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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
\makeheading{Real Analysis 6211 Autumn 2024 \hfill Homework problem list}
\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Topology}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Two metrics $\rho_1,\rho_2$ on $X$ are called \emph{equivalent} if there is a $C>0$ such that
$$
C^{-1}\rho_1(x, y) \leq \rho_2(x,y) \leq C \rho_1(x,y)
\qquad\qquad
\forall x,y\in X.
$$
Show that equivalent metrics induce the same topology on $X$.
That is, show that $U\subset X$ is open with respect to $\rho_1$ if and only if $U$ is open with respect to $\rho_2$.
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}[Sarason]
Let $(X,\rho)$ be a metric space.
\begin{enumerate}
\item
Let $\alpha : [0,\infty) \to [0,\infty)$ be a continuous non-decreasing function satisfying
\begin{itemize}
\item
$\alpha(s) = 0$ if and only if $s=0$, and
\item
$\alpha(s+t) \leq \alpha(s) + \alpha(t)$ for all $s,t\geq 0$.
\end{itemize}
Define $\sigma(x,y) := \alpha(\rho(x,y))$.
Show that $\sigma$ is a metric, and $\sigma$ induces the same topology on $X$ as $\rho$.
\item
Define $\rho_1,\rho_2: X\times X \to [0,\infty)$ by
\begin{align*}
\rho_1(x,y)
&:=
\begin{cases}
\rho(x,y) &\text{if }\rho(x,y)\leq 1
\\
1 &\text{otherwise.}
\end{cases}
\\
\rho_2(x,y)
&:=
\frac{\rho(x,y)}{1+\rho(x,y)}.
\end{align*}
Use part (1) to show that $\rho_1$ and $\rho_2$ are metrics on $X$ which induce the same topology on $X$ as $\rho$.
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
A collection of subsets of $(F_i)_{i\in I}$ of $X$ has the \emph{finite intersection property}
if for any finite $J\subset I$, we have $\bigcap_{j \in J} F_j \neq \emptyset$.
Prove that for a metric (or topological) space, the following are equivalent.
\begin{enumerate}
\item
Every open cover of $X$ has a finite subcover.
\item
For every collection of closed subsets $(F_i)_{i\in I}$ with the finite intersection property, $\bigcap_{i\in I} F_i \neq \emptyset$.
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}[Adapted from Wikipedia \url{https://en.wikipedia.org/wiki/Locally_compact_space}]
Consider the following conditions:
\begin{enumerate}
\item
Every point of $X$ has a compact neighborhood.
\item
Every point of $X$ has a closed compact neighborhood.
\item
Every point of $X$ has a relatively compact neighborhood.
\item
Every point of $X$ has a local base of relatively compact neighborhoods.
\item
Every point of $X$ has a local base of compact neighborhoods.
\item
For every point $x$ of $X$, every neighborhood of $x$ contains a compact neighborhood of $x$.
\end{enumerate}
Determine which conditions imply which other conditions.
Then show all the above conditions are equivalent when $X$ is Hausdorff.
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Suppose $(X,\tau)$ is a locally compact Hausdorff topological space and suppose $K\subset X$ is a non-empty compact set.
\begin{enumerate}
\item
Suppose $K\subset U$ is an open set.
Show there is a continuous function $f: X \to [0,1]$ with compact support such that $f|_K = 1$ and $f|_{U^c}=0$.
\item
Suppose $f:K \to \bbC$ is continuous.
Show there is a continuous function $F: X \to \bbC$ such that $F|_K = f$.
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Suppose $(X,\tau)$ is a locally compact topological space and $(f_n)$ is a sequence of continuous $\bbC$-valued functions on $X$.
Show that the following are equivalent:
\begin{enumerate}
\item
There is a continuous function $f: X \to \bbC$ such that $f_n|_K \to f|_K$ uniformly on every compact $K\subset X$.
\item
For every compact $K\subset X$, $(f_n|_K)$ is uniformly Cauchy.
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
\mbox{}
\begin{enumerate}
\item
Show that every open subset of $\bbR$ is a countable union of open intervals where both endpoints are rational.
\item
Suppose $U\subset \bbR$ is open and suppose $((a_j,b_j))_{j\in J}$ is a collection of open intervals which cover $U$:
$$
U \subset \bigcup_{j\in J} (a_j, b_j).
$$
Show there is a countable sub-cover, i.e., show that there is a countable subset $I\subset J$ such that
$$
U \subset \bigcup_{i\in I} (a_i, b_i).
$$
\item
Suppose $((a_j,b_j])_{j\in J}$ is a collection of half-open intervals which cover $(0,1]$:
$$
(0,1] \subset \bigcup_{j\in J} (a_j, b_j].
$$
Show there is a countable sub-cover, i.e., show that there is a countable subset $I\subset J$ such that
$$
(0,1] \subset \bigcup_{i\in I} (a_i, b_i].
$$
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Suppose $X$ is a locally compact Hausdorff space, $K\subset X$ is compact, and $\{U_1,\dots, U_n\}$ is an open cover of $K$.
Prove that there are $g_i \in C_c(X, [0,1])$ for $i=1,\dots, n$ such that $g_i=0$ on $U_i^c$ and $\sum_{i=1}^n g_i = 1$ everywhere on $K$.
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}[Pedersen \emph{Analysis Now}, E 1.3.4 and E 1.3.6]
A \emph{filter} on a set $X$ is a collection $\cF$ of non-empty subsets of $X$ satisfying
\begin{itemize}
\item
$A,B\in \cF$ implies $A\cap B \in \cF$, and
\item
$A\in \cF$ and $A\subset B$ implies $B\in \cF$.
\end{itemize}
Suppose $\tau$ is a topology on $X$.
We say a filter $\cF$ \emph{converges} to $x\in X$ if every open neighborhood $U$ of $x$ lies in $\cF$.
\begin{enumerate}
\item
Show that $A\subset X$ is open if and only if $A\in \cF$ for every filter $\cF$ that converges to a point in $A$.
\item
Show that if $\cF$ and $\cG$ are filters and $\cF\subset \cG$ ($\cG$ is a \emph{subfilter} of $\cF$), then $\cG$ converges to $x$ whenever $\cF$ converges to $x$.
\item
Suppose $(x_\lambda)$ is a net in $X$.
Let $\cF$ be the collection of sets $A$ such that $(x_\lambda)$ is eventually in $A$.
Show that $\cF$ is a filter.
Then show that $x_\lambda \to x$ if and only if $\cF$ converges to $x$.
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}[Pedersen \emph{Analysis Now}, E 1.3.5]
A filter $\cF$ on a set $X$ is called an \emph{ultrafilter} if it is not properly contained in any other filter.
\begin{enumerate}
\item
Show that a filter $\cF$ is an ultrafilter if and only if for every $A\subset X$, we have either $A \in \cF$ or $A^c \in \cF$.
\item
Use Zorn's Lemma to prove that every filter is contained in an ultrafilter.
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Let $(X,\tau)$ be a topological space.
A net $(x_\lambda)_{\lambda \in \Lambda}$ is called \emph{universal} if for every subset $Y\subset X$, $(x_\lambda)$ is either eventually in $Y$ or eventually in $Y^c$.
\begin{enumerate}
\item
%(optional)
Show that every net has a universal subnet.
\item
Show that $(X,\tau)$ is compact if and only if every universal net converges.
\\
\emph{Note: You may use part (1) to prove part (2) even if you choose not to prove part (1).}
\end{enumerate}
\emph{Hint for (1):
Let $(x_\lambda)$ be a net in $X$.
Define a \emph{filter for} $(x_\lambda)$ to be a collection $\cF$ of non-empty subsets of $X$ such that:
\begin{itemize}
\item
$\cF$ is closed under finite intersections,
\item
If $F\in \cF$ and $F\subset G$, then $G\in \cF$, and
\item
$(x_\lambda)$ is frequently in every $F\in \cF$.
\end{itemize}
\begin{enumerate}
\item
Show that the set of filters for $(x_\lambda)$ is non-empty.
\item
Order the set of filters for $(x_\lambda)$ by inclusion.
Show that if $(\cF_j)$ is a totally ordered set of filters for $(x_\lambda)$, then $\cup \cF_j$ is also a filter for $(x_\lambda)$.
\item
Use Zorn's Lemma to assert there is a maximal filter $\cF$ for $(x_\lambda)$.
\item
Show that $\cF$ is an ultrafilter.
%Show that for every $Y\subset X$, the net $(x_\lambda)$ is either
%\begin{enumerate}
%\item
%frequently in $E\cap Y$ for every $E\in \cF$, or
%\item
%frequently in $F\setminus Y$ for every $F\in \cF$.
%\end{enumerate}
%\item
%In case (a) above, deduce that $\cF' := \set{F\in \cF}{E\cap Y \subset F \,\,\forall\, E\in \cF}$
%is a filter for $(x_\lambda)$ containing $\cF$, and thus $\cF=\cF'$.
%Deduce $Y\in \cF$.
%\item
%Do something analogous to the previous step for case (b) above to deduce $X\setminus Y \in \cF$.
\item
Find a subnet of $(x_\lambda)$ that is universal.
\end{enumerate}
}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Show the following collections of functions are uniformly dense in the appropriate algebras:
\begin{enumerate}
\item
For $a**0$, there is an $F\subset E$ such that $n < \nu(F) <\infty$.
\\
\emph{This is exactly Folland \S1.3, \#14 applied to $\nu$.}
\item
Show that if $\mu$ is semifinite, then $\mu = \nu$.
\item
Show there is a measure $\rho$ on $\cM$ (which is generally not unique) which assumes only the values $0$ and $\infty$ such that $\mu = \nu + \rho$.
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Suppose $(\mu_i^*)_{i \in I}$ is a family of outer measures on $X$.
Show that
$$\mu^*(E) := \sup_{i\in I} \mu_i^*(E)$$
is an outer measure on $X$.
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
\label{prob:h-invervals}
Define the \emph{h-intervals}
$$
\cH:=
\{\emptyset\}
\cup
\set{(-a, b]}{ -\infty \leq a < b < \infty}
\cup
\set{(a,\infty)}{a\in \bbR}.
$$
Let $\cA$ be the collection of finite disjoint unions of elements of $\cH$.
Show \emph{directly from the definitions} that $\cA$ is an algebra.
Deduce that the $\sigma$-algebra $\cM(\cA)$ generated by $\cA$ is equal to the Borel $\sigma$-algebra $\cB_\bbR$.
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Denote by $\overline{\bbR}$ the extended real numbers $[-\infty,\infty]$ with its usual topology.
Prove the following assertions.
\begin{enumerate}
\item
The
Borel $\sigma$-algebra on $\overline{\bbR}$ is
generated by the open rays $(a,\infty]$ for $a\in \bbR$.
\item
If $\cE\subset P(\bbR)$ generates the Borel $\sigma$-algebra on $\bbR$,
then $\cE \cup\{\{\infty\}\}$ generates the
Borel $\sigma$-algebra on $\overline{\bbR}$.
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}[Adapted from Folland \S1.4, \#18 and \#22]
\label{prob:OuterMeasureFromSigmaFinitePremeasure}
Suppose $\cA$ is an algebra on $X$, and let $\cM$ be the $\sigma$-algebra generated by $\cA$.
Let $\mu_0$ be a $\sigma$-finite premeasure on $\cA$, $\mu^*$ the induced outer measure, and $\cM^*$ the $\sigma$-algebra of $\mu^*$-measurable sets.
Show that the following are equivalent.
\begin{enumerate}
\item
$E\in \cM^*$
%\item
%There is an $F\in \cM$ with $E\subset F$ such that $\mu^*(F\setminus E)=0$.
%\item
%$E=F\cup G$ with $F\in \cM$ such that there exists $N\in \cM$ with $G\subset N$ and $\mu^*(N) = 0$.
\item
$E= F \setminus N$ where $F\in \cM$ and $\mu^*(N)=0$.
\item
$E=F\cup N$ where $F\in \cM$ and $\mu^*(N) = 0$.
\end{enumerate}
Deduce that if $\mu$ is a $\sigma$-finite measure on $\cM$, then $\mu^*|_{\cM^*}$ on $\cM^*$ is the completion of $\mu$ on $\cM$.
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}[Folland \S1.4, \#20]
Let $\mu^*$ be an outer measure on $P(X)$, $\cM^*$ the $\sigma$-algebra of $\mu^*$-measurable sets, and $\mu := \mu^*|_{\cM^*}$.
Let $\mu^+$ be the outer measure on $P(X)$ induced by the (pre)measure $\mu$ on the ($\sigma$-)algebra $\cM^*$.
\begin{enumerate}
\item
Show that $\mu^*(E) \leq \mu^+(E)$ for all $E\subset X$ with equality if and only if there is an $F\in \cM^*$ with $E\subset F$ and $\mu^*(E) = \mu^*(F)$.
\item
Show that if $\mu^*$ was induced from a premeasure $\mu_0$ on an algebra $\cA$, then $\mu^* = \mu^+$.
\item
Construct an outer measure $\mu^*$ on the two point set $X=\{0,1\}$ such that $\mu^* \neq \mu^+$.
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{prob}[Folland \S1.4, \#22a]
%Let $(X,\cM,\mu)$ be a measure space, $\mu^*$ the outer measure induced by $\mu$ on $\cM$, $\cM^*$ the $\sigma$-algebra of $\mu^*$-measurable sets, and $\overline{\mu}:= \mu^*|_{\cM^*}$.
%Show that if $\mu$ is $\sigma$-finite, then
%$\overline{\mu}$ on $\cM^*$ is the completion of $\mu$ on $\cM$.
%\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}[Sarason]
Suppose $\mu_0$ is a finite premeasure on the algebra $\cA\subset P(X)$, and let $\mu^*: P(X) \to [0,\infty]$ be the outer measure induced by $\mu_0$.
Prove that the following are equivalent for $E\subset X$.
\begin{enumerate}
\item
$E\in \cM^*$, the $\mu^*$-measurable sets.
\item
$\mu^*(E) + \mu^*(X\setminus E) = \mu(X)$.
\end{enumerate}
\emph{Hint: Use Problem \ref{prob:OuterMeasureFromSigmaFinitePremeasure}.}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Assume the notation of Problem \ref{prob:h-invervals}.
Suppose $F: \bbR \to \bbR$ is non-decreasing and right continuous, and extend $F$ to a function $[-\infty, \infty]\to [-\infty, \infty]$ still denoted $F$ by
$$
F(-\infty) := \lim_{a\to -\infty} F(a)
\qquad\text{and}\qquad
F(\infty) := \lim_{b\to \infty} F(b).
$$
Define $\mu_0 : \cH\to [0,\infty]$ by
\begin{itemize}
\item
$\mu_0(\emptyset) := 0$,
\item
$\mu_0((a,b]) := F(b) - F(a)$ for all $-\infty \leq a< b< \infty$, and
\item
$\mu_0((a,\infty)) := F(\infty) - F(a)$ for all $a\in \bbR$.
\end{itemize}
Suppose $(a,\infty) = \coprod_{j=1}^\infty H_j$ where $(H_j) \subset \cH$ is a sequence of disjoint h-intervals.
Show that
$$
\mu_0((a,\infty)) = \sum_{j=1}^\infty \mu_0(H_j).
$$
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}[Folland, \S1.5, \#28]
Let $F: \bbR \to \bbR$ be increasing and right continuous, and let $\mu_F$ be the associated Lebesgue-Stieltjes Borel measure on $\cB_\bbR$.
For $a\in \bbR$, define
$$
F(a-) :=
\lim_{r\nearrow a} F(r).
$$
Prove that:
\begin{enumerate}
\item
$\mu_F(\{a\}) = F(a) - F(a-)$,
\item
$\mu_F([a,b)) = F(b-) - F(a-)$,
\item
$\mu_F([a,b]) = F(b) - F(a-)$, and
\item
$\mu_F((a,b)) = F(b-) - F(a)$.
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Let $(X,\rho)$ be a metric (or simply a topological) space.
A subset $S \subset X$ is called \emph{nowhere dense} if $\overline{S}$ does not contain any open set in $X$.
A subset $T\subset X$ is called \emph{meager} if it is a countable union of nowhere dense sets.
Construct a meager subset of $\bbR$ whose complement is Lebesgue null.
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}[Steinhaus Theorem, Folland \S1.5, \#30 and 31]
Suppose $E\in \cL$ and $\lambda(E) >0$.
\begin{enumerate}
\item
Show that for any $0\leq \alpha <1$, there is an open interval $I\subset \bbR$ such that $\lambda(E\cap I) > \alpha \lambda(I)$.
\item
Apply (1) with $\alpha = 3/4$ to show that the set
$$
E-E = \set{x-y}{x,y\in E}
$$
contains the interval $(-\lambda(I)/2 , \lambda(I)/2)$.
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Let $\cB_\bbR$ be the Borel $\sigma$-algebra of $\bbR$.
Suppose $\mu$ is a translation invariant measure on $\cB_\bbR$ such that $\mu((0,1])=1$.
Prove that $\mu = \lambda|_{\cB_\bbR}$, the restriction of Lebesgue measure on $\cL$ to $\cB_\bbR$.
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}[Sarason]
Suppose $E\in \cL$ is Lebesgue null, and $\varphi: \bbR \to \bbR$ is a $C^1$ function (continuous with continuous derivative).
Prove that $\varphi(E)$ is also Lebesgue null.
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Find an uncountable subset of $\bbR$ with Hausdorff dimension zero.
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Integration}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Suppose $(X,\cM)$ is a measurable space and $(Y,\tau)$, $(Z, \theta)$ are topological spaces, $i: Y \to Z$ is a continuous injection which maps open sets to open sets, and $f: X\to Y$.
Show that $f$ is $\cM-\cB_\tau$ measurable if and only if $i\circ f$ is $\cM- \cB_\theta$ measurable.
Deduce that if $f: (X,\cM) \to \overline{\bbR}$ only takes values in $\bbR$,
then $f$ is $\cM - \cB_{\overline{\bbR}}$ measurable if and only if $f$ is $\cM - \cB_\bbR$ measurable.
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Prove the following assertions.
\begin{enumerate}
\item
Suppose $f: X\to Y$ is a function.
Define $\overset{\leftarrow}{f} : P(Y) \to P(X)$ by $\overset{\leftarrow}{f}(T) := \set{x\in X}{f(x) \in T}$.
Then $\overset{\leftarrow}{f}$ preserves unions, intersections, and complements.
\item
Suppose $f: X\to Y$ is a function.
Define $\overset{\rightarrow}{f} : P(X) \to P(Y)$ by $\overset{\rightarrow}{f}(S) := \set{f(s)}{s\in S}$.
Then $\overset{\rightarrow}{f}$ preserves unions, but not intersections nor complements.
\item
Given $f: X \to Y$ and a topology $\theta$ on $Y$, $\overset{\rightarrow}{\overset{\leftarrow}{f}}(\theta) = \set{f^{-1}(U)}{U\in \theta}$ is a topology on $X$.
Moreover it is the weakest topology on $X$ such that $f$ is continuous.
\item
Given $f: X \to Y$ and a topology $\tau$ on $X$, $\overset{\leftarrow}{\overset{\leftarrow}{f}}(\tau) = \set{U\subset Y}{f^{-1}(U)\in \tau}$ is a topology on $Y$.
Moreover it is the strongest topology on $Y$ such that $f$ is continuous.
\item
Given $f: X \to Y$ and a $\sigma$-algebra $\cN$ on $Y$, $\overset{\rightarrow}{\overset{\leftarrow}{f}}(\cN) = \set{f^{-1}(F)}{F\in \cN}$ is a $\sigma$-algebra on $X$.
Moreover it is the weakest $\sigma$-algebra on $X$ such that $f$ is measurable.
\item
Given $f: X \to Y$ and a $\sigma$-algebra $\cM$ on $X$, $\overset{\leftarrow}{\overset{\leftarrow}{f}}(\cM) = \set{F\subset Y}{f^{-1}(F)\in \cM}$ is a $\sigma$-algebra on $Y$.
Moreover it is the strongest $\sigma$-algebra on $Y$ such that $f$ is measurable.
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Let $(X,\cM)$ be a measurable space.
\begin{enumerate}
\item
Prove that the Borel $\sigma$-algebra $\cB_\bbC$ on $\bbC$ is generated by the `open rectangles'
$$
\set{z\in \bbC}{ a< \Re(z)< b \text{ and }c < \Im(z) 0$ for all $k=1,\dots, n$
is integrable if and only if $\mu(E_k) < \infty$ for all $k=1,\dots, n$.
\item
Show that if a simple function $\psi = \sum_{k=1}^n c_k \chi_{E_k}$ is integrable with $\mu(E_k)<\infty$ for all $k=1,\dots, n$, then
$\int \psi = \sum_{k=1}^n c_k \mu(E_k)$.
\end{enumerate}
\emph{In both parts of the question, we do not assume that $\psi$ is written in its standard form.}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Suppose $f: (X,\cM, \mu) \to [0,\infty]$ is $\cM$-measurable and $\{f>0\}$ is $\sigma$-finite.
Show that there exists a sequence of nonnegative simple functions $(\psi_n)$ such that
\begin{itemize}
\item
$\psi_n \nearrow f$,
\item
$\psi_n$ is integrable for every $n\in \bbN$.
\end{itemize}
\emph{Optional: In what sense can you say $\psi_n \nearrow f$ uniformly?}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Assume Fatou's Lemma and prove the Monotone Convergence Theorem from it.
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Let $(X,\cM,\mu)$ be a measure space.
\begin{enumerate}
\item
Suppose $f\in L^+$ and $\int f <\infty$.
Prove that $\{f=\infty\}$ is $\mu$-null and $\{f >0\}$ is $\sigma$-finite.
\item
Suppose $f\in L^1(\mu, \bbC)$.
Prove that $\{f\neq 0\}$ is $\sigma$-finite.
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Suppose $(X, \cM,\mu)$ is a measure space and $f\in L^1(\mu,\bbC)$.
Prove that for every $\varepsilon>0$, there exists a $\delta>0$ such that for every $E\in \cM$ with $\mu(E)<\delta$, $\int_E |f| < \varepsilon$.
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Let $(X,\cM,\mu)$ be a measure space.
\begin{enumerate}
\item
Prove that $\|\cdot\|_1 : \cL^1(\mu,\bbC) \to [0,\infty)$ given by $\|f\|_1 := \int |f|$ is a norm.
That is, prove the following axioms hold:
\begin{itemize}
\item
(definite)
$\| f\|_1 = 0$ if and only if $f=0$.
\item
(homogeneous)
$\|\lambda \cdot f\|_1 = |\lambda|\cdot \|f\|_1$ for all $\lambda \in \bbC$.
\item
(subadditive)
$\|f+g\|_1 \leq \|f\|_1 + \|g\|_1$.
\end{itemize}
\item
Suppose $(V, \|\cdot\|)$ is a $\bbC$-vector space with a norm (you may assume $V=\cL^1(\mu,\bbC)$ and $\|\cdot\|=\|\cdot\|_1$ if you wish).
Prove that $\rho(x,y) := \|x-y\|$ defines a metric on $V$.
\item
Prove that the metric $\rho_1$ on $\cL^1$ induced by $\|\cdot\|_1$ is complete.
That is, prove every Cauchy sequence converges in $\cL^1$.
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Suppose $(X,\cM,\mu)$ is a measure space, and let $(X,\overline{\cM}, \overline{\mu})$ be its completion.
Find a canonical $\bbC$-vector space isomorphism $\cL^1(\mu, \bbC) \cong \cL^1(\overline{\mu},\bbC)$ which preserves $\|\cdot\|_1$.
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Let $\mu$ be a Lebesgue-Stieltjes Borel measure on $\bbR$.
Show that $C_c(\bbR)$, the continuous functions of compact support ($\overline{\{f\neq 0\}}$ compact) is dense in $\cL^1(\mu,\bbR)$.
Does the same hold for $\overline{\bbR}$ and $\bbC$-valued functions?
\\
\emph{Hint:
You could proceed in this way:
\begin{enumerate}
\item
Reduce to the case $f\in L^1\cap L^+$.
\item
Reduce to the case $f\in L^1\cap SF^+$.
\item
Reduce to the case $f=\chi_E$ with $E\in \cB_\bbR$ and $\mu(E)<\infty$.
\item
Reduce to the case $f=\chi_U$ with $U\subset\bbR$ open and $\mu(U)<\infty$.
\item
Reduce to the case $f=\chi_{(a,b)}$ with $a****0$.
There is a compact set $E\subset [a,b]$ such that $\lambda(E^c)<\varepsilon$ and $f|_E$ is continuous.
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Suppose $f\in \cL^1([0,1], \lambda)$ is an integrable non-negative function.
\begin{enumerate}
\item
Show that for every $n\in \bbN$, $\sqrt[n]{f}\in \cL^1([0,1], \lambda)$.
\item
Show that $(\sqrt[n]{f})$ converges in $\cL^1$ and compute its limit.
\end{enumerate}
\emph{Hint for both parts: Consider $\{f\geq 1\}$ and $\{f<1\}$ separately.}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Suppose $(X,\cM,\mu)$ is a measure space and $f_n \to f$ in measure and $g_n \to g$ in measure
(these functions are assumed to be measurable).
Show that
\begin{enumerate}
\item
$|f_n|\to |f|$ in measure.
\item
$f_n+g_n \to f+g$ in measure.
\item
$f_n g_n \to fg$ if $\mu(X)<\infty$, but not necessarily if $\mu(X) = \infty$.
\\
\emph{Hint: First show $f_n g \to fg$ in measure.
To do so, one could follow the following steps.
\begin{enumerate}
\item
Show that for $g: X \to \bbC$ with $\mu(X)<\infty$, $\mu(\{|g|\geq n\}) \to 0$ as $n\to \infty$.
\item
Show that for any $\varepsilon>0$,
by step (a), $X=E\amalg E^c$ where $|g|_E|0$ and carefully chosen $M>0$ and $E$,
\begin{align*}
\{|f_ng-fg| > \delta\}
&=
(\{|f_ng-fg| > \delta\} \cap E)
\amalg
(\{|f_ng-fg| > \delta\} \cap E^c)
\\&\subseteq
\left\{|f_n-f| > \frac{\delta}{M}\right\}
\cup
E^c.
\end{align*}
\end{enumerate}
}
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}[Folland \S2.4, \#33 and 34]
Suppose $(X,\cM,\mu)$ is a measure space and $f_n \to f$ in measure (these functions are assumed to be measurable).
\begin{enumerate}
\item
Show that if $f_n\geq 0$ everywhere, then $\int f \leq \liminf \int f_n$.
\item
Suppose $|f_n|\leq g \in \cL^1$.
Prove that
$\int f = \lim \int f_n$ and $f_n \to f$ in $\cL^1$.
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Product measures and differentiation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
For the following statement, either provide a proof or a counterexample.
Let $X,Y$ be topological spaces with Borel $\sigma$-algebras $\cB_X,\cB_Y$ respectively and regular Borel measures $\mu, \nu$.
Then the product measure $\mu\times \nu$ is also regular.
\\
\emph{Optional: If you find a counterexample, can you find a weak modification under which it is true?}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Suppose $f: \bbR^2\to \bbR$ is such that each $x$-section $f_x$ is Borel measurable and $f^y$ is continuous.
Show $f$ is Borel measurable.
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Suppose $(X,\cM)$ and $(Y,\cN)$ are measurable spaces and $(E_n) \subset \cM\times \cN$.
Prove the following assertions about $x$-sections.
\begin{enumerate}
\item
$\left(\bigcup E_n\right)_x = \bigcup (E_n)_x$.
\item
$\left(\bigcap E_n\right)_x = \bigcap (E_n)_x$.
\item
$\left(E_m \setminus E_n\right)_x = (E_m)_x \setminus (E_n)_x$.
\item
$\chi_{E_n}(x,y) = \chi_{(E_n)_x}(y)$ for all $x\in X$ and $y\in Y$.
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}[Counterexamples: Folland \S2.5, \#46 and \#48]
\mbox{}
\begin{enumerate}
\item
Let $X=Y=[0,1]$, $\cM=\cN = \cB_{[0,1]}$, $\mu = \lambda$ Lebesgue measure, and $\nu$ counting measure.
Let $\Delta = \set{(x,x)}{x\in [0,1]}$ be the diagonal.
Prove that $\int\int \chi_\Delta \, d\mu \,d\nu$, $\int\int \chi_\Delta \, d\nu \,d\mu$, and $\int \chi_\Delta \,d(\mu\times\nu)$ are all unequal.
\item
Let $X=Y=\bbN$, $\cM = \cN = P(\bbN)$, and $\mu = \nu$ counting measure.
Define
$$
f(m,n) :=
\begin{cases}
1 & \text{if $m=n$}
\\
-1 &\text{if $m=n+1$}
\\
0 &\text{else.}
\end{cases}
$$
Prove that $\int |f| d(\mu\times \nu) = \infty$, and $\int\int f\,d\mu\,d\nu$ and $\int\int f\,d\nu\,d\mu$ both exist and are unequal.
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Show that the conclusions of the Fubini and Tonelli Theorems hold when $(X,\cM,\mu)$ is an arbitrary measure space (not necessarily $\sigma$-finite) and $Y$ is a countable set, $\cN = P(Y)$, and $\nu$ is counting measure.
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Suppose $f,g\in \cL^1(\bbR, \lambda)$.
\begin{enumerate}
\item
Show that $y \mapsto f(x-y)g(y)$ is measurable for all $x\in \bbR$ and in $\cL^1(\bbR, \lambda)$ for a.e.~$x\in \bbR$.
\item
Define the \emph{convolution} of $f$ and $g$ by
$$
(f * g)(x) := \int_\bbR f(x-y) g(y)\, d\lambda.
$$
Show that $f*g \in \cL^1(\bbR, \lambda)$.
\item
Show that $\cL^1(\bbR, \lambda)$ is a commutative $\bbC$-algebra under $\cdot, +, *$.
\item
Show that $\int_\bbR |f*g| \leq \int_\bbR |f| \int_\bbR |g|$, i.e., $\|\cdot\|_1$ is submultiplicative.
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Suppose $f\in \cL^1(\lambda^n)$.
Prove that for all $T\in GL_n(\bbR):=\set{T\in M_n(\bbR)}{\det(T)\neq 0}$, $f\circ T \in \cL^1(\lambda^n)$ and
$$
\int f(x) \, d\lambda^n(x) = |\det(T)| \cdot \int f(Tx) \,d\lambda^n(x).
$$
Does this also hold when $\det(T)=0$?
Find a proof or counterexample.
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}[Sarason]
For $f\in \cL^1(\lambda^n)$, let $M$ be the Hardy-Littlewood maximal function
$$
(Mf)(x) := \sup\set{\frac{1}{\lambda^n(Q)}\int_Q |f| \,d\lambda^n}{ Q\in \cC(x)}
$$
where $\cC(x)$ is the set of all cubes of finite length which contain $x$.
Define
$$f(x):=
\begin{cases}
\displaystyle
\frac{1}{|x|(\ln|x|)^2} &\text{if }|x|\leq \frac{1}{2}
\\
0 & \text{if }|x|>\frac{1}{2}
\end{cases}
$$
Show that $f\in \cL^1(\lambda)$, but $Mf\notin \cL^1_{\rm loc}$.
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}[Sarason]
Suppose $E\subset \bbR^n$ (not assumed to be Borel measurable) and let $\cC$ be a family of cubes covering $E$
such that
$$
\sup\set{\ell(Q)}{ Q\in \cC} < \infty.
$$
Show there exists a sequence $(Q_k)\subset \cC$ of disjoint cubes such that
$$
\sum_{k=1}^\infty \lambda^n(Q_k)
\geq 5^{-n} (\lambda_n)^*(E).
$$
\emph{Hint: Inductively choose $Q_k$ such that $2\ell(Q_k)$ is larger than the sup of the lengths of all cubes which do not intersect $Q_1,\dots, Q_{k-1}$, with $Q_0 = \emptyset$ by convention.}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
In this exercise, we will show that
$$
M:=M(X,\cM,\bbR):=\{\text{finite signed measures on $(X,\cM)$}\}
$$
is a Banach space with $\|\nu\|:=|\nu|(X)$.
\begin{enumerate}
\item
Prove $\|\nu\|:=|\nu|(X)$ is a norm on $M$.
\item
Show that $(\nu_n)\subset M$ Cauchy implies $(\nu_n(E))\subset \bbR$ is uniformly Cauchy for all $E\in \cM$.
That is, show that for all $\varepsilon>0$, there is an $N\in \bbN$ such that for all $n\geq N$ and $E\in \cM$, $|\nu_m(E)-\nu_n(E)|<\varepsilon$.
\item
Use part (2) to define a candidate limit signed measure $\mu$ on $\cM$.
Prove that $\nu$ is $\sigma$-additive.
\\
\emph{Hint: first prove $\nu$ is finitely additive.}
\item
Prove that $\sum \nu(E_n)$ converges absolutely when $(E_n)\subset \cM$ is disjoint, and thus $\nu$ is a finite signed measure.
\item
Show that $\nu_n \to \nu$ in $M$.
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}[Folland \S3.1, \#3 and \S3.2, \#8]
Suppose $\mu$ is a positive measure on $(X,\cM)$ and $\nu$ is a signed measure on $(X,\cM)$.
\begin{enumerate}[(1)]
\item
Prove that the following are equivalent.
\begin{enumerate}[(a)]
\item
$\nu \perp \mu$
\item
$|\nu| \perp \mu$
\item
$\nu_+ \perp \mu$ and $\nu_- \perp \mu$.
\end{enumerate}
\item
Prove that the following are equivalent.
\begin{enumerate}[(a)]
\item
$\nu \ll \mu$
\item
$|\nu|\ll\mu$
\item
$\nu_+ \ll \mu$ and $\nu_- \ll \mu$.
\end{enumerate}
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}[Folland \S3.1, \#3]
Let $\nu$ be a signed measure on $(X,\cM)$.
Prove the following assertions:
\begin{enumerate}[(a)]
\item
$\cL^1(\nu) = \cL^1(|\nu|)$.
\item
If $f\in \cL^1(\nu)$, $\left|\int f \, d\nu \right| \leq \int |f| d |\nu|$.
\item
If $E\in \cM$, $|\nu|(E) = \sup\set{\left|\int_E f \, d\nu\right|}{-1\leq f \leq 1}$.
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}[Folland \S3.1, \#6]
Suppose
$$
\nu(E) := \int_E f \, d\mu
\qquad\qquad
E\in \cM
$$
where $\mu$ is a positive measure on $(X,\cM)$ and and $f$ is an extended $\mu$-integrable function.
Describe the Hahn decompositions of $\nu$ and the positive, negative, and total variations of $\nu$ in terms of $f$ and $\mu$.
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}[Adapted from Folland \S3.2, \#9]
Suppose $\mu$ is a positive measure on $(X,\cM)$.
Suppose $\{\nu_j\}$ is a sequence of positive measures on $(X,\cM)$ and $\mu$ is a positive measure on $(X,\cM)$.
Prove the following assertions.
\begin{enumerate}[(a)]
\item
If $\{\nu_j\}$ is a sequence of positive measures on $(X,\cM)$ with $\nu_j \perp \mu$ for all $j$, then $\sum_{j=1}^\infty \nu_j \perp \mu$.
\item
If $\nu_1,\nu_2$ are positive measures on $(X,\cM)$ with at least one of $\nu_1,\nu_2$ is finite and $\nu_j\perp \mu$ for $j=1,2$, then $(\nu_1-\nu_2)\perp \mu$.
\item
If $\{\nu_j\}$ is a sequence of positive measures on $(X,\cM)$ with $\nu_j \ll \mu$ for all $j$, then $\sum_{j=1}^\infty \nu_j \ll \mu$.
\item
If $\nu_1,\nu_2$ are positive measures on $(X,\cM)$ with at least one of $\nu_1,\nu_2$ is finite and $\nu_j \ll \mu$ for $j=1,2$, then $(\nu_1-\nu_2) \ll\mu$.
\end{enumerate}
\end{prob}
\begin{prob}
Suppose $F: [a,b]\to \bbC$.
\begin{enumerate}
\item
Show that if $F$ is continuous on $[a,b]$, differentiable on $(a,b)$, and $F'$ is bounded, then $F\in \BV[a,b]$.
\item
Show that if $F$ is absolutely continuous, then $F\in \BV[a,b]$.
\end{enumerate}
\end{prob}
\begin{prob}
Suppose $F\in \NBV$, and let $\nu_F$ be the corresponding Lebesgue-Stieltjes complex Borel measure.
\begin{enumerate}
\item
Prove that $\nu_F$ is regular.
\item
Prove that $|\nu_F|=\nu_{T_F}$.
\\
\emph{One could proceed as follows.
\begin{enumerate}
\item
Define $G(x):= |\nu_F|((-\infty,x])$.
Show that $|\nu_F| = \nu_{T_F}$ if and only if $G=T_F$.
\item
Show $T_F\leq G$.
\item
Show that $|\nu_F(E)| \leq \nu_{T_F}(E)$ whenever $E$ is an interval.
\item
Show that $|\nu_F|\leq \nu_{T_F}$.
\end{enumerate}
}
\end{enumerate}
\end{prob}
\begin{prob}[cf.~Folland Thm.~3.22]
Denote by $\lambda^n$ Lebesgue measure on $\bbR^n$.
Suppose $\nu$ is a regular signed or complex Borel measure on $\bbR^n$ which is finite on compact sets (and thus Radon and $\sigma$-finite).
Let $d\nu = d\rho + fd\lambda^n$ be its Lebesgue-Radon-Nikodym representation.
Then for $\lambda^n$-a.e.~$x\in \bbR^n$,
$$
\lim_{\substack{\ell(Q)\to 0 \\ Q\in \cC(x)}} \frac{\nu(Q)}{\lambda^n(Q)} = f(x).
$$
\\
\emph{Hint:
One could proceed as follows.
\begin{enumerate}
\item
Show that $d|\nu| = d|\rho|+|f|d\lambda^n$.
Deduce that $\rho$ and $fd\lambda^n$ are regular, and $f\in L^1_{\loc}$.
\item
Use the Lebesgue Differentiation Theorem to reduce the problem to showing
$$
\lim_{\substack{\ell(Q)\to 0 \\ Q\in \cC(x)}} \frac{|\rho|(Q)}{\lambda^n(Q)} = 0
\qquad\qquad
\text{$\lambda^n$-a.e.~$x\in \bbR^n$.}
$$
Thus we may assume $\rho$ is positive.
\item
Since $\rho\perp \lambda^n$, pick $P\subset \bbR^n$ Borel measurable such that $\rho(P)= \lambda^n(P^c)=0$.
For $a>0$, define
$$
E_a
:=
\set{x\in P}{\lim_{\substack{\ell(Q)\to 0 \\ Q\in \cC(x)}} \frac{|\rho|(Q)}{\lambda^n(Q)} >a}.
$$
Let $\varepsilon>0$.
Since $\rho$ is regular, there is an open $U_\varepsilon \supset P$ such that $\rho(U_\varepsilon)<\varepsilon$.
Adapt the proof of the HLMT to show there is a constant $c>0$, depending only on $n$, such that
for all $a>0$,
$$
\lambda^n(E_a)
\leq
c \cdot \frac{\rho(U_\varepsilon)}{a}
=
c \cdot \frac{\varepsilon}{a}
$$
(Choose your family of cubes to be contained in $U_\varepsilon$.)
Deduce that $\lambda^n(E_a)=0$.
\end{enumerate}
}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Let $F: \bbR \to \bbR$ be a bounded, non-decreasing continuously differentiable function, and let $\mu_F$ be the corresponding Lebesgue-Stieltjes measure on $\bbR$.
\begin{enumerate}
\item
Denoting Lebesgue measure by $\lambda$, prove that
$$
\mu_F(E) = \int_E F'\, d\lambda
\qquad\qquad
\forall E \in \cB_\bbR.
$$
\\\emph{Hint: First prove the above equality for intervals. Then use Problem \ref{prob:pi lambda part 3}.}
\item
Deduce that $\mu_F \ll \lambda$ and $\frac{d\mu_F}{d\lambda} = F'$ a.e.
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Functional analysis}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
\label{prob:QuotientBanach}
Suppose $X$ is a normed space and $Y\subset X$ is a subspace.
Define $Q: X \to X/Y$ by $Q x = x+Y$.
Define
$$
\|Q x \|_{X/Y} = \inf\set{\|x - y\|_X }{y\in Y}.
$$
\begin{enumerate}
\item
Prove that $\|\cdot \|_{X/Y}$ is a well-defined seminorm.
\item
Show that if $Y$ is closed, then $\|\cdot \|_{X/Y}$ is a norm.
\item
Show that in the case of (2) above, $Q: X \to X/Y$ is continuous and open.
\\
\emph{Optional: is $Q$ continuous or open only in the case of (1)?}
\item
Show that if $X$ is Banach, so is $X/Y$.
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Suppose $F$ is a finite dimensional vector space.
\mbox{}
\begin{enumerate}
\item
Show that for any two norms $\|\cdot\|_1$ and $\|\cdot\|_2$ on $F$, there is a $c>0$ such that $\|f\|_1 \leq c\|f\|_2$ for all $f\in F$.
Deduce that all norms on $F$ induce the same vector space topology on $F$.
\\
\emph{Note: You need only prove the result for one of $\bbR$ or $\bbC$.
You may use that the unit sphere in $\bbK^n$ is compact with respect to the usual Euclidean topology.}
\item
Show that for any two finite dimensional normed spaces $F_1$ and $F_2$, all linear maps $T: F_1 \to F_2$ are continuous.
\\
\emph{Optional:
Show that for any two finite dimensional vector spaces $F_1$ and $F_2$ endowed with their vector space topologies from part (1), all linear maps $T: F_1 \to F_2$ are continuous.
}
\item
Let $X,F$ be normed spaces with $F$ finite dimensional, and let $T: X \to F$ be a linear map.
Prove that the following are equivalent:
\begin{enumerate}[(a)]
\item
$T$ is bounded
(there is an $R>0$ such that $T(B_1(0_X))\subset B_R(0_F)$), and
\item
$\ker(T)$ is closed.
\end{enumerate}
\emph{Hint: One way to do (b) implies (a) uses Problem \ref{prob:QuotientBanach} part (3) and part (2) of this problem.}
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}[Folland \S5.1, \#7]
Suppose $X$ is a Banach space and $T\in \cL(X) = \cL(X,X)$.
Let $I \in \cL(X)$ be the identity map.
\begin{enumerate}
\item
Show that if $\|I-T\|<1$, then $T$ is invertible.
\\
\emph{Hint: Show that $\sum_{n\geq 0} (I-T)^n$ converges in $\cL(X)$ to $T^{-1}$.}
\item
Show that if $T\in \cL(X)$ is invertible and $\|S-T\| < \|T^{-1}\|^{-1}$, then $S$ is invertible.
\item
Deduce that the set of invertible operators $GL(X)\subset \cL(X)$ is open.
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}[Folland \S5.2, \#19]
Let $X$ be an infinite dimensional normed space.
\begin{enumerate}
\item
Construct a sequence $(x_n)$ such that $\|x_n\|=1$ for all $n$ and $\|x_m-x_n\|\geq 1/2$ for all $m\neq n$.
\item
Deduce $X$ is not locally compact.
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}[]
\label{prob:IntersectionOfKernels}
Suppose $\varphi,\varphi_1,\dots, \varphi_n$ are linear functionals on a vector space $X$.
Prove that the following are equivalent.
\begin{enumerate}
\item
$\varphi \in \sum_{k=1}^n \alpha_k \varphi_k$ where $\alpha_1,\dots, \alpha_n \in \mathbb{F}$.
\item
There is an $\alpha>0$ such that for all $x\in X$, $|\varphi(x)|\leq \alpha \max_{k=1,\dots,n}|\varphi_k(x)|$.
\item
$\bigcap_{k=1}^n \ker(\varphi_k) \subset \ker(\varphi)$.
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Consider the following sequence spaces.
\begin{align*}
\ell^1 &:= \set{(x_n)\subset \bbC^\infty}{ \sum |x_n| <\infty}
&&
\|x\|_1:= \sum |x_n|
\\
c_0 &:= \set{(x_n)\subset \bbC^\infty}{ x_n \to 0\text{ as }n\to\infty}
&&
\|x\|_\infty:= \sup |x_n|
\\
c &:= \set{(x_n)\subset \bbC^\infty}{ \lim_{n\to \infty}x_n\text{ exists}}
&&
\|x\|_\infty:= \sup |x_n|
%\\
%\ell^2 &:= \set{(x_n)\subset \bbC^\infty}{ \sum |x_n|^2 <\infty}
%&&
%\|x\|_2:= \left(\sum |x_n|^2\right)^{1/2}
\\
\ell^\infty &:= \set{(x_n)\subset \bbC^\infty}{ \sup |x_n| <\infty}
&&
\|x\|_\infty:= \sup |x_n|
\end{align*}
\begin{enumerate}
\item
Show that every space above is a Banach space.
\\
\emph{Hint: First show $\ell^1$ and $\ell^\infty$ are Banach.
Then show $c_0, c$ are closed in $\ell^\infty$.}
\item
Construct isometric isomorphisms $c_0^* \cong \ell^1 \cong c^*$
%, $(\ell^2)^* \cong \ell^2$,
and $(\ell^1)^*\cong \ell^\infty$.
\item
Which of the above spaces are separable?
\item (Folland \S5.2, \#25)
Prove that if $X$ is a Banach space such that $X^*$ is separable, then $X$ is separable.
\item
Find a separable Banach space $X$ such that $X^*$ is not separable.
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}[Folland \S5.3, \#42]
Let $E_n \subset C([0,1])$ be the space of all functions $f$ such that there is an $x_0\in [0,1]$ such that $|f(x)-f(x_0)| < n |x-x_0|$ for all $x\in [0,1]$.
\begin{enumerate}
\item
Prove that $E_n$ is nowhere dense in $C([0,1])$.
\item
Show that the subset of nowhere differentiable functions is residual in $C([0,1])$.
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Provide examples of the following:
\begin{enumerate}
\item
Normed spaces $X,Y$ and a discontinuous linear map $T: X\to Y$ with closed graph.
%\\
%\emph{This shows that $X$ Banach is a necessary assumption for the Closed Graph Theorem.}
\item
Normed spaces $X,Y$ and a family of linear operators $\{T_\lambda\}_{\lambda \in \Lambda}$ such that $(T_\lambda x)_{\lambda \in \Lambda}$ is bounded for every $x\in X$, but $(\|T_\lambda\|)_{\lambda \in \Lambda}$ is not bounded.
%\\
%\emph{This shows that $X$ Banach is a necessary assumption for the Uniform Boundedness Principle.}
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Suppose $X$ and $Y$ are Banach spaces and $T: X \to Y$ is a continuous linear map.
Show that the following are equivalent.
\begin{enumerate}[(a)]
\item
There exists a constant $c > 0$ such that $\|Tx\|_Y\geq c \|x\|_X$ for all $x\in X$.
\item
$T$ is injective and has closed range.
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}[]
Let $X$ be a normed space.
\begin{enumerate}
\item
Show that every weakly convergent sequence in $X$ is norm bounded.
\item
Suppose in addition that $X$ is Banach.
Show that every weak* convergent sequence in $X^*$ is norm bounded.
\item
Give a counterexample to (2) when $X$ is not Banach.
\\
\emph{Hint: Under $\|\cdot\|_\infty$, $c_{c}^*\cong \ell^1$, where $c_{c}$ is the space of sequences which are eventually zero.}
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{prob}
%Then deduce that for a normed space $X$, any linear functional $\varphi : X\to \bbC$ is continuous with respect to the weak topology if and only if $\varphi \in X^*$.
%\end{prob}
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\begin{prob}[Goldstine's Theorem]
Let $X$ be a normed vector space with closed unit ball $B$.
Let $B^{**}$ be the unit ball in $X^{**}$, and let $i: X \to X^{**}$ be the canonical inclusion.
Show that $i(B)$ is weak* dense in $B^{**}$.
\\
\emph{Note: recall that the weak* topology on $X^{**}$ is the weak topology induced by $X^*$.}
\\
\emph{Hint: You could use a Hahn-Banach separation theorem that we did not discuss in class.
Or you could proceed as follows.
\begin{enumerate}
\item
Show that for every $z\in B^{**}$, $\varphi_1,\dots, \varphi_n\in X^*$, and $\delta>0$, there is an $x\in (1+\delta)B$ such that
$\varphi_i(x)=z(\varphi_i)$ for all $1\leq i\leq n$.
\item
Suppose $U$ is a basic open neighborhood of $z\in B^{**}$.
Deduce that for every $\delta>0$, $(1+\delta)i(B) \cap U \neq \emptyset$.
That is, there is an $x_\delta \in (1+\delta)B$ such that $i(x_\delta) \in U$.
\item
By part (2), $(1+\delta)^{-1}x_\delta \in B$.
Show that for $\delta$ sufficiently small (which can be expressed in terms of the basic open neighborhood $U$), $(1+\delta)^{-1}i(x_\delta) \in i(B) \cap U$.
\end{enumerate}
}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}[Banach Limits]
Let $\ell^\infty(\bbN,\bbR)$ denote the Banach space of bounded functions $\bbN \to \bbR$.
Show that there is a $\varphi \in \ell^\infty(\bbN, \bbR)^*$ satisfying the following two conditions:
\begin{enumerate}
\item
Letting $S: \ell^\infty(\bbN,\bbR) \to \ell^\infty(\bbN,\bbR)$ be the shift operator $(Sx)_n = x_{n+1}$ for $x=(x_n)_{n\in\bbN}$, $\varphi = \varphi \circ S$.
\item
For all $x\in \ell^\infty$, $\liminf x_n \leq \varphi(x) \leq \limsup x_n$.
\end{enumerate}
\emph{Hint: One could proceed as follows.
\begin{enumerate}
\item
Consider the subspace $Y = \operatorname{im}(S-I) = \set{Sx-x}{x\in \ell^\infty}$.
Prove that for all $y\in Y$ and $r\in \bbR$,
$\|y+r\cdot \mathbf{1}\| \geq |r|$,
where $\mathbf{1} = (1)_{n\in \bbN}\in \ell^\infty$.
\item
Show that the linear map $f : Y\oplus \bbR\mathbf{1} \to \bbR$ given by $f(y+r\cdot \mathbf{1}):= r$ is well-defined, and $|f(z)|\leq \|z\|$ for all $z\in Y\oplus \bbR\mathbf{1} $.
\item
Use the Real Hahn-Banach Theorem to extend $f$ to a $g\in \ell^\infty(\bbN,\bbR)^*$ which satisfies the desired properties.
\end{enumerate}
}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Let $X$ be a compact Hausdorff topological space.
For $x\in X$, define $\operatorname{ev}_x : C(X) \to \bbF$ by $\operatorname{ev}_x(f) = f(x)$.
\begin{enumerate}
\item
Prove that $\ev_x \in C(X)^*$, and find $\|\ev_x\|$.
\item
Show that the map $\operatorname{ev}:X\to C(X)^*$ given by $x\mapsto \operatorname{ev}_x$ is a homeomorphism onto its image, where the image has the relative weak* topology.
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
\label{prob:Weak-Weak-Continuous}
Suppose $X,Y$ are Banach spaces and $T:X \to Y$ is a linear transformation.
\begin{enumerate}
\item
Show that if $T\in \cL(X,Y)$, then $T$ is weak-weak continuous.
That is, if $x_\lambda \to x$ in the weak topology on $X$ induced by $X^*$, then $Tx_\lambda \to Tx$ in the weak topology on $Y$ induced by $Y^*$.
\item
Show that if $T$ is norm-weak continuous, then $T\in \cL(X,Y)$.
\item
Show that if $T$ is weak-norm continuous, then $T$ has finite rank, i.e., $TX$ is finite dimensional.
\end{enumerate}
\emph{Hint:
For part (3), one could proceed as follows.
\begin{enumerate}[(a)]
\item
First, reduce to the case that $T$ is injective by replacing $X$ with $Z=X/\ker(T)$ and $T$ with $S: Z\to Y$ given by $x+\ker(T) \mapsto Tx$.
(You must show $S$ is weak-norm continuous on $Z$.)
\item
Take a basic open set $\cU = \set{z\in Z}{|\varphi_i(z)|<\varepsilon \text{ for all }i=1,\dots, n} \subset S^{-1}B_1(0_Y)$.
Use that $S$ is injective to prove that $\bigcap_{i=1}^n \ker(\varphi_i) = (0)$.
\item
Use Problem \ref{prob:IntersectionOfKernels} to deduce that $Z^*$ is finite dimensional, and thus that $Z$ and $TX=SZ$ are finite dimensional.
\end{enumerate}
}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
\label{prob:SeparableSpace}
Suppose $X$ is a Banach space.
Prove the following are equivalent:
\begin{enumerate}
\item
$X$ is separable.
\item
The relative weak* topology on the closed unit ball of $X^*$ is metrizable.
\end{enumerate}
Deduce that the closed unit ball of $X^*$ is weak* sequentially compact.
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
\label{prob:SeparableDualSpace}
Suppose $X$ is a Banach space.
Prove the following are equivalent:
\begin{enumerate}
\item
$X^*$ is separable.
\item
The relative weak topology on the closed unit ball of $X$ is metrizable.
\end{enumerate}
Prove that in this case, $X$ is also separable.
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}[Eberlein-Smulian]
\label{prob:ReflexiveSpace}
Suppose $X$ is a Banach space.
Prove the following are equivalent:
\begin{enumerate}
\item
$X$ is reflexive.
\item
The closed unit ball of $X$ is weakly compact.
\item
The closed unit ball of $X$ is weakly sequentially compact.
\end{enumerate}
\emph{\underline{Optional:}
How do you reconcile Problems \ref{prob:SeparableSpace}, \ref{prob:SeparableDualSpace}, and \ref{prob:ReflexiveSpace}?
That is, how do you reconcile the fact that there exist separable Banach spaces which are not reflexive?
}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Consider the space $L^2(\bbT):=L^2(\bbR/\bbZ)$ of $\bbZ$-periodic functions $\bbR\to \bbC$ such that $\int_{[0,1]} |f|^2 <\infty$.
Define
$$
\langle f, g\rangle := \int_{[0,1]} f\overline{g}.
$$
\begin{enumerate}
\item
Prove that $L^2(\bbT)$ is a Hilbert space.
\item
Show that the subspace $C(\mathbb{T})\subset L^2(\mathbb{T})$ of continuous $\bbZ$-periodic functions is dense.
\item
Prove that $\left\{e_n(x):=\exp(2\pi i n x)\middle| n\in\mathbb{Z}\right\}$ is an orthonormal basis for $L^2(\mathbb{T})$.
\\
\emph{Hint: Orthonormality is easy. Use (2) and the Stone-Weierstrass Theorem to show the linear span is dense.}
\item
Define $\mathcal{F}: L^2(\mathbb{T}) \to \ell^2(\mathbb{Z})$ by $\mathcal{F}(f) _n := \langle f,e_n\rangle_{L^2(\mathbb{T})}= \int_{0}^{1} f(x) \exp(-2\pi i n x)\, dx$.
Show that if $f\in L^2(\mathbb{T})$ and $\mathcal{F}(f)\in \ell^1(\mathbb{Z})$, then $f\in C(\mathbb{T})$, i.e., $f$ is a.e.~equal to a continuous function.
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Suppose $H$ is a Hilbert space, $E\subset H$ is an orthonormal set, and $\{e_1,\dots, e_n\}\subset E$.
Prove the following assertions.
\begin{enumerate}
\item
If $x=\sum_{i=1}^n c_i e_i$, then $c_i = \langle x, e_i\rangle$.
\item
The set $E$ is linearly independent.
\item
For every $x\in H$, $\sum_{i=1}^n \langle x, e_i\rangle e_i$ is the unique element of $\operatorname{span}\{e_1,\dots, e_n\}$ minimizing the distance to $x$.
\item
(Bessel's Inequality)
For every $x\in H$, $\|x\|^2 \geq \sum_{i=1}^n |\langle x, e_i\rangle|^2$.
\item
If $H$ is separable, then $E$ is countable.
\item
The set $E$ can be extended to an orthonormal basis for $H$.
\item
If $E$ is an orthonormal basis, then the map $H \to \ell^2(E)$ given by $x\mapsto (\langle x, \cdot\rangle : E \to \bbC)$ is a unitary isomorphism of Hilbert spaces.
\end{enumerate}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Radon measures}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Let $X$ be a locally compact Hausdorff space and suppose $\varphi : C_0(X) \to \bbC$ is a linear functional such that $\varphi(f) \geq 0$ whenever $f\geq 0$.
Prove that $\varphi$ is bounded.
\\
\emph{Hint:
Prove that $\set{\varphi(f)}{0\leq f\leq 1}$ is bounded.}
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Suppose $X$ is an LCH space, $K\subset X$ is compact, and $U_1,\dots, U_n$ are open sets such that $K\subset \bigcup_{i=1}^n U_i$.
Show there exist $g_1,\dots, g_n \in C_c(X)$ such that $g_i \prec U_i$ for all $i$ and $\sum_{i=1}^n g_i = 1$ on $K$.
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Suppose $X$ is an LCH space, $\mu$ is a $\sigma$-finite Radon measure on $X$, and $E$ is a Borel set.
Prove that for every $\varepsilon>0$, there is an open set $U$ and a closed set $F$ with $F \subset E \subset U$ such that $\mu(U\setminus F)<\varepsilon$.
\end{prob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prob}
Suppose $X$ is an LCH space and $\varphi \in C_0(X)^*$.
Prove there are finite Radon measures $\mu_0,\mu_1, \mu_2, \mu_3$ on $X$ such that
$$
\varphi(f)
=
\sum_{k=1}^3 i^k \int f\, d\mu_k
\qquad\qquad
\forall\, f\in C_0(X).
$$
\end{prob}
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\end{document}**