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# 7.9.E: Problems on Lebesgue-Stieltjes Measures

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## Exercise $$\PageIndex{1}$$

Do Problems 7 and 8 in §4 and Problem 3' in §5, if not done before.

## Exercise $$\PageIndex{2}$$

Prove in detail Theorems 1 to 3 in §8 for LS measures and outer measures.

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

Do Problem 2 in §8 for LS-outer measures in $$E^{1}$$.

## Exercise $$\PageIndex{4}$$

Prove that $$f : E^{1} \rightarrow(S, \rho)$$ is right (left) continuous at $$p$$ iff
$\lim _{n \rightarrow \infty} f\left(x_{n}\right)=f(p) \text { as } x_{n} \searrow p\left(x_{n} \nearrow p\right).$
[Hint: Modify the proof of Theorem 1 in Chapter 4, §2.]

## Exercise $$\PageIndex{5}$$

Fill in all proof details in Theorem 2.
[Hint: Use Problem 4.]

## Exercise $$\PageIndex{6}$$

In Problem 8(iv) of §4, describe $$m_{\alpha}^{*}$$ and $$M_{\alpha}^{*}$$.

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

Show that if $$\alpha=c$$constant on an open interval $$I \subseteq E^{1}$$ then
$(\forall A \subseteq I) \quad m_{\alpha}^{*}(A)=0.$
Disprove it for nonopen intervals $$I$$ (give a counterexample).

## Exercise $$\PageIndex{8}$$

Let $$m^{\prime}: \mathcal{M} \rightarrow E^{*}$$ be a topological, translation-invariant measure in $$E^{1}$$, with $$m^{\prime}(0,1]=c<\infty.$$ Prove the following.
(i) $$m^{\prime}=c m$$ on the Borel field $$\mathcal{B}.$$ (Here $$m: \mathcal{M}^{*} \rightarrow E^{*}$$ is Lebesgue measure in $$E^{1}$$.)
*(ii) If $$m^{\prime}$$ is also complete, then $$m^{\prime}=c m$$ on $$\mathcal{M}^{*}$$.
(iii) If $$0<c<\infty,$$ some set $$Q \subset[0,1]$$ is not $$m^{\prime}$$-measurable.
*(iv) If $$\mathcal{M}^{\prime}=\mathcal{B},$$ then $$c m$$ is the completion of $$m^{\prime}$$ (Problem 15 in §6).
[Outline: (i) By additivity and translation invariance,
$m^{\prime}(0, r]=c m(0, r]$
for rational
$r=\frac{n}{k}, \quad n, k \in N$
(first take $$r=n,$$ then $$r=\frac{1}{k},$$ then $$r=\frac{n}{k}$$).
By right continuity (Theorem 2 in §4), prove it for real $$r>0$$ (take rationals $$r_{i} \searrow r$$).
By translation, $$m^{\prime}=c m$$ on half-open intervals. Proceed as in Problem 13 of §8.
(iii) See Problems 4 to 6 in §8. Note that, by Theorem 2, one may assume $$m^{\prime}=m_{\alpha}$$ (a translation-invariant $$L S$$ measure). As $$m_{\alpha}=c m$$ on half-open intervals, Lemma 2 in §2 yields $$m_{\alpha}=c m$$ on $$\mathcal{G}$$ (open sets). Use $$\mathcal{G}$$-regularity to prove $$m_{\alpha}^{*}=c m^{*}$$ and $$\mathcal{M}_{\alpha}^{*}=\mathcal{M}^{*}$$.]

## Exercise $$\PageIndex{9*}$$

(LS measures in $$E^{n}.$$) Let
$\mathcal{C}^{*}=\left\{\text {alf-open intervals in } E^{n}\right\}.$
For any $$\operatorname{map} G : E^{n} \rightarrow E^{1}$$ and any $$(\overline{a}, \overline{b}] \in \mathcal{C}^{*},$$ set
\begin{aligned} \Delta_{k} G(\overline{a}, \overline{b}] &=G\left(x_{1}, \ldots, x_{k-1}, b_{k}, x_{k+1}, \ldots, x_{n}\right) \\ &-G\left(x_{1}, \ldots, x_{k-1}, a_{k}, x_{k+1}, \ldots, x_{n}\right), \quad 1 \leq k \leq n. \end{aligned}
Given $$\alpha : E^{n} \rightarrow E^{1},$$ set
$s_{\alpha}(\overline{a}, \overline{b}]=\Delta_{1}\left(\Delta_{2}\left(\cdots\left(\Delta_{n} \alpha(\overline{a}, \overline{b}]\right) \cdots\right)\right).$
For example, in $$E^{2}$$,
$s_{\alpha}(a, b]=\alpha\left(b_{1}, b_{2}\right)-\alpha\left(b_{1}, a_{2}\right)-\left[\alpha\left(a_{1}, b_{2}\right)-\alpha\left(a_{1}, a_{2}\right)\right].$
Show that $$s_{\alpha}$$ is additive on $$\mathcal{C}^{*}$$. Check that the order in which the $$\Delta_{k}$$ are applied is immaterial. Set up a formula for $$s_{\alpha}$$ in $$E^{3}$$.
[Hint: First take two disjoint intervals
$(\overline{a}, \overline{q}] \cup(\overline{p}, \overline{b}]=(\overline{a}, \overline{b}],$
as in Figure 2 in Chapter 3, §7. Then use induction, as in Problem 9 of Chapter 3, §7.]

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

If $$s_{\alpha}$$ in Problem 9 is nonnegative, and $$\alpha$$ is right continuous in each variable $$x_{k}$$ separately, we call $$\alpha$$ a distribution function, and $$s_{\alpha}$$ is called the $$\alpha$$-induced $$L S$$ premeasure in $$E^{n};$$ the $$L S$$ outer measure $$m_{\alpha}^{*}$$ and measure
$m_{\alpha} : \mathcal{M}_{\alpha}^{*} \rightarrow E^{*}$
in $$E^{n}$$ (obtained from $$s_{\alpha}$$ as shown in } §§5 and 6) are said to be induced by $$\alpha.$$
For $$s_{\alpha}, m_{\alpha}^{*},$$ and $$m_{\alpha}$$ so defined, redo Problems 1-3 above.