9.2.E: Problems on L-Integrals and Absolute Continuity

Exercise \(\PageIndex{1}\)

Fill in all details in the proof of Lemma 1 and of Theorems 3 and 4 from §1.

Exercise \(\PageIndex{2}\)

Prove Theorem 1 and Corollaries 1, 2, and 5.

Exercise \(\PageIndex{2'}\)

Disprove the converse to Corollary 4. (Give an example!)

Exercise \(\PageIndex{3}\)

\(\Rightarrow 3\). Show that if \(F: E^{1} \rightarrow E\) is L-integrable on \(A=[a, b]\) and continuous at \(p \in A,\) then \(p\) is an L-point of \(F .\)
[Hint: Use the \(\varepsilon, \delta \text { definition of continuity. }]\)

Exercise \(\PageIndex{4}\)

Complete all proof details for Lemma \(2,\) Theorems 3 and \(4,\) and Corollary \(3 .\)

Exercise \(\PageIndex{5}\)

Let \(F=1\) on \(R(=\text { rationals })\) and \(F=0\) on \(E^{1}-R\) (Dirichlet function).
Show that \(F\) has exactly three derivates \((0,+\infty, \text { and }-\infty)\) at every \(p \in E^{1} .\)

Exercise \(\PageIndex{6}\)

\(\Rightarrow 6\). We say that \(F\) is a Lipschitz map, or has the uniform Lipschitz property on \(A,\) iff
\[
\left(\exists K \in E^{1}\right)(\forall x, y \in A) \quad|F(x)-F(y)| \leq K|x-y| .
\]
Prove the following:
(i) Any such \(F\) is absolutely continuous on \(A=[a, b]\).
(ii) If all derivates of \(f\) satisfy
\[
|D f(x)| \leq k<\infty, \quad x \in A=[a, b] ,
\]
then \(f\) is a Lipschitz map on \(A\).

Exercise \(\PageIndex{7}\)

\(\Rightarrow 7\). Let \(g: E^{1} \rightarrow E^{1}\) and \(f: E^{1} \rightarrow E\) (real or not) be absolutely continuous on \(A=[a, b]\) and \(g[A],\) respectively.
Prove that \(h=f \circ g\) is absolutely continuous on \(A,\) provided that either \(f\) is as in Problem \(6,\) or \(g\) is strictly monotone on \(A .\)

Exercise \(\PageIndex{8}\)

Prove that if \(F: E^{1} \rightarrow E^{1}\) is absolutely continuous on \(A=[a, b],\) if \(Q \subseteq A,\) and if \(m Q=0,\) then \(m^{*} F[Q]=0(m=\) Lebesgue measure).
[Outline: We may assume \(Q \subseteq(a, b) .\) (Why?)
\(\quad\) Fix \(\varepsilon>0\) and take \(\delta\) as in Definition 2. As \(m\) is regular, there is an open \(G\),
\[
Q \subseteq G \subseteq(a, b) ,
\]
with \(m G<\delta .\) By Lemma 2 of Chapter 7, §2,
\[
G=\bigcup_{k=1}^{\infty} I_{k}(\text {disjoint})
\]
for some \(I_{k}=\left(a_{k}, b_{k}\right]\).
\(\quad\) Let \(u_{k}=\inf F\left[I_{k}\right], v_{k}=\sup F\left[I_{k}\right] ;\) so
\[
F\left[I_{k}\right] \subseteq\left[u_{k}, v_{k}\right]
\]
and
\[
m^{*} F\left[I_{k}\right] \leq v_{k}-u_{k} .
\]
Also,
\[
\sum\left(b_{k}-a_{k}\right)=\sum m I_{k}=m G<\delta .
\]
From Definition \(2,\) show that
\[
\sum_{k=1}^{\infty}\left(v_{k}-u_{k}\right) \leq \varepsilon
\]
(first consider partial sums). As
\[
F[Q] \subseteq F[G] \subseteq \bigcup_{k} F\left[I_{k}\right] ,
\]
get
\[
\left.m^{*} F[Q] \leq \sum_{k} m^{*} F\left[I_{k}\right]=\sum_{k}\left(v_{k}-u_{k}\right) \leq \varepsilon \rightarrow 0 .\right]
\]

Exercise \(\PageIndex{9}\)

Show that if \(F\) is as in Problem 8 and if
\[
A=[a, b] \supseteq B, \quad B \in \mathcal{M}^{*}
\]
(L-measurable sets), then
\[
F[B] \in \mathcal{M}^{*} .
\]
\(\left(" F \text { preserves } \mathcal{M}^{*} \text {-sets." }\right)\)
[Outline: (i) If \(B\) is closed, it is compact, and so is \(F[B]\) (Theorems 1 and 4 of Chapter 4, §6).
(ii) If \(B \in \mathcal{F}_{\sigma},\) then
\[
B=\bigcup_{i} B_{i}, \quad B_{i} \in \mathcal{F} ;
\]
so by (i),
\[
F[B]=\bigcup_{i} F\left[B_{i}\right] \in \mathcal{F}_{\sigma} \subseteq \mathcal{M}^{*} .
\]
(iii) If \(B \in \mathcal{M}^{*},\) then by Theorem 2 of Chapter 7, §8,
\[
\left(\exists K \in \mathcal{F}_{\sigma}\right) \quad K \subseteq B, m(B-K)=0 .
\]
Now use Problem \(8, \text { with } Q=B-K .]\)

Exercise \(\PageIndex{10}\)

\(\Rightarrow 10\). (Change of variable.) Suppose \(g: E^{1} \rightarrow E^{1}\) is absolutely continuous and one-to-one on \(A=[a, b],\) while \(f: E^{1} \rightarrow E^{*}\left(E^{n}, C^{n}\right)\) is L-integrable on \(g[A] .\)
Prove that \((f \circ g) g^{\prime}\) is L-integrable on \(A\) and
\[
L \int_{a}^{b}(f \circ g) g^{\prime}=L \int_{p}^{q} f ,
\]
where \(p=g(a)\) and \(q=g(b)\).
[Hints: Let \(F=L \int f\) and \(H=F \circ g\) on \(A .\)
By Theorems 2 and 3 and Problem 7 (end), \(F\) and \(H\) are absolutely continuous on \(g[A]\) and \(A,\) respectively; and \(H^{\prime}\) is L-integrable on \(A .\) So by Theorem 3
\[
H=L \int H^{\prime}=L \int(f \circ g) g^{\prime} ,
\]
as \(\left.H^{\prime}=(f \circ g) g^{\prime} \text { a.e. on } A .\right]\)

Exercise \(\PageIndex{11}\)

Setting \(f(x)=0\) if not defined otherwise, find the intervals (if any) on which \(f\) is absolutely continuous if \(f(x)\) is defined by
(a) \(\sin x\);
(b) \(\cos 2 x\);
(c) \(1 / x\);
(d) \(\tan x\);
(e) \(x^{x}\);
(f) \(x \sin (1 / x)\);
(g) \(x^{2} \sin x^{-2}(\text { Problem } 5 \text { in } §1)\);
(h) \(\left.\sqrt{x^{3}} \cdot \sin (1 / x) \text { (verify that }\left|f^{\prime}(x)\right| \leq \frac{3}{2}+x^{-\frac{1}{2}}\right)\);
[Hint: Use Problems 6 and 7.]