7.4.E: Problems on Set Functions

Exercise $\PageIndex{1}$

Prove Theorem 2 in detail for semirings.
[Hint: We know that
$$X_{n}-X_{n-1}=\bigcup_{i=1}^{m_{n}} Y_{ni} \text { (disjoint)}$$
for some $Y_{ni} \in \mathcal{C},$ so
$$\overline{s}\left(X_{n}-X_{n-1}\right)=\sum_{i=1}^{m_{n}} s Y_{ni},$$
with $\overline{s}$ as in Theorem 1.]

Exercise $\PageIndex{2}$

Let $s$ be additive on $\mathcal{M},$ a ring. Prove that $s$ is also $\sigma$-additive provided $s$ is either
(i) left continuous, or
(ii) finite on $\mathcal{M}$ and right-continuous at $\emptyset;$ i.e.,
$$\lim _{n \rightarrow \infty} s X_{n}=0$$
when $X_{n} \searrow \emptyset$ $\left(X_{n} \in \mathcal{M}\right)$.
[Hint: Let
$$A=\bigcup_{n} A_{n} \text { (disjoint)}, \quad A, A_{n} \in \mathcal{M}.$$
Set
$$X_{n}=\bigcup_{k=1}^{n} A_{k}, Y_{n}=A-X_{n}.$$
Verify that $X_{n}, Y_{n} \in \mathcal{M}, X_{n} \nearrow A, Y_{n} \searrow \emptyset$.
In case (i),
$$s A=\lim s X_{n}=\sum_{k=1}^{\infty} s A_{k}.$$
(Why?)
For (ii), use the $Y_{n}$.]

Exercise $\PageIndex{3}$

Let
$$\mathcal{M}=\left\{\text {all intervals in the rational field } R \subset E^{1}\right\}.$$
Let
$$s X=b-a$$
if $a, b$ are the endpoints of $X \in \mathcal{M}$ $(a, b \in R, a \leq b).$ Prove that
(i) $\mathcal{M}$ is a semiring;
(ii) $s$ is continuous;
(iii) $s$ is additive but not $\sigma$-additive; thus Problem 2 fails for semirings.
[Hint: $R$ is countable. Thus each $X \in \mathcal{M}$ is a countable union of singletons $\{x\}=[x, x];$ hence $s X=0$ if $s$ were $\sigma$-additive.]

Exercise $\PageIndex{3'}$

Let $N=$ {naturals}. Let
$$\mathcal{M}=\{\text {all finite subsets of } N \text { and their complements in } N\}.$$
If $X \in \mathcal{M},$ let $s X=0$ if $X$ is finite, and $s X=\infty$ otherwise. Show that
(i) $\mathcal{M}$ is a set field;
(ii) $s$ is right continuous and additive, but not $\sigma$-additive.
Thus Problem 2 (ii) fails if $s$ is not finite.

Exercise $\PageIndex{4}$

Let
$$\mathcal{C}=\left\{\text {finite and infinite intervals in } E^{1}\right\}.$$
If $a, b$ are the endpoints of an interval $X$ $\left(a, b \in E^{*}, a<b\right),$ set
$$s X=\left\{\begin{array}{ll}{b-a,} & {a<b,} \\ {0,} & {a=b.}\end{array}\right.$$
Show that $s$ is $\sigma$-additive on $\mathcal{C},$ a semiring.
Let
$$X_{n}=(n, \infty);$$
so $s X_{n}=\infty-n=\infty$ and $X_{n} \searrow \emptyset.$ (Verify!) Yet
$$\lim s X_{n}=\infty \neq s \emptyset.$$
Does this contradict Theorem 2?

Exercise $\PageIndex{5}$

Fill in the missing proof details in Theorem 1.

Exercise $\PageIndex{6}$

Let $s$ be additive on $\mathcal{M}.$ Prove the following.
(i) If $\mathcal{M}$ is a ring or semiring, so is
$$\mathcal{N}=\{X \in \mathcal{M}| | s X |<\infty\}$$
if $\mathcal{N} \neq \emptyset$.
(ii) If $\mathcal{M}$ is generated by a set family $\mathcal{C},$ with $|s|<\infty$ on $\mathcal{C},$ then $|s|<\infty$ on $\mathcal{M}.$
[Hint: Use Problem 16 in §3.]

Exercise $\PageIndex{7}$

$\Rightarrow$ (Lebesgue-Stieltjes set functions.) Let $\alpha$ and $s_{\alpha}$ be as in Example (d). Prove the following.
(i) $s_{\alpha} \geq 0$ on $\mathcal{C}$ iff $\alpha \uparrow$ on $E^{1}$ (see Theorem 2 in Chapter 4, §5).
(ii) $s_{\alpha}\{p\}=s_{\alpha}[p, p]=0$ iff $\alpha$ is continuous at $p$.
(iii) $s_{\alpha}$ is additive.
[Hint: If
$$A=\bigcup_{i=1}^{n} A_{i} \text { (disjoint),}$$
the intervals $A_{i-1}, A_{i}$ must be adjacent. For two such intervals, consider all cases like
$$(a, b] \cup(b, c),[a, b) \cup[b, c], \text { etc.}$$
Then use induction on $n$.]
(iv) If $\alpha$ is right continuous at $a$ and $b,$ then
$$s_{\alpha}(a, b]=\alpha(b)-\alpha(b).$$
If $\alpha$ is continuous at $a$ and $b,$ then
$$s_{\alpha}[a, b]=s_{\alpha}(a, b]=s_{\alpha}[a, b)=s_{\alpha}(a, b).$$
(v) If $\alpha \uparrow$ on $E^{1}$, then $s_{\alpha}$ satisfies Lemma 1 and Corollary 2 in §1 (same proof), as well as Lemma 1, Theorem 1, Corollaries 1-4, and Note 3 in §2 (everything except Corollaries 5 and 6).
[Hint: Use (i) and (iii). For Lemma 1 in §2, take first a half-open $B=(a, b];$ use the definition of a right-side limit along with Theorems 1 and 2 in Chapter 4, §5, to prove
$$(\forall \varepsilon>0)(\exists c>b) \quad 0 \leq \alpha(c-)-\alpha(b+)<\varepsilon;$$
then set $C=(a, c).$ Similarly for $B=[a, b),$ etc. and for the closed interval $A \subseteq B$.]
(vi) If $\alpha(x)=x$ then $s_{\alpha}=v,$ the volume (or length) function in $E^{1}$.

Exercise $\PageIndex{8}$

Construct LS set functions (Example (d)), with $\alpha \uparrow$ (see Problem 7(v)), so that
(i) $s_{\alpha}[0,1] \neq s_{\alpha}[1,2]$;
(ii) $s_{\alpha} E^{1}=1$ (after extending $s_{\alpha}$ to $\mathcal{C}_{\sigma}-sets in \(E^{1}$);
(ii') $s_{\alpha} E^{1}=c$ for a fixed $c \in(0, \infty)$;
(iii) $s_{\alpha}\{0\}=1$ and $s_{\alpha}[0,1]>s_{\alpha}(0,1]$.
Describe $s_{\alpha}$ if $\alpha(x)=[x]$ (the integral part of $x$).
[Hint: See Figure 16 in Chapter 4, §1.]

Exercise $\PageIndex{9}$

For an arbitrary $\alpha : E^{1} \rightarrow E^{1},$ define $\sigma_{\alpha} : \mathcal{C} \rightarrow E^{1}$ by
$$\sigma_{\alpha}[a, b]=\sigma_{\alpha}(a, b]=\sigma_{\sigma}[a, b)=\sigma_{\alpha}(a, b)=\alpha(b)-\alpha(a)$$
(the original Stieltjes method). Prove that $\sigma_{\alpha}$ is additive but not $\sigma$-additive unless $\alpha$ is continuous (for Theorem 2 fails).