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).]