Integration: More Exercises

1. If $\mu$ is a complex measure on a σ-algebra $ℳ$, and if $E\in ℳ$, define $\lambda \left(E\right)=\mathrm{sup}\sum |\mu \left({E}_{i}\right)|,$ the supremum being taken over all finite partitions $\left\{{E}_{i}\right\}$ of $E$. Does it follow that $\lambda =|\mu |$?
2. Prove that the example given at the end of [Ru, Sec. 6.10] has the stated properties.
3. Prove that the vector space $M\left(X\right)$ of all complex regular Borel measures on a locally compact Hausdorff space $X$ is a Banch space if $‖\mu ‖=|\mu |\left(X\right)$. Hint: Compare [Ru, Ch. 5, Exercise 8]. [That the difference of any two members of $M\left(X\right)$ is in $M\left(X\right)$ was used in the first paragraph of the proof of [Ru, Theorem 6.19]; supply a proof of this fact.]
4. Suppose $1\le p\le \infty$ and $q$ is the exponent conjugate to $p$. Suppose $\mu$ is a positive σ-finite measure and $g$ is a measurable function such that $fg\in {L}^{1}\left(\mu \right)$ for every $f\in {L}^{p}\left(\mu \right)$. Prove that then $g\in {L}^{q}\left(\mu \right)$.
5. Suppose $X$ consists of two points $a$ and $b$; define $\mu \left(\left\{a\right\}\right)=1$, $\mu \left(\left\{b\right\}\right)=\mu \left(X\right)=\infty$, and $\mu \left(\varnothing \right)=0$. Is it true, for this $\mu$ that ${L}^{\infty }\left(\mu \right)$ is the dual space of ${L}^{1}\left(\mu \right)$?
6. Suppose $1 and $q$ is the exponent conjugate to $p$. Prove that ${L}^{q}\left(\mu \right)$ is the dual space of ${L}^{p}\left(\mu \right)$ even if $\mu$ is not σ-finite.
7. Suppose $\mu$ is a complex Borel measure on $\left[0,2\pi \right)$ (or on the unit circle $T$), and define the Fourier coefficients of $\mu$ by
 $\stackrel{^}{\mu }\left(n\right)=\int {e}^{-i\pi t}d\mu \left(t\right)\phantom{\rule{2em}{0ex}}\left(n=0,±1,±2,\dots \right)$.
Assume that $\stackrel{^}{\mu }\left(n\right)\to 0$ as $n\to +\infty$ and prove that then $\stackrel{^}{\mu }\left(n\right)\to 0$ as $n\to -\infty$. Hint: The assumption also holds with $f\phantom{\rule{0.1em}{0ex}}d\mu$ in place of $d\mu ;$ if $f$ is any trigonometric polynomial, hence if $f$ is continuous, hence if $f$ is any bounded Borel function, hence if $d\mu$ is replaced by $d|\mu |$.
8. In the terminology of Exercise 7, find all $\mu$ such that $\stackrel{^}{\mu }$ is periodic, with period $k$. [This means that $\stackrel{^}{\mu }\left(n+k\right)=\stackrel{^}{\mu }\left(n\right)$ for all integers $n$; of course, $k$ is also assumed to be an integer.]
9. Suppose that $\left\{{g}_{n}\right\}$ is a sequence of positive continuous functions on $I=\left[0,1\right]$, that $\mu$ is a positive Borel measure on $I$, and that
(a) $\underset{n\to \infty }{\mathrm{lim}}\phantom{\rule{0.2em}{0ex}}{g}_{n}\left(x\right)=0,\phantom{\rule{2em}{0ex}}\mathrm{a.e.}\left[m\right]$,
(b) ${\int }_{I}\phantom{\rule{0.2em}{0ex}}{g}_{n}\phantom{\rule{0.2em}{0ex}}dm=1$,    for all $n$,
(c)$\underset{n\to \infty }{\mathrm{lim}}{\int }_{I}\phantom{\rule{0.2em}{0ex}}f{g}_{n}\phantom{\rule{0.2em}{0ex}}dm={\int }_{I}\phantom{\rule{0.2em}{0ex}}f\phantom{\rule{0.2em}{0ex}}d\mu$,    for every $f\in C\left(I\right)$.
Does it follow that $\mu \perp m$?
10. Let $\left(X,ℳ,\mu \right)$ be a positive measure space. Call a set $\Phi \subseteq {L}^{1}\left(\mu \right)$ uniformly integrable if to each $\epsilon <0$ corresponds a $\delta >0$ such that
 $|{\int }_{E}\phantom{\rule{0.2em}{0ex}}f\phantom{\rule{0.2em}{0ex}}d\mu |<\epsilon$,
whenever $f\in \Phi$ and $\mu \left(E\right)<\delta$.
(a)   Prove that every finite subset of ${L}^{1}\left(\mu \right)$ is uniformly integrable.
(b)   Prove the following convergence theorem of Vitali: If
(i)   $\mu \left(X\right)<\infty$,
(ii)   $\left\{{f}_{n}\right\}$ is uniformly integrable,
(iii)   ${f}_{n}\to f\left(x\right)$ a.e. as $n\to \infty$,
(iv)   $|f\left(x\right)|<\infty$ a.e.,
then $f\in {L}^{1}\left(\mu \right)$ and
 $\underset{n\to \infty }{\mathrm{lim}}{\int }_{X}\phantom{\rule{0.2em}{0ex}}|f-{f}_{n}|\phantom{\rule{0.2em}{0ex}}d\mu =0$.
Suggestion: Use Egoroff's theorem.
(c)   Show that (b) fails if $\mu$ is Lebesgue measure on $\left(-\infty ,\infty \right)$, even if $\left\{{‖{f}_{n}‖}_{1}\right\}$ is assumed to be bounded. Hypothesis (i) can therefore not be omitted in (b).
(d)   Show that hypothesis (iv) is redundant in (b) for some $\mu$ ( for instance, for Lebesgue measure on a bounded interval), but that there are finite measures for which the omission of (iv) would make (b) false.
(e)   Show that Vitali's theorem implies Lebesgue's dominated convergence theorem, for finite measure spaces. Construct an example in which Vitali's theorem applies although the hypotheses of Lebesgue's theorem do not hold.
(f)   Construct a sequence $\left\{{f}_{n}\right\}$, say on $\left[0,1\right]$, so that ${f}_{n}\left(x\right)\to 0$, but $\left\{{f}_{n}\right\}$ is not uniformly integrable (with respect to Lebesgue measure).
(g)   However, the following converse of Vitali's theorem is true: If $\mu \left(X\right)<\infty$, ${f}_{n}\in {L}^{1}\left(\mu \right)$ and $\underset{n\to \infty }{\mathrm{lim}}{\int }_{E}\phantom{\rule{0.2em}{0ex}}{f}_{n}\phantom{\rule{0.2em}{0ex}}d\mu$ exists for every $E\in ℳ$, then $\left\{{f}_{n}\right\}$ is uniformly integrable.
Prove this by completing the following outline.
Define $\rho \left(A,B\right)=\int |{\chi }_{A}-{\chi }_{B}|\phantom{\rule{0.2em}{0ex}}d\mu$. Then $\left(ℳ,\rho \right)$ is a complete metric space (modulo sets of measure $0$), and $E\to {\int }_{E}\phantom{\rule{0.2em}{0ex}}{f}_{n}\phantom{\rule{0.2em}{0ex}}d\mu$ is continuous for each $n$. If $\epsilon >0$, there exist ${E}_{0},\delta ,N$ (Exercise 13, Chapt. 5) so that
 $|{\int }_{X}\phantom{\rule{0.2em}{0ex}}\left(f-{f}_{n}\right)\phantom{\rule{0.2em}{0ex}}d\mu |<\epsilon$,    if   $\rho \left(E,{E}_{0}\right)<\delta$,     $n>N$. (*)
If $\mu <\delta$, (*) holds with $B={E}_{0}-A$ and $C={E}_{0}\cup A$ in place of $E$. Thus (*) holds with $A$ in place of $E$ and $2\epsilon$ in place of $\epsilon$. Now apply (a) to $\left\{{f}_{1},{f}_{2},\dots {f}_{n}\right\}$: There exists $\delta \prime >0$ such that
 $|{\int }_{X}\phantom{\rule{0.2em}{0ex}}{f}_{n}\phantom{\rule{0.2em}{0ex}}d\mu |<3\epsilon$,    if   $\mu \left(A\right)<\delta \prime$,     $n=1,2,3,\dots$.
11. Suppose that $\mu$ is a positive measure on $X$, $\mu \left(X\right)\le \infty$, ${f}_{n}\in {L}^{1}\left(\mu \right)$ for $n=1,2,3,\dots$, ${f}_{n}\to f\left(x\right)$ a.e., and there exists $p>1$ and $C<\infty$ such that ${\int }_{X}\phantom{\rule{0.2em}{0ex}}{|{f}_{n}|}^{p}\phantom{\rule{0.2em}{0ex}}d\mu for all $n$. Prove that
 $\underset{n\to \infty }{\mathrm{lim}}{\int }_{X}\phantom{\rule{0.2em}{0ex}}|f-{f}_{n}|\phantom{\rule{0.2em}{0ex}}d\mu =0$.
Hint: $\left\{{f}_{n}\right\}$ is uniformly integrable.
12. Let $ℳ$ be the collection of all sets $E$ in the unit interval $\left[0,1\right]$ such that either $E$ or its complement is at most countable. Let $\mu$ be the counting measure on this σ-algebra $ℳ$. If $g\left(x\right)=x$ for $0\le x\le 1$, show that $g$ is not $ℳ$-measurable, although the mapping
 $f↦\sum xf\left(x\right)=\int fg\phantom{\rule{0.1em}{0ex}}d\mu$
makes sense for every $f\in {L}^{1}\left(\mu \right)$ and defines a bounded linear functional on ${L}^{1}\left(\mu \right)$. Thus ${\left({L}^{\infty }\right)}^{*}\ne {L}^{1}$ in this situation.
13. Let ${L}^{\infty }={L}^{\infty }\left(m\right)$, where $m$ is Lebesgue measure on $I=\left[0,1\right]$. Show that there is a bounded linear functional $\Lambda \ne 0$ on ${L}^{\infty }$ that is $0$ on $C\left(I\right)$, and that therefore there is no $g\in {L}^{1}\left(m\right)$ that satisfies $\Lambda f={\int }_{I}\phantom{\rule{0.2em}{0ex}}fg\phantom{\rule{0.2em}{0ex}}dm$ for every $f\in {L}^{\infty }$. Thus ${\left({L}^{\infty }\right)}^{*}\ne {L}^{1}$.

Notes and References

These exercises are taken from [Ru, Chapt. 6] for a course in "Measure Theory" at the Masters level at University of Melbourne.

References

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