Last updates: 20 March 2011

- Let $\left\{{f}_{n}\right\}$ be a sequence of real nonnegative
functions on ${\mathbb{R}}^{1}$ 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,

- Let $f$ be an arbitrary function on
${\mathbb{R}}^{1}$, and define
$\varphi (x,\delta )=\mathrm{sup}\left\{\right|f\left(s\right)-f\left(t\right)\phantom{\rule{0.2em}{0ex}}|\phantom{\rule{0.2em}{0ex}}s,t\in (x-\delta ,x+\delta \left)\right\}$, $\varphi \left(x\right)=\mathrm{inf}\left\{\varphi \right(x,\delta \left)\phantom{\rule{0.2em}{0ex}}\right|\phantom{\rule{0.2em}{0ex}}\delta >0\}$,

Formulate and prove an analogous statement for general topological spaces in place of ${\mathbb{R}}^{1}$. - Let $X$ be a metric space, with metric $\rho $. For any nonempty
$E\subseteq X$, define
${\rho}_{E}\left(x\right)=\mathrm{inf}\left\{\rho (\right(x,y\left)\phantom{\rule{0.2em}{0ex}}\right|\phantom{\rule{0.2em}{0ex}}y\in E\}$. $f\left(x\right)={\displaystyle \frac{{\rho}_{A}\left(x\right)}{{\rho}_{A}\left(x\right)+{\rho}_{B}\left(x\right)}}$ - 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 $\mu ({E}_{1}\cup {E}_{2})=\mu ({E}_{1}+{E}_{2})$, even if ${E}_{1}$ and ${E}_{2}$ are not in $\mathcal{M}$.
- (b) If $E\in {\mathcal{M}}_{F}$, 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 $\mu \left(N\right)=0$.

- Let $m$ denote Lebesgue measure on ${\mathbb{R}}^{1}$. Let $E$ be Cantor's familiar "middle thirds" set. Show that $m\left(E\right)=0$ even though $E$ and ${\mathbb{R}}^{1}$ have the same cardinality.
- Let $m$ denote Lebesgue measure on ${\mathbb{R}}^{1}$.
Construct a totally disconnected compact set
$K\subseteq {\mathbb{R}}^{1}$
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. - Let $m$ denote Lebesgue measure on ${\mathbb{R}}^{1}$. If $0<\u03f5<1$, construct an open set $E\subseteq [0,1]$ which is dense in $[0,1]$, such that $m\left(E\right)=\u03f5$. (To say that $A$ is dense in $B$ means that the closure of $A$ contains $B$.)
- Let $m$ denote Lebesgue measure on ${\mathbb{R}}^{1}$.
Construct a Borel set $E\subseteq {\mathbb{R}}^{1}$ such
that
$0<m(E\cap I)<m\left(I\right)$ -
Construct a sequence of continuous functions ${f}_{n}$
on $[0,1]$ such that
$0\le {f}_{n}\le 1$, such that
$\underset{n\to \infty}{\mathrm{lim}}{\int}_{0}^{1}{f}_{n}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx=0$, - If $\left\{{f}_{n}\right\}$
is a sequence of continuous functions ${f}_{n}$
on $[0,1]$ such that
$0\le {f}_{n}\le 1$ and such that
${f}_{n}\left(x\right)\to 0$
as $n\to \infty $ for every $x\in [0,1]$, then
$\underset{n\to \infty}{\mathrm{lim}}{\int}_{0}^{1}{f}_{n}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx=0$. *Communications on Pure and Applied Mathematics*, vol. X, pp. 357-360, 1957.) - 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 $0$. - Show that every compact subset of ${\mathbb{R}}^{1}$ is the support of a Borel measure.
- Is it true that every compact subset of ${\mathbb{R}}^{1}$ is the support of a continuous function? If not, can you describe the class of all compact sets in ${\mathbb{R}}^{1}$ which are supports of continuous functions? Is your description valid in other topological spaces?
- Let $f$ be a real-valued Lebesgue measurable function on ${\mathbb{R}}^{k}$. 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 $x\in {\mathbb{R}}^{k}$.
- It is easy to guess the limits of
${\int}_{0}^{n}{(1-\frac{x}{n})}^{n}{e}^{x/2}\phantom{\rule{0.2em}{0ex}}dx$ and ${\int}_{0}^{n}{(1+\frac{x}{n})}^{n}{e}^{-2x}\phantom{\rule{0.2em}{0ex}}dx$ - Why is $m\left(Y\right)=0$ in the proof of Theorem 2.20(e)?
- Define the distance between points
$({x}_{1},{y}_{1})$
and
$({x}_{2},{y}_{2})$
in the plane to be
$|{y}_{1}-{y}_{2}|$ if ${x}_{1}={x}_{2}$, $1+|{y}_{1}-{y}_{2}|$ if ${x}_{1}\ne {x}_{2}$.

If $f\in {C}_{c}\left(X\right)$, let ${x}_{1},\dots {x}_{n}$ be those values of $x$ for which $f(x,y)\ne 0$ for at least one $y$ (there are only finitely many such $x$!), and define$\Lambda f={\displaystyle \sum _{j=1}^{n}{\int}_{-\mathrm{\infty}}^{\infty}f({x}_{j},y)\phantom{\rule{0.2em}{0ex}}dy}$. -
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.)

*not*regular (every neighborhood of ${\omega}_{1}$ has measure 1), and that$f\left({\omega}_{1}\right)={\displaystyle {\int}_{X}f\phantom{\rule{0.2em}{0ex}}d\lambda}$. - 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.
- Find continuous functions ${f}_{n}:[0,1]\to [0,\infty )$ such that ${f}_{n}\left(x\right)\to 0$ for all $x\in [0,1]$ 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.)
- If $X$ is compact and $f:\to (-\infty ,\infty )$ is upper semicontinuous, prove that $f$ attains its maximum at some point of $X$.
- Suppose that $X$ is a metric space, with metric $d$,
and that $f:X\to [0,\infty ]$ 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
${g}_{n}\left(x\right)=\mathrm{inf}\left\{f\right(p)+nd(x,p)\phantom{\rule{0.2em}{0ex}}|\phantom{\rule{0.2em}{0ex}}p\in X\}$ - (i) $|{g}_{n}\left(x\right)-{g}_{n}\left(y\right)|\le nd(x,y)$,
- (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 $x\in X$.

- Suppose that $V$ is open in ${\mathbb{R}}^{k}$ and $\mu $ is a finite positive Borel measure on ${\mathbb{R}}^{k}$. Is the function that sends $x$ to $\mu (V+x)$ necessarily continuous? lower semicontinuous? upper semicontinuous?
- A
*step function*is, by definition, a finite linear combination of characteristic functions of bounded intervals in ${\mathbb{R}}^{1}$. Assume $f\in {L}^{1}\left({\mathbb{R}}^{1}\right)$, and prove that there is a sequence $\left\{{g}_{n}\right\}$ of step functions so that$\underset{n\to \infty}{\mathrm{lim}}{\int}_{-\infty}^{\infty}|f\left(x\right)-{g}_{n}\left(x\right)|\phantom{\rule{0.2em}{0ex}}dx=0$. - (i)
Find the smallest constant $c$ such that
$\mathrm{log}(1+{e}^{t})<c+t\phantom{\rule{2em}{0ex}}(0<t<\infty )$. $\underset{n\to \infty}{\mathrm{lim}}\frac{1}{n}{\int}_{0}^{1}\mathrm{log}(1+{e}^{nf\left(x\right)})\phantom{\rule{0.2em}{0ex}}dx$.

These exercises are taken from [Ru, Chapt. 2] for a course in "Measure Theory" at the Masters level at University of Melbourne.

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