## Integration: Exercises Chapter 2

Last updates: 20 March 2011

## Integration: Exercises Chapter 2

1. Let $\left\{{f}_{n}\right\}$ be a sequence of real nonnegative functions on ${ℝ}^{1}$ and consider the following four statements:
(a)   If ${f}_{1}$ and ${f}_{2}$ are upper semicontinuous, then ${f}_{1}+{f}_{2}$ is upper semicontinuous,
(b)   If ${f}_{1}$ and ${f}_{2}$ are lower semicontinuous, then ${f}_{1}+{f}_{2}$ is lower semicontinuous,
(c)   If each ${f}_{n}$ is upper semicontinuous, then ${\sum }_{1}^{\infty }{f}_{n}$ is upper semicontinuous,
(d)   If each ${f}_{n}$ is lower semicontinuous, then ${\sum }_{1}^{\infty }{f}_{n}$ is lower semicontinuous,
Show that three of these are true and one is false. What hapens if the word "nonnegative" is omitted? Is the truth of the statements affected if ${ℝ}^{1}$ is replaced by a general topological space?
2. Let $f$ be an arbitrary function on ${ℝ}^{1}$, and define
 $\varphi \left(x,\delta \right)=\mathrm{sup}\left\{|f\left(s\right)-f\left(t\right)\phantom{\rule{0.2em}{0ex}}|\phantom{\rule{0.2em}{0ex}}s,t\in \left(x-\delta ,x+\delta \right)\right\}$, $\varphi \left(x\right)=\mathrm{inf}\left\{\varphi \left(x,\delta \right)\phantom{\rule{0.2em}{0ex}}|\phantom{\rule{0.2em}{0ex}}\delta >0\right\}$,
Prove that $\varphi$ is upper semicontinuous, that $f$ is continuous at a point $x$ if and only if $\varphi \left(x\right)=0$, and hence that the set of points of continuity of an arbitrary complex function is a ${G}_{\delta }$.
Formulate and prove an analogous statement for general topological spaces in place of ${ℝ}^{1}$.
3. Let $X$ be a metric space, with metric $\rho$. For any nonempty $E\subseteq X$, define
 ${\rho }_{E}\left(x\right)=\mathrm{inf}\left\{\rho \left(\left(x,y\right)\phantom{\rule{0.2em}{0ex}}|\phantom{\rule{0.2em}{0ex}}y\in E\right\}$.
Show that ${\rho }_{E}$ is a uniformly continuous function on $X$. If $A$ and $B$ are disjoint nonempty closed subsets of $X$, examine the relevance of the function
 $f\left(x\right)=\frac{{\rho }_{A}\left(x\right)}{{\rho }_{A}\left(x\right)+{\rho }_{B}\left(x\right)}$
to Urysohn's lemma.
4. Examine the proof of the Riesz theorem and prove the following two statements:
(a)   If ${E}_{1}\subseteq {V}_{1}$ and ${E}_{2}\subseteq {V}_{2}$, where ${V}_{1}$ and ${V}_{2}$ are disjoint open sets, then $\mu \left({E}_{1}\cup {E}_{2}\right)=\mu \left({E}_{1}+{E}_{2}\right)$, even if ${E}_{1}$ and ${E}_{2}$ are not in $ℳ$.
(b)   If $E\in {ℳ}_{F}$, then $E=N\cup {K}_{1}\cup {K}_{2}\cup \cdots$, where $\left\{{K}_{i}\right\}$ is a disjoint countable collection of compact sets and $\mu \left(N\right)=0$.
5. Let $m$ denote Lebesgue measure on ${ℝ}^{1}$. Let $E$ be Cantor's familiar "middle thirds" set. Show that $m\left(E\right)=0$ even though $E$ and ${ℝ}^{1}$ have the same cardinality.
6. Let $m$ denote Lebesgue measure on ${ℝ}^{1}$. Construct a totally disconnected compact set $K\subseteq {ℝ}^{1}$ such that $m\left(K\right)>0$. ($K$ is to have no connected subset consisting of more than one point.)
If $v$ is lower semicontinuous and $v\le {\chi }_{K}$, show that actually $v\le 0$. Hence ${\chi }_{K}$ cannot be approximated from below by lower semicontinuous functions, in the sense of the Vitali-Carathéodory theorem.
7. Let $m$ denote Lebesgue measure on ${ℝ}^{1}$. If $0<ϵ<1$, construct an open set $E\subseteq \left[0,1\right]$ which is dense in $\left[0,1\right]$, such that $m\left(E\right)=ϵ$. (To say that $A$ is dense in $B$ means that the closure of $A$ contains $B$.)
8. Let $m$ denote Lebesgue measure on ${ℝ}^{1}$. Construct a Borel set $E\subseteq {ℝ}^{1}$ such that
 $0
for every nonempty segment $I$. Is it possible to have $m\left(E\right)<\infty$ for such a set?
9. Construct a sequence of continuous functions ${f}_{n}$ on $\left[0,1\right]$ such that $0\le {f}_{n}\le 1$, such that
 $\underset{n\to \infty }{\mathrm{lim}}{\int }_{0}^{1}{f}_{n}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx=0$,
but such that the sequence $\left\{{f}_{n}\left(x\right)\right\}$ converges for no $x\in \left[0,1\right]$.
10. If $\left\{{f}_{n}\right\}$ is a sequence of continuous functions ${f}_{n}$ on $\left[0,1\right]$ such that $0\le {f}_{n}\le 1$ and such that ${f}_{n}\left(x\right)\to 0$ as $n\to \infty$ for every $x\in \left[0,1\right]$, then
 $\underset{n\to \infty }{\mathrm{lim}}{\int }_{0}^{1}{f}_{n}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx=0$.
Try to prove this without using any measure theory or any theorems about Lebesgue integration. (This is to impress you with the power of the Lebesgue integral. A nice proof was given by W.F. Oberlein in Communications on Pure and Applied Mathematics, vol. X, pp. 357-360, 1957.)
11. Let $\mu$ be a regular Borel measure on a compact Hausdorff space $X$; assume that $\mu \left(X\right)=1$. Prove that there is a compact set $K\subseteq X$ (the carrier or support of $\mu$) such that $\mu \left(K\right)=1$ but $\mu \left(H\right)<1$ for every proper comapct subset $H$ of $K$. Hint: Let $K$ be the intersection of all compact ${K}_{\alpha }$ with $\mu \left({K}_{\alpha }\right)=1$; show that every open set $V$ which contains $K$ also contains some ${K}_{\alpha }$. Regularity of $\mu$ is needed; compare exercise 18. Show that ${K}^{c}$ is the largest open set in $X$ whose measure is $0$.
12. Show that every compact subset of ${ℝ}^{1}$ is the support of a Borel measure.
13. Is it true that every compact subset of ${ℝ}^{1}$ is the support of a continuous function? If not, can you describe the class of all compact sets in ${ℝ}^{1}$ which are supports of continuous functions? Is your description valid in other topological spaces?
14. Let $f$ be a real-valued Lebesgue measurable function on ${ℝ}^{k}$. Prove that there exist Borel functions $g$ and $h$ such that $g\left(x\right)=h\left(x\right)$ a.e. $\left[m\right]$, and $g\left(x\right)\le f\left(x\right)\le h\left(x\right)$ for every $x\in {ℝ}^{k}$.
15. It is easy to guess the limits of
 ${\int }_{0}^{n}{\left(1-\frac{x}{n}\right)}^{n}{e}^{x/2}\phantom{\rule{0.2em}{0ex}}dx$       and       ${\int }_{0}^{n}{\left(1+\frac{x}{n}\right)}^{n}{e}^{-2x}\phantom{\rule{0.2em}{0ex}}dx$
as $n\to \infty$. Prove that your guesses are correct.
16. Why is $m\left(Y\right)=0$ in the proof of Theorem 2.20(e)?
17. Define the distance between points $\left({x}_{1},{y}_{1}\right)$ and $\left({x}_{2},{y}_{2}\right)$ in the plane to be
 $|{y}_{1}-{y}_{2}|$    if ${x}_{1}={x}_{2}$,        $1+|{y}_{1}-{y}_{2}|$    if ${x}_{1}\ne {x}_{2}$.
Show that this is indeed a metric, and that the resulting metric space $X$ is locally compact.
If $f\in {C}_{c}\left(X\right)$, let ${x}_{1},\dots {x}_{n}$ be those values of $x$ for which $f\left(x,y\right)\ne 0$ for at least one $y$ (there are only finitely many such $x$!), and define
 $\Lambda f=\sum _{j=1}^{n}{\int }_{-\infty }^{\infty }f\left({x}_{j},y\right)\phantom{\rule{0.2em}{0ex}}dy$.
Let $\mu$ be the measure associated with this $\Lambda$ by Theorem 2.14. If $E$ is the $x$-axis, show that $\mu \left(E\right)=\infty$ although $\mu \left(K\right)=0$ for every compact $K\subseteq E$.
18. Let $X$ be a well-ordered uncountable set which has a last element ${\omega }_{1}$, such that every predecessor of ${\omega }_{1}$ has at most countably many predecessors. ("Construction": Take any well-ordered set which has elements with uncountably many predecessors, and let ${\omega }_{1}$ be the first of these; ${\omega }_{1}$ is called the first countable ordinal.) For $\alpha \in X$, let ${P}_{\alpha }$ (resp. ${S}_{\alpha }$) be the set of all predecessors (resp. successors) of $\alpha$, and call a subset of $X$ open if it is a ${P}_{\alpha }$ or an ${S}_{\beta }$ or a ${P}_{\alpha }\cap {S}_{\beta }$ or a union of such sets. Prove that $X$ is a compact Hausdorff space. (Hint: No well ordered set contains an infinite decreasing sequence.)
(a)   Prove that the complement of the point ${\omega }_{1}$ is an open set which is not σ-compact.
(b)   Prove that to every $f\in C\left(X\right)$ there corresponds an $\alpha \ne {\omega }_{1}$ such that $f$ is constant on ${S}_{\alpha }$.
(c)   Prove that the intersection of every countable collection $\left\{{K}_{n}\right\}$ of uncountable compact subsets of $X$ is uncountable. (Hint: Consdier limits of increasing countable sequences in $X$ which intersect each ${K}_{n}$ in infinitely many points.)
Let $ℳ$ be the collection of all $E\subseteq X$ such that either $E\cup \left\{{\omega }_{1}\right\}$ or ${E}^{c}\cup \left\{{\omega }_{1}\right\}$ contains an uncountable compact set; in the first case, define $\lambda \left(E\right)=1$; in the second case, define $\lambda \left(E\right)=0$. Prove that $ℳ$ is a σ-algebra which contains all Borel sets in $X$, that $\lambda$ is a measure on $ℳ$ which is not regular (every neighborhood of ${\omega }_{1}$ has measure 1), and that
 $f\left({\omega }_{1}\right)={\int }_{X}f\phantom{\rule{0.2em}{0ex}}d\lambda$.
for every $f\in C\left(X\right)$. Describe the regular $\mu$ which Theorem 2.14 associates with this linear functional.
19. Go through the proof of Theorem 2.14, assuming $X$ to be compact (or even compact metric) rather than just locally compact, and see what simplifications you can find.
20. Find continuous functions ${f}_{n}:\left[0,1\right]\to \left[0,\infty \right)$ such that ${f}_{n}\left(x\right)\to 0$ for all $x\in \left[0,1\right]$ as $n\to \infty$, ${\int }_{0}^{1}{f}_{n}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\to 0$, but ${\mathrm{sup}}_{n}{f}_{n}$ is not in ${L}^{1}$. (This shows that the conclusion of the dominated convergence theorem may hold even when part of its hypothesis is violated.)
21. If $X$ is compact and $f:\to \left(-\infty ,\infty \right)$ is upper semicontinuous, prove that $f$ attains its maximum at some point of $X$.
22. Suppose that $X$ is a metric space, with metric $d$, and that $f:X\to \left[0,\infty \right]$ is lower semicontinuous $f\left(p\right)<\infty$ for at least one $p\in X$. For $n=1,2,3,\dots$ and $x\in X$, define
 ${g}_{n}\left(x\right)=\mathrm{inf}\left\{f\left(p\right)+nd\left(x,p\right)\phantom{\rule{0.2em}{0ex}}|\phantom{\rule{0.2em}{0ex}}p\in X\right\}$
and prove that
(i)   $|{g}_{n}\left(x\right)-{g}_{n}\left(y\right)|\le nd\left(x,y\right)$,
(ii)   $0\le {g}_{1}\le {g}_{2}\le \cdots \le f$,
(iii)   ${g}_{n}\left(x\right)\to {f}_{n}\left(x\right)$ as $n\to \infty$, for all $x\in X$.
Thus $f$ is the pointwise limit of an increasing sequence of continuous functions. (Note that the converse is almost trivial.)
23. Suppose that $V$ is open in ${ℝ}^{k}$ and $\mu$ is a finite positive Borel measure on ${ℝ}^{k}$. Is the function that sends $x$ to $\mu \left(V+x\right)$ necessarily continuous? lower semicontinuous? upper semicontinuous?
24. A step function is, by definition, a finite linear combination of characteristic functions of bounded intervals in ${ℝ}^{1}$. Assume $f\in {L}^{1}\left({ℝ}^{1}\right)$, and prove that there is a sequence $\left\{{g}_{n}\right\}$ of step functions so that
 $\underset{n\to \infty }{\mathrm{lim}}{\int }_{-\infty }^{\infty }|f\left(x\right)-{g}_{n}\left(x\right)|\phantom{\rule{0.2em}{0ex}}dx=0$.
25. (i)   Find the smallest constant $c$ such that
 $\mathrm{log}\left(1+{e}^{t}\right).
(ii)   Does
 $\underset{n\to \infty }{\mathrm{lim}}\frac{1}{n}{\int }_{0}^{1}\mathrm{log}\left(1+{e}^{nf\left(x\right)}\right)\phantom{\rule{0.2em}{0ex}}dx$.
exist for every real $f\in {L}^{1}$? If it exists, what is it?

## Notes and References

These exercises are taken from [Ru, Chapt. 2] 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.