7.9.E: Problems on Lebesgue-Stieltjes Measures
Do Problems 7 and 8 in §4 and Problem 3' in §5, if not done before.
Prove in detail Theorems 1 to 3 in §8 for LS measures and outer measures.
Do Problem 2 in §8 for LS-outer measures in \(E^{1}\).
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.]
Fill in all proof details in Theorem 2.
[Hint: Use Problem 4.]
In Problem 8(iv) of §4, describe \(m_{\alpha}^{*}\) and \(M_{\alpha}^{*}\).
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).
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 \(\mathcal{M}_{\alpha}^{*}=\mathcal{M}^{*}\).]
(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.]
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.