## Integration: Exercises R Ch 3

1. Prove that the supremum of any collection of convex functions on $\left(a,b\right)$ is convex on $\left(a,b\right)$ (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?
2. If $\phi$ is convex on $\left(a,b\right)$ and if $\psi$ is convex and nondecreasing on the range of $\phi$, prove that $\psi \circ \phi$ is convex on $\left(a,b\right)$. For $\phi >0$, show that the convexity of $\mathrm{log}\phi$ implies the convexity of $\phi$, but not vice versa.
3. Assume that $\phi$ is a continuous real function on $\left(a,b\right)$ such that
 $\phi \left(\frac{x+y}{2}\right)\le \frac{1}{2}\phi \left(x\right)+\frac{1}{2}\phi \left(y\right)$
for all $x$ and $y\in \left(a,b\right)$. Prove that $\phi$ is convex. (The conclusion does not follow if continuity is omitted from the hypothesis.)
4. Suppose $f$ is a complex measurable function on $X$, and
 $\phi \left(p\right)={\int }_{X}{|f|}^{p}\phantom{\rule{0.2em}{0ex}}d\mu ={‖f‖}_{p}^{p}\phantom{\rule{2em}{0ex}}\left(0.
Let $E=\left\{p\phantom{\rule{0.2em}{0ex}}|\phantom{\rule{0.2em}{0ex}}\phi \left(p\right)<\infty \right\}$. Assume ${‖f‖}_{\infty }>0$.
(a)   If $r, $r\in E$, 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 $\left(0,\infty \right)$?
(d)   If $r, prove that ${‖f‖}_{p}\le \mathrm{max}\left({‖f‖}_{r},{‖f‖}_{s}\right)$. 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 ${‖f‖}_{p}<\infty$ for some $r<\infty$ and prove that
 ${‖f‖}_{p}\to {‖f‖}_{\infty }\phantom{\rule{2em}{0ex}}\text{as}\phantom{\rule{0.5em}{0ex}}p\to \infty$.
5. Assume, in addition to the hypotheses of Exercise 4, that
 $\mu \left(X\right)=1$.
(a)   Prove that ${‖f‖}_{r}\le {‖f‖}_{s}$ if $0.
(b)   Under what conditions does it happen that $0 and ${‖f‖}_{r}={‖f‖}_{s}<\infty$?
(c)   Prove that ${L}^{r}\left(\mu \right)\supseteq {L}^{s}\left(\mu \right)$ if $0. Under what conditions do these two spaces contain the same functions?
(d)   Assume that ${‖f‖}_{r}<\infty$ for some $r>0$, and prove that
 $\underset{p\to 0}{\mathrm{lim}}{‖f‖}_{p}=\mathrm{exp}\left\{{\int }_{X}\mathrm{log}|f|\phantom{\rule{0.2em}{0ex}}d\mu \right\}$
if $\mathrm{exp}\left\{-\infty \right\}$ is defined to be 0.
6. Let $m$ be Lebesgue measure on $\left[0,1\right]$, and define ${‖f‖}_{p}$ with respect to $m$. Find all functions $\Phi$ on $\left[0,\infty \right)$ such that the relations
 $\Phi \left(\underset{p\to 0}{\mathrm{lim}}{‖f‖}_{p}\right)={\int }_{0}^{1}\left(\Phi \circ f\right)\phantom{\rule{0.2em}{0ex}}dm$
holds for every bounded, measurable, positive, $f$. Show first that
 $c\Phi \left(x\right)+\left(1-c\right)\Phi \left(1\right)=\Phi \left({x}^{c}\right)\phantom{\rule{2em}{0ex}}\left(x>0,0\le c\le 1\right)$.
Compare with Exercise 5(d).
7. For some measures, the relation $r 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.
8. If $g$ is a positive function on $\left(0,1\right)$ such that $g\left(x\right)\to \infty$ as $x\to 0$, then there is a convex function $h$ on $\left(0,1\right)$ such that $h\le g$ and $h\left(x\right)\to \infty$ as $x\to 0$. True or false? Is the problem changed if $\left(0,1\right)$ is replaced by $\left(0,\infty \right)$ and $x\to 0$ is replaced by $x\to \infty$.
9. Suppose $f$ is Lebesgue measurable on $\left(0,1\right)$ and not essentially bounded. By Exercise 4(e), ${‖f‖}_{p}\to \infty$ as $p\to \infty$. Can ${‖f‖}_{p}$ tend to $\infty$ arbitrarily slowly? More precisely, is it true that to every positive function $\Phi$ on $\left(0,\infty \right)$ such that $\Phi \left(p\right)\to \infty$ as $p\to \mathrm{infin;}$ one can find an $f$ such that ${‖f‖}_{p}\to \infty$ as $p\to \infty$, but ${‖f‖}_{p}\le \Phi \left(p\right)$ for all sufficiently large $p$?
10. Suppose ${f}_{n}$ is in ${L}^{p}\left(\mu \right)$, for $n=1,2,3,\dots$, and ${‖{f}_{n}-f‖}_{p}\to 0$ and ${f}_{n}\to g$ a.e., as $n\to \infty$. What relation exists between $f$ and $g$?
11. Suppose $\mu \left(\Omega \right)=1$, and suppose $f$ and $g$ are positive measurable functions on $\Omega$ such that $fg\ge 1$. Prove that
 ${\int }_{\Omega }f\phantom{\rule{0.2em}{0ex}}d\mu \cdot {\int }_{\Omega }g\phantom{\rule{0.2em}{0ex}}d\mu \ge 1$.
12. Suppose $\mu \left(\Omega \right)=1$ and $h:\Omega \to \left[0,\infty \right]$ 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 $\left[0,1\right]$ 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.
13. 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.
14. Suppose $1, $f\in {L}^{p}={L}^{p}\left(\left(0,\infty \right)\right)$, 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}}\left(0.
(a)   Prove Hardy's inequality
 ${‖F‖}_{p}\le \frac{p}{p-1}{‖f‖}_{p}$.
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 $p/\left(p-1\right)$ 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 $f\in {C}_{c}\left(\left(0,\infty \right)\right)$. Integration by parts gives
 ${\int }_{0}^{\infty }{F}^{p}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx=-p{\int }_{0}^{\infty }{F}^{p-1}\left(x\right)xF\prime \left(x\right)\phantom{\rule{0.2em}{0ex}}dx$.
Note that $xF\prime =f-F$, and apply Hölder's inequality to $\int {F}^{p-1}f$. Then derive the general case. (c) Take $f\left(x\right)={x}^{-1/p}$ on $\left[1,A\right]$, $f\left(x\right)=0$ elsewhere, for large $A$. See also Excercise 14, Chap. 8.
15. Suppose $\left\{{a}_{n}\right\}$ is a sequence of positive numbers. Prove that
 $\sum _{N=1}^{\infty }{\left(\frac{1}{N}\sum _{n=1}^{N}{a}_{n}\right)}^{p}\le {\left(\frac{p}{p-1}\right)}^{p}\sum _{n=1}^{\infty }{a}_{n}^{p}$
if $1. 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.
16. 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 $\mu \left(X-E\right)<\epsilon$ 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
 $S\left(n,k\right)=\bigcap _{i,j>n}\left\{x\phantom{\rule{0.2em}{0ex}}|\phantom{\rule{0.2em}{0ex}}|{f}_{i}\left(x\right)-{f}_{j}\left(x\right)|<\frac{1}{k}$,
show that $\mu \left(S\left(n,k\right)\right)\to \mu \left(X\right)$ as $n\to \infty$, for each $k$, and hence that there is a suitably increasing sequence $\left\{{n}_{k}\right\}$ such that $E=\bigcap S\left({n}_{k},k\right)$ 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.
17. (a)   If $0, put ${\gamma }_{p}=\mathrm{max}\left(1,{2}^{p-1}\right)$, and show that
 ${|\alpha -\beta |}^{p}\le {\gamma }_{p}\left({|\alpha |}^{p}+{|\beta |}^{p}\right)$
for arbitrary complex numbers $\alpha$ and $\beta$.
(b)   Suppose $\mu$ is a positive measure on $X$, $0, $f\in {L}^{p}\left(\mu \right)$, ${f}_{n}\in {L}^{p}\left(\mu \right)$, ${f}_{n}\left(x\right)\to f\left(x\right)$ a.e., and ${‖{f}_{n}‖}_{p}\to {‖f‖}_{p}$ as $n\to \infty$. Show that then $\mathrm{lim}{‖f-{f}_{n}‖}_{p}=0$, by completing the two proofs that are sketched below.
(i)   By Egoroff's theorem, $X=A\cup B$ in such a way that ${\int }_{A}{|f|}^{p}<\epsilon$, $\mu \left(B\right)<\infty$, and ${f}_{n}\to f$ uniformly on $B$. Fatou's lemma, applied to ${\int }_{B}{|f|}^{p}$, leads to
 $\mathrm{limsup}{\int }_{A}{|{f}_{n}|}^{p}\phantom{\rule{0.2em}{0ex}}d\mu \le \epsilon$.
(ii)   Put ${h}_{n}={\gamma }_{p}\left({|f|}^{p}+{|{f}_{n}|}^{p}\right)-{|f-{f}_{n}|}^{p}$, 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 ${‖{f}_{n}‖}_{p}\to {‖f‖}_{p}$ is omitted, even if $\mu \left(X\right)<\infty$.
18. 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
 $\mu \left(\left\{x\phantom{\rule{0.2em}{0ex}}|\phantom{\rule{0.2em}{0ex}}|{f}_{n}\left(x\right)-f\left(x\right)|>\epsilon \right\}\right)<\epsilon$
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 ${‖{f}_{n}-f‖}_{p}\to 0$, 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 ${ℝ}^{1}$?
19. 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
 $\mu \left(\left\{x\phantom{\rule{0.2em}{0ex}}|\phantom{\rule{0.2em}{0ex}}|f\left(x\right)-w|<\epsilon \right\}\right)>0$
for every $\epsilon >0$. Prove that ${R}_{f}$ is compact. What relation exists between the set ${R}_{f}$ and the number ${‖f‖}_{\infty }$?
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 $E\in ℳ$ 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?
20. Suppose $\phi$ is a real function on ${ℝ}^{1}$ 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.
21. 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.
22. 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.)
23. Suppose $\mu$ is a positive measure on $X$, $\mu \left(X\right)<\infty$, $f\in {L}^{\infty }\left(\mu \right)$, ${‖f‖}_{\infty }>0$, and
 ${\alpha }_{n}={\int }_{X}{|f|}^{n}\phantom{\rule{0.2em}{0ex}}d\mu \phantom{\rule{2em}{0ex}}\left(n=1,2,3,\dots \right)$.
Prove that
 $\underset{n\to \infty }{\mathrm{lim}}\frac{{\alpha }_{n+1}}{{\alpha }_{n}}={‖f‖}_{\infty }$.
24. Suppose $\mu$ is a positive measure, $f\in {L}^{p}\left(\mu \right)$, $g\in {L}^{p}\left(\mu \right)$.
(a)   If $0, prove that
 $\int |{|f|}^{p}-{|g|}^{p}|\phantom{\rule{0.2em}{0ex}}d\mu \le \int {|f-g|}^{p}\phantom{\rule{0.2em}{0ex}}d\mu$,
that $\Delta \left(f,g\right)=\int {|f-g|}^{p}\phantom{\rule{0.2em}{0ex}}d\mu$ defines a metric on ${L}^{p}\left(\mu \right)$, and that the resulting metric space is complete.
(b)   If $1\le p<\infty$ and ${‖f‖}_{p}\le R$, ${‖g‖}_{p}\le R$, prove that
 $\int |{|f|}^{p}-{|g|}^{p}|\phantom{\rule{0.2em}{0ex}}d\mu \le 2p{R}^{p-1}{‖f-g‖}_{p}$.
Hint: Prove first, for $x\ge 0$, $y\ge 0$, that Note that (a) and (b) establish the continuity of the mapping $f\to {|f|}^{p}$ that carries ${L}^{p}\left(\mu \right)$ into ${L}^{1}\left(\mu \right)$.
25. Suppose $\mu$ is a positive measure on $X$ and $f:X\to \left(0,\infty \right)$ 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
 ${\int }_{E}\left(\mathrm{log}f\right)\phantom{\rule{0.2em}{0ex}}d\mu \le \mu \left(E\right)\mathrm{log}\frac{1}{\mu \left(E\right)}$
and, when $0,
 ${\int }_{E}{f}^{p}\phantom{\rule{0.2em}{0ex}}d\mu \le {\mu \left(E\right)}^{1-p}$.
26. If $f$ is a positive measurable function on $\left[0,1\right]$, 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, McGraw-Hill, 1976. MR??????.

[Ru] W. Rudin, Real and complex analysis, Third edition, McGraw-Hill, 1987. MR0924157.