9.1.E: Problems on L-Integrals and Antiderivatives
Fill in proof details in Theorems 1 and 2, Lemma 1, and Corollaries 1-3.
Verify Note 2.
Let \(F\) be Cantor's function (Problem 6 in Chapter 4, §5). Let
\[
G=\bigcup_{k, i} G_{k t}
\]
\(\left(G_{k t} \text { as in that problem }\right) .\) So \([0,1]-G=P(\text { Cantor's set }) ; m P=0\) (Problem \(10 \text { in Chapter } 7, §8) .\)
Show that \(F\) is differentiable \(\left(F^{\prime}=0\right)\) on \(G .\) By Theorems 2 and 3 of Chapter 8, §9,
\[
R \int_{0}^{1} F^{\prime}=L \int_{0}^{1} F^{\prime}=L \int_{G} F^{\prime}=0
\]
exists, yet \(F(1)-F(0)=1-0 \neq 0\).
Does this contradict Corollary \(1 ?\) Is \(F\) a genuine antiderivative of \(f ?\) If not, find one.
Let
\[
F=\left\{\begin{array}{ll}{0} & {\text { on }\left[0, \frac{1}{2}\right), \text { and }} \\ {1} & {\text { on }\left[\frac{1}{2}, 1\right] .}\end{array}\right.
\]
Show that
\[
R \int_{0}^{1} F^{\prime}=0
\]
exists, yet
\[
F(1)-F(0)=1-0=1 .
\]
What is wrong?
[Hint: A genuine primitive of \(\left.F^{\prime} \text { (call it } \phi\right)\) has to be relatively continuous on \([0,1] ;\) find \(\phi \text { and show that } \phi(1)-\phi(0)=0 .]\)
What is wrong with the following computations?
(i) \(L \int_{-1}^{\frac{1}{2}} \frac{d x}{x^{2}}=-\left.\frac{1}{x}\right|_{-1} ^{\frac{1}{2}}=-1\).
(ii) \(L \int_{-1}^{1} \frac{d x}{x}=\left.\ln |x|\right|_{-1} ^{1}=0 .\) Is there a primitive on the whole interval?
[Hint: See hint to Problem 3.]
(iii) How about \(L \int_{-1}^{1} \frac{|x|}{x} d x\) (cf. examples (a) and (b) of Chapter 5, §5)?
Let
\[
F(x)=x^{2} \cos \frac{\pi}{x^{2}}, \quad F(0)=1 .
\]
Prove the following:
(i) \(F\) is differentiable on \(A=[0,1]\).
(ii) \(f=F^{\prime}\) is bounded on any \([a, b] \subset(0,1),\) but not on \(A\).
(iii) Let
\[
a_{n}=\sqrt{\frac{2}{4 n+1}} \text { and } b_{n}=\frac{1}{\sqrt{2 n}} \text { for } n=1,2, \ldots
\]
Show that
\[
A \supseteq \bigcup_{n=1}^{\infty}\left[a_{n}, b_{n}\right](\text {disjoint})
\]
and
\[
L \int_{a_{n}}^{b_{n}} f=\frac{1}{2 n} ;
\]
so
\[
L \int_{a}^{b} f \geq L \int_{\cup_{n=1}^{\infty}\left[a_{n}, b_{n}\right]} f \geq \sum_{n=1}^{\infty} \frac{1}{2 n}=\infty ,
\]
and \(f=F^{\prime}\) is not L-integrable on \(A\).
What is wrong? Is there a contradiction to Theorem \(2 ?\)
Consider both
(a) \(f(x)=\frac{\sin x}{x}, f(0)=1,\) and
(b) \(f(x)=\frac{1-e^{-x}}{x}, f(0)=1\).
In each case, show that \(f\) is continuous on \(A=[0,1]\) and
\[
R \int_{A} f \leq 1
\]
exists, yet it does not "work out" via primitives. What is wrong? Does a primitive exist?
To use Corollary \(1,\) first expand \(\sin x\) and \(e^{-x}\) in a Taylor series and find the series for
\[
\int f
\]
by Theorem 3 of Chapter 5, §9.
Find
\[
R \int_{A} f
\]
approximately, to within \(1 / 10,\) using the remainder term of the series to estimate accuracy.
[Hint: Primitives exist, by Theorem 2 of Chapter 5, §11, even though they are none of the known "calculus functions." \(]\)
Take \(A, G_{n}=\left(a_{n}, b_{n}\right),\) and \(P(m P>0)\) as in Problem 17(iii) of Chapter 7, §8.
Define \(F=0\) on \(P\) and
\[
F(x)=\left(x-a_{n}\right)^{2}\left(x-b_{n}\right)^{2} \sin \frac{1}{\left(b_{n}-a_{n}\right)\left(x-a_{n}\right)\left(x-b_{n}\right)} \quad \text { if } x \notin P .
\]
Prove that \(F\) has a bounded derivative \(f,\) yet \(f\) is not R-integrable on \(A ;\) so Theorem 2 applies, but Corollary 1 does not.
[Hints: If \(p \notin P,\) compute \(F^{\prime}(p)\) as in calculus.
If \(p \in P\) and \(x \rightarrow p+\) over \(A-P,\) then \(x\) is always in some \(\left(a_{n}, b_{n}\right), p \leq a_{n}<x .\) (Why?) Deduce that \(\Delta x=x-p>x-a_{n}\) and
\[
\left|\frac{\Delta F}{\Delta x}\right| \leq\left(x-a_{n}\right)(b-a)^{2} \leq|\Delta x|(b-a)^{2} ;
\]
so \(F_{+}^{\prime}(p)=0 .\) (What if \(x \rightarrow p+\) over \(P\) ?) Similarly, show that \(F_{-}^{\prime}=0\) on \(P\).
Prove however that \(F^{\prime}(x)\) oscillates from 1 to \(-1\) as \(x \rightarrow a_{n}+\) or \(x \rightarrow b_{n}-\), hence also as \(x \rightarrow p \in P\) (why?); so \(F^{\prime}\) is discontinuous on all of \(P,\) with \(m P>0 .\) Now use Theorem 3 in Chapter 8, §9.]
\(\Rightarrow 8\). If
\[
Q \subseteq A=[a, b]
\]
and \(m Q=0,\) find a continuous map \(g: A \rightarrow E^{1}, g \geq 0, g \uparrow,\) with
\[
g^{\prime}=+\infty \quad \text { on } Q .
\]
[Hints: By Theorem 2 of Chapter 7, §8, fix ( \(\forall n\) ) an open \(G_{n} \supseteq Q,\) with
\[
m G_{n}<2^{-n} .
\]
Set
\[
g_{n}(x)=m\left(G_{n} \cap[a, x]\right)
\]
and
\[
g=\sum_{n=1}^{\infty} g_{n}
\]
on \(A ; \sum g_{n}\) converges uniformly on \(A .\) (Why?)
By Problem 4 in Chapter 7, §9, and Theorem 2 of Chapter 7, §4, each \(g_{n}\) (hence g) is continuous. (Why?) If \([p, x] \subseteq G_{n},\) show that
\[
g_{n}(x)=g_{n}(p)+(x-p) ,
\]
so
\[
\frac{\Delta g_{n}}{\Delta x}=1
\]
and
\[
\left.\frac{\Delta g}{\Delta x}=\sum_{n=1}^{\infty} \frac{\Delta g_{n}}{\Delta x} \rightarrow \infty .\right]
\]
(i) Prove Corollary 4.
(ii) State and prove earlier analogues for Corollary 5 of Chapter 5, §5, and Theorems 3 and 4 from Chapter 5, §10.
[Hint for (i): For primitives, this is Problem 3 in Chapter 5, §5. As \(g[Q]\) is countable (Problem 2 in Chapter 1, §9) and \(f\) is bounded on
\[
g[A]-g[Q] \subseteq g[A-Q] ,
\]
\(\left.f \text { satisfies Theorem 2 on } g[A], \text { with } P=g[Q], \text { while }(f \circ g) g^{\prime} \text { satisfies it on } A .\right]\)
\(\Rightarrow 10\). Show that if \(h: E^{1} \rightarrow E^{*}\) is L-integrable on \(A=[a, b],\) and
\[
(\forall x \in A) \quad L \int_{a}^{x} h=0 ,
\]
then \(h=0\) a.e. on \(A\).
[Hints: Let \(K=A(h>0)\) and \(H=A-K,\) with, say, \(m K=\varepsilon>0 .\)
Then by Corollary 1 in Chapter 7, §1 and Definition 2 of Chapter 7, §5,
\[
H \subseteq \bigcup_{n} B_{n}(d i s j o i n t)
\]
for some intervals \(B_{n} \subseteq A,\) with
\[
\sum_{n} m B_{n}<m H+\varepsilon=m H+m K=m A .
\]
(Why?) Set \(B=\cup_{n} B_{n} ;\) so
\[
\int_{B} h=\sum_{n} \int_{B_{n}} h=0
\]
(for \(L \int h=0\) on intervals \(B_{n}\)). Thus
\[
\int_{A-B} h=\int_{A} h-\int_{B} h=0 .
\]
But \(B \supseteq H ;\) so
\[
A-B \subseteq A-H=K ,
\]
where \(h>0,\) even though \(m(A-B)>0 .\) (Why?)
Hence find a contradiction to Theorem \(1(\mathrm{h})\) of Chapter 8, §5. Similarly, disprove that \(m A(h<0)=\varepsilon>0 .]\)
\(\Rightarrow 11\). Let \(F \uparrow\) on \(A=[a, b],|F|<\infty,\) with derived function \(F^{\prime}=f .\) Taking Theorem 3 from Chapter 7, §10, for granted, prove that
\[
L \int_{a}^{x} f \leq F(x)-F(a), \quad x \in A ,
\]
[Hints: With \(f_{n}\) as in \((3), F\) and \(f_{n}\) are bounded on \(A\) and measurable by Theorem 1 of Chapter 8, §2. (Why?) Deduce that \(f_{n} \rightarrow f\) (a.e.) on \(A .\) Argue as in Lemma 1 using Fatou's lemma (Chapter 8, §6, Lemma 2).]
("Truncation.") Prove that if \(g: S \rightarrow E\) is m-integrable on \(A \in \mathcal{M}\) in a measure space \((S, \mathcal{M}, m),\) then for any \(\varepsilon>0,\) there is a bounded, \(M\)-measurable and integrable on \(A \operatorname{map} g_{0}: S \rightarrow E\) such that
\[
\int_{A}\left|g-g_{0}\right| d m<\varepsilon .
\]
[Outline: Redefine \(g=0\) on a null set, to make \(g \mathcal{M}\)-measurable on \(A .\) Then for \(n=1,2, \ldots\) set
\[
g_{n}=\left\{\begin{array}{ll}{g} & {\text { on } A(|g|<n), \text { and }} \\ {0} & {\text { elsewhere. }}\end{array}\right.
\]
(The function \(g_{n}\) is called the \(n\)th truncate of \(g\).)
\(\quad\) Each \(g_{n}\) is bounded and \(\mathcal{M}\) -measurable on \(A\) (why?), and
\[
\int_{A}|g| d m<\infty
\]
by integrability. Also, \(\left|g_{n}\right| \leq|g|\) and \(g_{n} \rightarrow g(\text {pointwise})\) on \(A .\) (Why?)
Now use Theorem 5 from Chapter 8, §6, to show that one of the \(g_{n}\) may serve as the desired \(\left.g_{o} .\right]\)
Fill in all proof details in Lemma \(2 .\) Prove it for unbounded \(g\).
[Hints: By Problem \(12,\) fix a bounded \(g_{o}\left(\left|g_{o}\right| \leq B\right),\) with
\[
L \int_{A}\left|g-g_{o}\right|<\frac{1}{2} \frac{\varepsilon}{f(a)-f(b)} .
\]
Verify that
\[
\begin{aligned}\left|s_{n}\right| \leq \sum_{i=1}^{q_{n}} \int_{A_{n i}} w_{n i}|g| & \leq \sum_{i} \int_{A_{n i}} w_{n i}\left|g_{o}\right|+\sum_{i} \int_{A_{n i}} w_{n i}\left|g-g_{o}\right| \\ & \leq B \sum_{i} w_{n i} m A_{n i}+\sum_{i} \int_{A_{n i}}[f(a)-f(b)]\left|g-g_{o}\right| \\ &<\frac{1}{n}+\int_{A}[f(a)-f(b)]\left|g-g_{o}\right|<\frac{1}{n}+\frac{1}{2} \varepsilon . \end{aligned}
\]
For all \(n>2 / \varepsilon,\) we get \(\left|s_{n}\right|<\frac{1}{2} \varepsilon+\frac{1}{2} \varepsilon=\varepsilon .\) Hence \(s_{n} \rightarrow 0 .\) Now finish as in the text.]
Show that Theorem 4 fails if \(F\) is not differentiable at some \(p \in A .\)
[Hint: See Problems 2 and 3.]