
# 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
$\lim s X_{n}=\infty \neq s \emptyset.$

## 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).

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