Integration: Exercises R Ch 3
Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au
Last updates: 18 April 2011
Integration: Exercises R Ch 3
 Prove that the supremum of any collection of convex functions on
$(a,b)$
is convex on
$(a,b)$ (if it is
finite) and that pointwise limits of sequences of convex functions are convex.
What can you say about upper and lower limits of sequences of convex functions?

If $\phi $ is convex on
$(a,b)$
and if $\psi $ is convex and nondecreasing on the range of
$\phi $, prove that
$\psi \circ \phi $ is convex on
$(a,b)$. For
$\phi >0$, show that the convexity of
$\mathrm{log}\phi $ implies the convexity of $\phi $,
but not vice versa.

Assume that $\phi $ is a continuous real function on
$(a,b)$
such that
$\phi \left({\displaystyle \frac{x+y}{2}}\right)\le \frac{1}{2}\phi \left(x\right)+\frac{1}{2}\phi \left(y\right)$
 
for all $x$ and $y\in (a,b)$.
Prove that $\phi $ is convex. (The conclusion
does not follow if continuity is omitted from the hypothesis.)
 Suppose $f$ is a complex measurable function on $X$,
and
$\phi \left(p\right)={\int}_{X}{\leftf\right}^{p}\phantom{\rule{0.2em}{0ex}}d\mu ={\Vert f\Vert}_{p}^{p}\phantom{\rule{2em}{0ex}}(0<p<\infty )$.
 
Let $E=\left\{p\phantom{\rule{0.2em}{0ex}}\right\phantom{\rule{0.2em}{0ex}}\phi \left(p\right)<\infty \}$.
Assume
${\Vert f\Vert}_{\infty}>0$.
 (a)
If $r<p<s$,
$r\in E$, and
$s\in E$, prove that
$p\in E$.
 (b)
Prove that $\mathrm{log}\phi $ is convex in the interior of $E$
and that $\phi $ is continuous on $E$.
 (c)
By (a), $E$ is connected. Is $E$ necessarily open? Closed?
Can $E$ consist of a single point? Can $E$ be any connected subset of
$(0,\infty )$?
 (d)
If $r<p<s$,
prove that
${\Vert f\Vert}_{p}\le \mathrm{max}({\Vert f\Vert}_{r},{\Vert f\Vert}_{s})$.
Show that this implies the inclusion
${L}^{r}\left(\mu \right)\cap {L}^{s}\left(\mu \right)\subseteq {L}^{p}\left(\mu \right)$.
 (e)
Assume that
${\Vert f\Vert}_{p}<\infty $ for some $r<\infty $
and prove that
${\Vert f\Vert}_{p}\to {\Vert f\Vert}_{\infty}\phantom{\rule{2em}{0ex}}\text{as}\phantom{\rule{0.5em}{0ex}}p\to \infty $.
 

Assume, in addition to the hypotheses of Exercise 4, that
 (a)
Prove that
${\Vert f\Vert}_{r}\le {\Vert f\Vert}_{s}$ if
$0<r<s\le \infty $.
 (b)
Under what conditions does it happen that
$0<r<s\le \infty $
and
${\Vert f\Vert}_{r}={\Vert f\Vert}_{s}<\infty $?
 (c)
Prove that
${L}^{r}\left(\mu \right)\supseteq {L}^{s}\left(\mu \right)$
if $0<r\&t;s$. Under what conditions
do these two spaces contain the same functions?
 (d)
Assume that
${\Vert f\Vert}_{r}<\infty $ for some $r>0$, and prove that
$\underset{p\to 0}{\mathrm{lim}}{\Vert f\Vert}_{p}=\mathrm{exp}\left\{{\int}_{X}\mathrm{log}\leftf\right\phantom{\rule{0.2em}{0ex}}d\mu \right\}$
 
if $\mathrm{exp}\{\infty \}$ is
defined to be 0.
 Let $m$ be Lebesgue measure on
$[0,1]$, and define
${\Vert f\Vert}_{p}$
with respect to $m$. Find all functions $\Phi $ on
$[0,\infty )$ such that the
relations
$\Phi \left(\underset{p\to 0}{\mathrm{lim}}{\Vert f\Vert}_{p}\right)={\int}_{0}^{1}(\Phi \circ f)\phantom{\rule{0.2em}{0ex}}dm$
 
holds for every bounded, measurable, positive, $f$. Show first
that
$c\Phi \left(x\right)+(1c)\Phi \left(1\right)=\Phi \left({x}^{c}\right)\phantom{\rule{2em}{0ex}}(x>0,0\le c\le 1)$.
 
Compare with Exercise 5(d).
 For some measures, the relation $r<s$ implies
${L}^{r}\left(\mu \right)\subseteq {L}^{s}\left(\mu \right)$;
for others the inclusion is reversed; and there are some for which
${L}^{r}\left(\mu \right)$
does not contain
${L}^{s}\left(\mu \right)$
if $r\ne s$. Give examples of these situations,
and find conditions on $\mu $ under which these situations will occur.
 If $g$ is a positive function on
$(0,1)$ such that
$g\left(x\right)\to \infty $
as $x\to 0$, then there is a convex function $h$
on $(0,1)$ such that
$h\le g$ and
$h\left(x\right)\to \infty $
as $x\to 0$. True or false? Is the problem changed if
$(0,1)$ is replaced by
$(0,\infty )$ and
$x\to 0$
is replaced by
$x\to \infty $.
 Suppose $f$ is Lebesgue measurable on
$(0,1)$ and not essentially bounded.
By Exercise 4(e),
${\Vert f\Vert}_{p}\to \infty $
as $p\to \infty $. Can
${\Vert f\Vert}_{p}$ tend to $\infty $ arbitrarily slowly? More precisely, is it
true that to every positive function $\Phi $ on
$(0,\infty )$ such that
$\Phi \left(p\right)\to \infty $
as $p\to \mathrm{infin;}$ one can find an $f$
such that
${\Vert f\Vert}_{p}\to \infty $
as $p\to \infty $,
but
${\Vert f\Vert}_{p}\le \Phi \left(p\right)$
for all sufficiently large $p$?
 Suppose ${f}_{n}$ is in
${L}^{p}\left(\mu \right)$,
for $n=1,2,3,\dots $, and
${\Vert {f}_{n}f\Vert}_{p}\to 0$ and
${f}_{n}\to g$ a.e., as
$n\to \infty $. What relation exists between $f$
and $g$?
 Suppose $\mu \left(\Omega \right)=1$,
and suppose $f$ and $g$ are positive measurable functions on
$\Omega $ such that $fg\ge 1$.
Prove that
${\int}_{\Omega}f\phantom{\rule{0.2em}{0ex}}d\mu \cdot {\int}_{\Omega}g\phantom{\rule{0.2em}{0ex}}d\mu \ge 1$.
 
 Suppose $\mu \left(\Omega \right)=1$
and $h:\Omega \to [0,\infty ]$ is measurable. If
$A={\int}_{\Omega}h\phantom{\rule{0.2em}{0ex}}d\mu $,
 
prove that
$\sqrt{1+{A}^{2}}\le {\int}_{\Omega}\sqrt{1+{h}^{2}}\phantom{\rule{0.2em}{0ex}}d\mu \le 1+A$.
 
If $\mu $ is Lebesgue measure on
$[0,1]$
and if $h$ is continuous, $h=f\prime $,
the above inequalities have a simple geometric interpretation. From this, conjecture (for general
$\Omega $) under what conditions on $h$ equality can hold in
either of the above inequalities, and prove your conjecture.
 Under what conditions on $f$ and $g$ does equality
hold in the conclusions of Theorems 3.8 and 3.9?
You may have to treat the cases $p=1$ and
$p=\infty $ separately.
 Suppose $1<p<\infty $,
$f\in {L}^{p}={L}^{p}\left(\right(0,\infty \left)\right)$,
relative to Lebesgue measure, and
$F\left(x\right)=\frac{1}{x}{\int}_{0}^{x}f\left(t\right)\phantom{\rule{0.2em}{0ex}}dt\phantom{\rule{2em}{0ex}}(0<x<\infty )$.
 
 (a)
Prove Hardy's inequality
${\Vert F\Vert}_{p}\le {\displaystyle \frac{p}{p1}}{\Vert f\Vert}_{p}$.
 
which shows that the mapping $f\to F$ carries
${L}^{p}$ into
${L}^{p}$.
 (b)
Prove that equality holds only if $f=0$ a.e.
 (c)
Prove that the constant $p/(p1)$
cannot be replaced by a smaller one.
 (d)
If $f>0$ and $f\in {L}^{1}$, prove that $F\notin {L}^{1}$.
Suggestions: (a) Assume first that $f\ge 0$
and $f\in {C}_{c}\left(\right(0,\infty \left)\right)$. Integration
by parts gives
${\int}_{0}^{\infty}{F}^{p}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx=p{\int}_{0}^{\infty}{F}^{p1}\left(x\right)xF\prime \left(x\right)\phantom{\rule{0.2em}{0ex}}dx$.
 
Note that $xF\prime =fF$, and apply Hölder's inequality
to $\int {F}^{p1}f$. Then derive the general case. (c) Take $f\left(x\right)={x}^{1/p}$
on $[1,A]$,
$f\left(x\right)=0$ elsewhere, for large
$A$. See also Excercise 14, Chap. 8.
 Suppose $\left\{{a}_{n}\right\}$ is a sequence of
positive numbers. Prove that
$\sum _{N=1}^{\infty}{({\displaystyle \frac{1}{N}\sum _{n=1}^{N}{a}_{n})}}^{p}\le {({\displaystyle \frac{p}{p1})}}^{p}\sum _{n=1}^{\infty}{a}_{n}^{p}$
 
if $1<p<\infty $. Hint:
If ${a}_{n}\ge {a}_{n+1}$,
the result can be made to follow from Exercise 14. This special case implies the general one.
 Prove Egoroff's theorem: If $\mu \left(X\right)<\infty $, if $\left\{{f}_{n}\right\}$
is a sequence of complex measurable functions which converges pointwise at every point of
$X$, and if $\epsilon >0$, there is a
measurable set $E\subseteq X$, with $\mu (XE)<\epsilon $
such that $\left\{{f}_{n}\right\}$ converges uniformly
on $E$.
(The conclusion is that by redefining the ${f}_{n}$ on a
set of arbitrarily small measure we can convert a pointwise convergent sequence to a uniformly
convergent one; note the similarity with Lusin's theorem.)
Hint: Put
$S(n,k)=\bigcap _{i,j>n}\left\{x\phantom{\rule{0.2em}{0ex}}\right\phantom{\rule{0.2em}{0ex}}{f}_{i}\left(x\right){f}_{j}\left(x\right)<\frac{1}{k}$,
 
show that $\mu \left(S\right(n,k\left)\right)\to \mu \left(X\right)$
as $n\to \infty $, for each $k$,
and hence that there is a suitably increasing sequence $\left\{{n}_{k}\right\}$ such that
$E=\bigcap S({n}_{k},k)$ has the desired property.
Show that the theorem does not extend to $\sigma $finite spaces.
Show that the theorem does extend, with essentially the same proof, to the situation in which the
sequence $\left\{{f}_{n}\right\}$ is replaced by a
family $\left\{{f}_{t}\right\}$, where $t$
ranges over the positive reals; the assumptions are now that, for all
$x\in X$,
 (i)
$\underset{t\to \infty}{\mathrm{lim}}{f}_{t}\left(x\right)=f\left(x\right)$ and
 (ii)
$t\to {f}_{t}\left(x\right)$
is continuous.
 (a)
If $0<p<\infty $,
put ${\gamma}_{p}=\mathrm{max}(1,{2}^{p1})$, and show that
${\alpha \beta }^{p}\le {\gamma}_{p}({\left\alpha \right}^{p}+{\left\beta \right}^{p})$
 
for arbitrary complex numbers $\alpha $ and $\beta $.
(b)
Suppose $\mu $ is a positive measure on $X$,
$0<p<\infty $,
$f\in {L}^{p}\left(\mu \right)$,
${f}_{n}\in {L}^{p}\left(\mu \right)$,
${f}_{n}\left(x\right)\to f\left(x\right)$ a.e., and
${\Vert {f}_{n}\Vert}_{p}\to {\Vert f\Vert}_{p}$
as $n\to \infty $. Show that then
$\mathrm{lim}{\Vert f{f}_{n}\Vert}_{p}=0$, by completing the two proofs that are sketched below.
 (i)
By Egoroff's theorem, $X=A\cup B$ in such
a way that
${\int}_{A}{\leftf\right}^{p}<\epsilon $, $\mu \left(B\right)<\infty $,
and ${f}_{n}\to f$ uniformly on $B$.
Fatou's lemma, applied to
${\int}_{B}{\leftf\right}^{p}$,
leads to
$\mathrm{limsup}{\int}_{A}{\left{f}_{n}\right}^{p}\phantom{\rule{0.2em}{0ex}}d\mu \le \epsilon $.
 
 (ii)
Put ${h}_{n}={\gamma}_{p}({\leftf\right}^{p}+{\left{f}_{n}\right}^{p}){f{f}_{n}}^{p}$, and use Fatou's lemma as in the proof of Theorem 1.34.
(c)
Show that the conclusion of (b) is false if the hypothesis
${\Vert {f}_{n}\Vert}_{p}\to {\Vert f\Vert}_{p}$
is omitted, even if $\mu \left(X\right)<\infty $.
 Let $\mu $ be a positive measure on $X$. A sequence
$\left\{{f}_{n}\right\}$ of complex measurable functions on
$X$ is said to converge in measure to the measurable function
$f$ if to every $\epsilon >0$ there
corresponds an $N$ such that
$\mu \left(\right\{x\phantom{\rule{0.2em}{0ex}}\left\phantom{\rule{0.2em}{0ex}}{f}_{n}\left(x\right)f\right(x\left)\right>\epsilon \left\}\right)<\epsilon $
 
for all $n>N$. (this notion is of importance in
probability theory.) Assume
$\mu \left(X\right)<\infty $ and prove the following statements:
 (a)
If ${f}_{n}\left(x\right)\to f\left(x\right)$ a.e.,
then ${f}_{n}\to f$
in measure.
 (b)
If ${f}_{n}\in {L}^{p}\left(\mu \right)$
and
${\Vert {f}_{n}f\Vert}_{p}\to 0$,
then ${f}_{n}\to f$
in measure; here $1\le p\le \infty $.
 (c)
If ${f}_{n}\to f$
in measure, then $\left\{{f}_{n}\right\}$
has a subsequence which converges to $f$ a.e.
Investigate the converses of (a) and (b). What happens to (a), (b), and (c) if
$\mu \left(X\right)=\infty $, for instance, if $\mu $ is Lebesgue
measure on ${\mathbb{R}}^{1}$?
 Define the essential range of a function
$f\in {L}^{\infty}\left(\mu \right)$ to be the set ${R}_{f}$
consisting of all complex numbers $w$ such that
$\mu \left(\right\{x\phantom{\rule{0.2em}{0ex}}\left\phantom{\rule{0.2em}{0ex}}f\right(x)w<\epsilon \left\}\right)>0$
 
for every $\epsilon >0$. Prove that
${R}_{f}$ is compact. What relation exists between the set
${R}_{f}$ and the number
${\Vert f\Vert}_{\infty}$?
Let ${A}_{j}$ be the set of all averages
$\frac{1}{\mu \left(E\right)}}{\int}_{E}f\phantom{\rule{0.2em}{0ex}}d\mu $
 
where $E\in \mathcal{M}$ and
$\mu \left(E\right)>0$. What relations
exist between
${A}_{f}$
and
${R}_{f}$? Is
${A}_{f}$
always closed?
Are there measures $\mu $ such that
${A}_{f}$
is convex for every $f\in {L}^{\infty}\left(\mu \right)$?
Are there measures $\mu $ such that
${A}_{f}$
fails to be convex for some $f\in {L}^{\infty}\left(\mu \right)$?
How are these results affected if
${L}^{\infty}\left(\mu \right)$ is
replaced by
${L}^{1}\left(\mu \right)$, for instance?
 Suppose $\phi $ is a real function on ${\mathbb{R}}^{1}$
such that
$\phi \left({\int}_{0}^{1}f\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)\le {\int}_{0}^{1}\phi \left(f\right)\phantom{\rule{0.2em}{0ex}}dx$
 
for every real bounded measurable $f$. Prove that $\phi $
is then convex.

Call a metric space $Y$ a completion of a metric space $X$
if $X$ is dense in $Y$ and $Y$ is complete.
In Sec. 3.15 reference was made to "the" completion of a metric space. State and prove a uniqueness theorem
which justifies this terminology.
 Suppose $X$ is a metric space in which every Cauchy sequence has a convergent
subsequence. Does it follow that $X$ is complete? (See the proof of Theorem 3.11.)

Suppose $\mu $ is a positive measure on $X$,
$\mu \left(X\right)<\infty $,
$f\in {L}^{\infty}\left(\mu \right)$,
${\Vert f\Vert}_{\infty}>0$, and
${\alpha}_{n}={\int}_{X}{\leftf\right}^{n}\phantom{\rule{0.2em}{0ex}}d\mu \phantom{\rule{2em}{0ex}}(n=1,2,3,\dots )$.
 
Prove that
$\underset{n\to \infty}{\mathrm{lim}}{\displaystyle \frac{{\alpha}_{n+1}}{{\alpha}_{n}}={\Vert f\Vert}_{\infty}}$.
 

Suppose $\mu $ is a positive measure,
$f\in {L}^{p}\left(\mu \right)$,
$g\in {L}^{p}\left(\mu \right)$.
 (a)
If $0<p<1$, prove that
$\int {\leftf\right}^{p}{\leftg\right}^{p}\phantom{\rule{0.2em}{0ex}}d\mu \le \int {fg}^{p}\phantom{\rule{0.2em}{0ex}}d\mu $,
 
that $\Delta (f,g)=\int {fg}^{p}\phantom{\rule{0.2em}{0ex}}d\mu $
defines a metric on
${L}^{p}\left(\mu \right)$, and that
the resulting metric space is complete.
 (b)
If $1\le p<\infty $
and
${\Vert f\Vert}_{p}\le R$,
${\Vert g\Vert}_{p}\le R$,
prove that
$\int {\leftf\right}^{p}{\leftg\right}^{p}\phantom{\rule{0.2em}{0ex}}d\mu \le 2p{R}^{p1}{\Vert fg\Vert}_{p}$.
 
Hint: Prove first, for
$x\ge 0$,
$y\ge 0$,
that
${x}^{p}{y}^{p}\le \{\begin{array}{ll}{xy}^{p},& \text{if}0p1,\\ pxy({x}^{p1}+{y}^{p1}),& \text{if}1\le p\infty .\end{array}$
 
Note that (a) and (b) establish the continuity of the mapping
$f\to {\leftf\right}^{p}$
that carries
${L}^{p}\left(\mu \right)$ into
${L}^{1}\left(\mu \right)$.
 Suppose $\mu $ is a positive measure on $X$ and
$f:X\to (0,\infty )$
satisfies
${\int}_{X}f\phantom{\rule{2em}{0ex}}d\mu =1$. Prove, for every $E\subseteq X$
with $0<\mu \left(E\right)<\infty $, that
${\int}_{E}\left(\mathrm{log}f\right)\phantom{\rule{0.2em}{0ex}}d\mu \le \mu \left(E\right)\mathrm{log}{\displaystyle \frac{1}{\mu \left(E\right)}}$
 
and, when $0<p<1$,
${\int}_{E}{f}^{p}\phantom{\rule{0.2em}{0ex}}d\mu \le {\mu \left(E\right)}^{1p}$.
 
 If $f$ is a positive measurable function on
$[0,1]$, which is
larger,
${\int}_{0}^{1}f\left(x\right)\mathrm{log}f\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\phantom{\rule{2em}{0ex}}\text{or}\phantom{\rule{2em}{0ex}}{\int}_{0}^{1}f\left(s\right)\phantom{\rule{0.2em}{0ex}}ds{\int}_{0}^{1}\mathrm{log}f\left(t\right)\phantom{\rule{0.2em}{0ex}}dt$?
 
Notes and References
These exercises are taken from [Ru, Chapt. 3] for a course in "Measure Theory" at the Masters level at University of Melbourne.
References
[RuB]
W. Rudin,
Principles of Mathematical Analysis, Third edition, McGrawHill, 1976.
MR??????.
[Ru]
W. Rudin,
Real and complex analysis, Third edition, McGrawHill, 1987.
MR0924157.
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