,$ respectively). Show that
$$\left.\sum s X_{i}=s U^{+}-s U^{-} \leq 2 \sup |s|<\infty.\right]$$

Exercise $\PageIndex{9}$

For any $s : \mathcal{M} \rightarrow E^{*}$ and $A \in \mathcal{M},$ set
$$s^{+} A=\sup \{s X | A \supseteq X \in \mathcal{M}\}$$
and
$$s^{-} A=\sup \{-s X | A \supseteq X \in \mathcal{M}\}.$$
Prove that if $s$ is additive and bounded on $\mathcal{M},$ a ring, so are $s^{+}$ and $s^{-};$ furthermore,
$$\begin{aligned} s^{+} &=\frac{1}{2}\left(v_{s}+s\right) \geq 0, \\ s^{-} &=\frac{1}{2}\left(v_{s}-s\right) \geq 0, \\ s &=s^{+}-s^{-}, \text{ and } \\ v_{s} &=s^{+}+s^{-}. \end{aligned}$$
[Hints: Use Problem 8. Set
$$s^{\prime}=\frac{1}{2}\left(v_{s}+s\right).$$
Then $(\forall X \in \mathcal{M} | X \subseteq A)$
$$\begin{aligned} 2 s X=s A+s X-s(A-X) & \leq s A+(|s X|+|s(A-X)|) \\ & \leq s A+v_{s} A=2 s^{\prime} A. \end{aligned}$$
Deduce that $s^{+} A \leq s^{\prime} A$.
To prove also that $s^{\prime} A \leq s^{+} A,$ let $\varepsilon>0.$ By Problems 2 and 8, fix $\left\{X_{i}\right\} \subseteq \mathcal{M}$, with
$$A=\bigcup_{i=1}^{n} X_{i} \text { (disjoint)}$$
and
$$v_{s} A-\varepsilon<\sum_{i=1}^{n}\left|s X_{i}\right|.$$
Show that
$$2 s^{\prime} A-\varepsilon=v_{s} A+s A-\varepsilon \leq s U^{+}-s U^{-}+s \bigcup_{i=1}^{n} X_{i}=2 s U^{+}$$
and
$$\left.2 s^{+} A \geq 2 s U^{+} \geq 2 s^{\prime} A-\varepsilon.\right]$$

Exercise $\PageIndex{10}$

Let
$$\mathcal{K}=\{\text {compact sets in a topological space }(S, \mathcal{G})\}$$
(adopt Theorem 2 in Chapter 4, §7, as a definition). Given
$$s : \mathcal{M} \rightarrow E, \quad \mathcal{M} \subseteq 2^{S},$$
we call $s$ compact regular (CR) iff
$$\begin{aligned}(\forall \varepsilon>0) &(\forall A \in \mathcal{M})(\exists F \in \mathcal{K})(\exists G \in \mathcal{G}) \\ F, G & \in \mathcal{M}, F \subseteq A \subseteq G, \text { and } v_{s} G-\varepsilon \leq v_{s} A \leq v_{s} F+\varepsilon. \end{aligned}$$
Prove the following.
(i) If $s, t : \mathcal{M} \rightarrow E$ are $\mathrm{CR},$ so are $s \pm t$ and $k s$ ($k$ as in Corollary 2).
(ii) If $s$ is additive and CR on $\mathcal{M},$ a semiring, so is its extension to the ring $\mathcal{M}_{s}$ (Theorem 1 in §4 and Theorem 4 of §3).
(iii) If $E=E^{n}\left(C^{n}\right)$ and $v_{s}<\infty$ on $\mathcal{M},$ a ring, then $s$ is CR iff its components $s_{k}$ are, or in the case $E=E^{1},$ iff $s^{+}$ and $s^{-}$ are (see Problem 9).
[Hint for (iii): Use (i) and Problem 4(iii). Consider $v_{s}(G-F)$.]

Exercise $\PageIndex{11}$

(Aleksandrov.) Show that if $s : \mathcal{M} \rightarrow E$ is CR (see Problem 10) and additive on $\mathcal{M},$ a ring in a topological space $S,$ and if $v_{s}<\infty$ on $\mathcal{M}$, then $v_{s}$ and $s$ are $\sigma$-additive, and $v_{s}$ has a unique $\sigma$-additive extension $\overline{v}_{s}$ to the $\sigma$-ring $\mathcal{N}$ generated by $\mathcal{M}.$
The latter holds for $s,$ too, if $S \in \mathcal{M}$ and $E=E^{n}\left(C^{n}\right)$.
[Proof outline: The $\sigma$-additivity of $v_{s}$ results as in Theorem 1 of §2 (first check Lemma 1 in §1 for $v_{s}$).
For the $\sigma$-additivity of $s,$ let
$$A=\bigcup_{i=1}^{\infty} A_{i} \text { (disjoint)}, \quad A, A_{i} \in \mathcal{M};$$
then
$$\left|s A-\sum_{i=1}^{r-1} s A_{i}\right| \leq \sum_{i=r}^{\infty} v_{s} A_{i} \rightarrow 0$$
as $r \rightarrow \infty,$ 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).]