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

1. 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?
2. 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 $\log \phi$ implies the convexity of $\phi$, but not vice versa.
3. 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.)
4. Suppose $f$ is a complex measurable function on $X$, and

   $\phi \left(p\right)={\int }_X{\left|f\right|}^{p}d\mu ={\Vert f\Vert }_p^p(0<p<\infty )$.

   Let $E=\left\{p\right|\phi \left(p\right)<\infty \}$. 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 $\log \phi$ 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 ${\Vert f\Vert }_p\le \max ({\Vert f\Vert }_r,{\Vert f\Vert }_s)$. 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

   ${\Vert f\Vert }_p\to {\Vert f\Vert }_{\infty }\text{as}p\to \infty$.

5. Assume, in addition to the hypotheses of Exercise 4, that

   $\mu \left(X\right)=1$.

   (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

   $\underset{p\to 0}{\lim }{\Vert f\Vert }_p=\exp \left\{{\int }_X\log \left|f\right|d\mu \right\}$

   if $\exp \{-\infty \}$ is defined to be 0.

6. 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

   $\Phi \left(\underset{p\to 0}{\lim }{\Vert f\Vert }_p\right)={\int }_0^1(\Phi \circ f)dm$

   holds for every bounded, measurable, positive, $f$. Show first that

   $c\Phi \left(x\right)+(1-c)\Phi \left(1\right)=\Phi \left(x^c\right)(x>0,0\le c\le 1)$.

   Compare with Exercise 5(d).
7. 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.
8. 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 $x\to \infty$.
9. 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 $p$?
10. 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 $g$?
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

    ${\int }_{\Omega }fd\mu \cdot {\int }_{\Omega }gd\mu \ge 1$.

12. Suppose $\mu \left(\Omega \right)=1$ and $h:\Omega \to [0,\infty ]$ is measurable. If

    $A={\int }_{\Omega }hd\mu$,

    prove that

    $\sqrt{1+A^2}\le {\int }_{\Omega }\sqrt{1+h^2}d\mu \le 1+A$.

    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.
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 $1<p<\infty$, $f\in L^p=L^p\left(\right(0,\infty \left)\right)$, relative to Lebesgue measure, and

    $F\left(x\right)=\frac{1}{x}{\int }_0^{x}f\left(t\right)dt(0<x<\infty )$.

    (a)   Prove Hardy's inequality

    ${\Vert F\Vert }_p\le {\displaystyle \frac{p}{p-1}}{\Vert f\Vert }_p$.

    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 $p/(p-1)$ 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

    ${\int }_0^{\infty }{F}^p\left(x\right)dx=-p{\int }_0^{\infty }{F}^{p-1}\left(x\right)xF\prime \left(x\right)dx$.

    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 $[1,A]$, $f\left(x\right)=0$ elsewhere, for large $A$. See also Excercise 14, Chap. 8.
15. Suppose $\left\{a_n\right\}$ is a sequence of positive numbers. Prove that

    $\sum _{N=1}^{\infty }{({\displaystyle \frac{1}{N}\sum _{n=1}^N{a}_n)}}^p\le {({\displaystyle \frac{p}{p-1})}}^p\sum _{n=1}^{\infty }{a}_n^p$

    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.
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 $\mu (X-E)<\epsilon$ 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

    $S(n,k)=\bigcap _{i,j>n}\left\{x\right||f_i\left(x\right)-f_j\left(x\right)|<\frac{1}{k}$,

    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)   $\underset{t\to \infty }{\lim }{f}_t\left(x\right)=f\left(x\right)$ and

    (ii)   $t\to f_t\left(x\right)$ is continuous.

17. (a)   If $0<p<\infty$, put ${\gamma }_p=\max (1,2^{p-1})$, 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 $\lim {\Vert f-f_n\Vert }_p=0$, by completing the two proofs that are sketched below.

    (i)   By Egoroff's theorem, $X=A\cup B$ in such a way that ${\int }_A{\left|f\right|}^p<\epsilon$, $\mu \left(B\right)<\infty$, and $f_n\to f$ uniformly on $B$. Fatou's lemma, applied to ${\int }_B{\left|f\right|}^p$, leads to

    $\mathrm{limsup}{\int }_A{\left|f_n\right|}^{p}d\mu \le \epsilon$.

    (ii)   Put $h_n={\gamma }_p({\left|f\right|}^p+{\left|f_n\right|}^p)-{|f-f_n|}^p$, 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 $\mu \left(X\right)<\infty$.
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

    $\mu \left(\right\{x\left||f_n\left(x\right)-f\right(x\left)\right|>\epsilon \left\}\right)<\epsilon$

    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 ${ℝ}^1$?
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

    $\mu \left(\right\{x\left||f\right(x)-w|<\epsilon \left\}\right)>0$

    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

    ${\displaystyle \frac{1}{\mu \left(E\right)}}{\int }_{E}fd\mu$

    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

    $\phi \left({\int }_0^{1}f\left(x\right)dx\right)\le {\int }_0^1\phi \left(f\right)dx$

    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 $X$, $\mu \left(X\right)<\infty$, $f\in L^{\infty }\left(\mu \right)$, ${\Vert f\Vert }_{\infty }>0$, and

    ${\alpha }_n={\int }_X{\left|f\right|}^{n}d\mu (n=1,2,3,\dots )$.

    Prove that

    $\underset{n\to \infty }{\lim }{\displaystyle \frac{{\alpha }_{n+1}}{{\alpha }_n}={\Vert f\Vert }_{\infty }}$.

24. Suppose $\mu$ is a positive measure, $f\in L^p\left(\mu \right)$, $g\in L^p\left(\mu \right)$.

    (a)   If $0<p<1$, prove that

    $\int |{\left|f\right|}^p-{\left|g\right|}^p|d\mu \le \int {|f-g|}^{p}d\mu$,

    that $\Delta (f,g)=\int {|f-g|}^{p}d\mu$ 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

    $\int |{\left|f\right|}^p-{\left|g\right|}^p|d\mu \le 2p{R}^{p-1}{\Vert f-g\Vert }_p$.

    Hint: Prove first, for $x\ge 0$, $y\ge 0$, that

    $|x^p-y^p|\le \{\begin{array}{ll}{|x-y|}^p,& \text{if }0<p<1,\\ p|x-y|(x^{p-1}+y^{p-1}),& \text{if }1\le p<\infty .\end{array}$

    Note that (a) and (b) establish the continuity of the mapping $f\to {\left|f\right|}^p$ that carries $L^p\left(\mu \right)$ into $L^1\left(\mu \right)$.
25. Suppose $\mu$ is a positive measure on $X$ and $f:X\to (0,\infty )$ satisfies ${\int }_{X}fd\mu =1$. Prove, for every $E\subseteq X$ with $0<\mu \left(E\right)<\infty$, that

    ${\int }_E\left(\log f\right)d\mu \le \mu \left(E\right)\log {\displaystyle \frac{1}{\mu \left(E\right)}}$

    and, when $0<p<1$,

    ${\int }_E{f}^{p}d\mu \le {\mu \left(E\right)}^{1-p}$.

26. If $f$ is a positive measurable function on $[0,1]$, which is larger,

    ${\int }_0^{1}f\left(x\right)\log f\left(x\right)dx\text{or}{\int }_0^{1}f\left(s\right)ds{\int }_0^1\log f\left(t\right)dt$?

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.

page history