\documentclass[letterpaper]{exam}
\usepackage{calculus,bm}

\iftrue % Two columns.
  \usepackage[landscape,includehead,headheight=.25cm,headsep=.05cm,left=1cm,right=1cm,top=.7cm,bottom=1cm,]{geometry}
  \twocolumn
\else
  % Put your parameters here for 1up printing, and change \iftrue above to
  % \iffalse
\fi

\noaddpoints
\newcommand{\secref}[1]{\textbf{Sec.\ #1.\xspace}}

\begin{document}
\title{Math 720: Homework.}
\thispagestyle{plain}
\maketitle
Do, but don't turn in optional problems.
\textit{There is a firm `no late homework' policy.}

\begin{assignment}{Wed 08/28}{Wed 09/04}%{{{HW1
  \uplevel{\note{%
    Keep in mind there is a firm ``no late homework'' policy.
    Starred problems are optional; but I'd recommend looking at them. They often involve results I will use later in class.}}

  \item
    Let $\mu$ be a positive measure on $(X, \Sigma)$.
    \begin{parts}
      \item
	If $A_i \in \Sigma$ are such that $A_i \subseteq A_{i+1}$, show that $\mu(\bigcup_{i=1}^\infty A_i) = \lim_{i \to \infty} \mu(A_i)$.
      \item
	If $A_i \in \Sigma$ are such that $A_i \supseteq A_{i+1}$, show that $\mu(\bigcap_{i=1}^\infty A_i) = \lim_{i \to \infty} \mu(A_i)$, provided $\mu(A_1) < \infty$. Show by example this is false true if $\mu(A_1) = \infty$.
    \end{parts}

  \item
    Prove any open subset of $\R^d$ is a countable union of cells.

  \item
    For each of the following sets, compute the Lebesgue outer measure.
    \begin{mcparts}[3]
      \item Any countable set.
      \item The Cantor set.
      \item $\{x \in [0, 1] \st x \not \in \Q \}$.
    \end{mcparts}

  \item
    \begin{parts}
      \item If $V \subset \R^d$ is a subspace with $\dim(V) < d$, then show that $\lambda^*(V) = 0$.
      \item If $P \subset \R^2$ is a polygon show that $\operatorname{area}(P) = \lambda^*(P)$.
    \end{parts}

  \item
    Does there exist a $\sigma$-algebra whose cardinality is countably infinite?
    Disprove, or find an example.


  \fullwidth{\noindent\emph{Optional problems, and details in class I left for you to check.}}

  \itemopt
    Define $\mu(A)$ to be the number of elements in $A$.
    Show that $\mu$ is a measure on $(X, \mc P(X))$.
    (This is called the counting measure.)

  \itemopt
    Let $x_0 \in X$ be fixed.
    Define $\delta_{x_0}(A) = 1$ if $x_0 \in A$ and $0$ otherwise.
    Show that $\delta_{x_0}$ is a measure on $(X, \mc P(X))$.
    (This is called the delta measure at $x_0$.)
    
  \itemopt Show that $\lambda^*(a + E)  = \lambda^*(E)$ for all $a \in \R^d$, $E \subset \R^d$.

  \itemopt
    Show that $\lambda^*(I) = \ell(I)$ for all cells. (I only proved it for closed cells in class.)

  \itemopt
    Show that $\mc B(\R)$ has the same cardinality as $\R$.


  \itemopt
    \emph{(Challenge)}
    Suppose $f_n:[0,1] \to [0, 1]$ are all Riemann integrable, $0 \leq f_n \leq 1$ and $(f_n) \to 0$ pointwise.
    Show that $\dlim_{n \to \infty} \int_0^1 f_n = 0$, using only standard tools from Riemann integration.



\end{assignment}%}}}
\newpage
\begin{assignment}{Wed 09/04}{Wed 09/11}%{{{HW2
  \item
    \begin{parts}
      \item Say $\mu$ is a \emph{translation invariant} measure on $(\R^d, \mathcal L)$ (i.e. $\mu(x + A) = \mu(A)$ for all $A \in \mc L$, $x \in \R^d$) which is finite on bounded sets.
	Show that $\exists c \geq 0$ such that $\mu(A) = c \lambda(A)$.
      \item Let $T:\R^d \to \R^d$ be a linear transformation, and $A \in \mc L$.
	Show that $T(A) \in \mc L$ and $\lambda(T(A)) = \abs{\det(T)} \lambda(A)$.
	\hintb{Express $T$ in terms of elementary transformations.}
    \end{parts}

  \item
    \begin{parts}
      \item
	Let $\mathcal E \subset \mathcal P(X)$, and $\rho: \mathcal E \to [0, \infty]$ be such that $\emptyset \in \mathcal E$, $X \in \mathcal E$ and $\rho(\emptyset) = 0$.
	For any $A \subset X$ define
	$$
	  \mu^*(A) = \inf\Bigl\{ \sum_1^\infty \rho(E_i) \mathop\Big| E_i \in \mathcal E, \text{ and } A \subset \bigcup_{1}^\infty E_j \Bigr\}.
	$$
	Show that $\mu^*$ is an outer measure.

      \item
	Let $(X, d)$ be any metric space, $\delta > 0$, $\alpha \geq 0$ and define
	$$
	  \mc E_\delta = \{A \subset X \st \diam(A) < \delta \}
	  \quad\text{and}\quad
	  \rho_\alpha (A) = \frac{ \pi^{\alpha/2} }{ \Gamma( 1 + \frac{\alpha}{2} ) } \paren[\Big]{ \frac{\diam(A)}{2} }^\alpha.
	$$
	Let $H_{\alpha,\delta}^*$ be the outer measure obtained with $\rho = \rho_\alpha$ and the collection of sets $\mc E_\delta$.
	Define $H^*_\alpha = \lim_{\delta \to 0} H^*_{\alpha, \delta}$.
	Show $H^*_\alpha$ is an outer measure and restricts to a measure $H_\alpha$ on a $\sigma$-algebra that contains all Borel sets.
	The measure $H_\alpha$ is called the \emph{Hausdorff measure of dimension $\alpha$.}

      \item
	If $X = \R^d$, and $\alpha = d$ show that $H_d$ is a non-zero, finite constant multiple of the Lebesgue measure. \noteb{In fact $H_d = \lambda$ because of our choice of normalization constant, but the proof is much harder.}
	%If $X = \R^d$, and $\alpha = d$ show that $H_d$ is the Lebesgue measure.

      \item
	Let $S \in \mc B(X)$.
	Show that there exists (a unique) $d \in [0, \infty]$ such that $H_\alpha(S) = \infty$ for all $\alpha \in (0, d)$, and $H_\alpha(S) = 0$ for all $\alpha \in (d, \infty)$. 
	This number is called the \emph{Hausdorff dimension} of the set $S$.

      \item Compute the Hausdorff dimension of the Cantor set.
    \end{parts}

  \item
    Using notation from the previous question, let $\mc S_\delta = \{ B(x, r) \st x \in X, r \in (0, \delta) \}$.
    Using the collection of sets $\mc S_\delta$ and the function $\rho = \rho_\alpha$, we obtain an outer measure $S_{\alpha, \delta}^*$.
    As before one can show that $S_\alpha^* = \lim_{\delta \to 0} S_{\alpha, \delta}^*$ is an outer measure, and gives a Borel measure $S_\alpha$.
    \begin{parts}
      \item Show by example $S_\alpha \neq H_\alpha$ in general.
      \item If $X = \R^d$ with the standard metric show that $S_d = \lambda$. \noteb{You may assume $\rho_d( B_r ) = \lambda(B_r)$.}
    \end{parts}



  \fullwidth{\noindent\emph{Details in class I left for you to check. (Do it, but don't turn it in.)}}

    \itemopt
      Using notation from the proof of Caratheodory, show that $\mu^*( A \cap (\cup_1^\infty E_i) ) = \sum_1^\infty \mu^*(A \cap E_i)$. \noteb{We only proved it for $A = X$ in class.}

\end{assignment}%}}}
\newpage

\begin{assignment}{Wed 09/11}{Wed 09/18}%{{{ HW3
  \item
    Let $\mu, \nu$ be two measures on $(X, \Sigma)$. Suppose $\mc C \subset \Sigma$ is a $\pi$-system such that $\mu = \nu$ on $\mc C$.
  \begin{parts}
    \item Suppose $\exists C_i \in \mc C$ such that $\bigcup_1^\infty C_i = X$ and $\mu(C_i) = \nu(C_i) < \infty$. Show that $\mu = \nu$ on $\sigma(\mc C)$.
    \item If we drop the finiteness condition $\mu(C_i) < \infty$ is the previous subpart still true? Prove or find a counter example.
  \end{parts}

  \item
    \begin{parts}
      \item
	Let $X$ be a metric space and $\mu$ a Borel measure on $X$. Suppose there exists a sequence of sets $B_n \subset X$ such that $\bar B_n \subset \mathring B_{n+1}$, $\bar B_n$ is compact, $X = \cup_1^\infty B_n$ and $\mu(B_n) < \infty$.
      Show that $\mu$ is regular.
      \item
	Show directly that for all $A \in \mc L$, $\lambda(A) = \sup\{ \lambda(K) \}$ where $K \subset A $ is compact, and $\lambda(A) = \inf\{ \lambda(U) \}$ where $U \supset A$ is open.
	\noteb{Note: The previous subpart will \emph{only} show this for all $A \in \mc B(\R^d)$.}
    \end{parts}

  \item
    \begin{parts}
      \item
	Find $E \in \mc B(\R)$ so that for all $a < b$, we have $0 < \lambda(E \cap (a, b)) < b - a$.
      \item
	Let $\kappa \in (0, 1/2)$.
	Does there exist $E \in \mc B(\R)$ such that for all $a < b \in \R$, we have $\kappa(b - a) \leq \lambda(I \cap (a, b)) \leq (1-\kappa) (b - a)$?
	Prove it.
    \end{parts}

  \item
    Let $A \in \mc L(\R^d)$.
    Prove every subset of $A$ is Lebesgue measurable $\iff \lambda(A) = 0$.

  \item
    \begin{parts}
      \item Prove $\mc B(\R^{m+n}) =  \sigma( \{ A \times B \st A \in \mc B(\R^m) \mathop\& B \in \mc B(\R^n) \})$.
      \item Prove $\mc L(\R^{m+n}) \supsetneq  \sigma( \{ A \times B \st A \in \mc L(\R^m) \mathop\& B \in \mc L(\R^n) \})$.
      \item Show $\mc L(\R^2) \supsetneq \mc B(\R^2)$.
    \end{parts}

  \fullwidth{\noindent\emph{Optional problems, and details in class I left for you to check.}}

  \itemopt Let $\mu$ be a finite Borel measure on a compact metric space.
    Let
    $$
      \mc C = \{A \st \sup_{\substack{K \subset A\\K \text{ compact}} } \mu(K) = \mu(A) = \inf_{\substack{U \supset A\\U \text{ open}} } \mu(U) \}.
    $$
    We saw in class that $\mc C$ is closed under countable increasing unions.
    Show $\mc C$ is closed under relative compliments.

  \itemopt Is any $\sigma$-finite Borel measure on $\R^d$ regular?

  \itemopt Show that there exists $A \subset \R$ such that if $B \subset A$ and $B \in \mc L$ then $\lambda(B) = 0$, \emph{and further}, if $B \subset A^c$ and $B \in \mc L$ then $\lambda(B) = 0$.

  \fullwidth{\vspace{-\baselineskip}We say $\mc A \subset \mc P(X)$ is an algebra if $\emptyset \in \mc A$, and $\mc A$ is closed under complements and \emph{finite} unions.
    We say $\mu_0: \mc A \to [0, \infty]$ is a (positive) \emph{pre-measure} on $\mc A$ if $\mu_0(\emptyset) = 0$, and for any countable disjoint sequence of sets sequence $A_i \in \mc A$ \emph{such that $\bigcup_1^\infty  A_i \in \mc A$}, we have $\mu_0(\bigcup_1^\infty A_i) = \sum_1^\infty \mu_0(A_i)$.

  Namely, a pre-measure is a finitely additive measure on an algebra $\mc A$, which is also countably additive for disjoint unions \emph{that belong to the algebra}.}
  \itemopt \emph{(Caratheodory extension)} If $\mc A$ is an algebra, and $\mu_0$ is a pre-measure on $\mc A$, show that there exists a measure $\mu$ defined on $\sigma(\mc A)$ that extends $\mu_0$.


\end{assignment}%}}}
\newpage
\begin{assignment}{Wed 09/18}{Wed 09/25}%{{{ HW4
  \item Let $C \subset \R^d$ be convex.
    Must $C$ be Lebesgue measurable?
    Must $C$ be Borel measurable?
    Prove or find counter examples.
    \noteb{The cases $d = 1$ and $d > 1$ are different.}

  \item Let $(X, \Sigma, \mu)$ be a measure space. For $A \in P(X)$ define $\mu^*(A) = \inf\{\mu(E) \st E \supset A \mathop \& E \in \Sigma \}$, and $\mu_*(A) = \sup\{ \mu(E) \st E \subset A \mathop \& E \in \Sigma \}$.
    \begin{parts}
      \item
	Show that $\mu^*$ is an outer measure.

      \item
	Let $A_1, A_2, \dots \in \mc P(X)$ be disjoint. Show that $\mu_*(\bigcup_1^\infty A_i) \geq \sum_1^\infty \mu_*(A_i)$. \noteb{The set function $\mu_*$ is called an \emph{inner measure}.}

      \item Show that for all $A \subset X$, $\mu^*(A) + \mu_*(A^c) = \mu(X)$.

      \item
	Let $A \subset \mc P(X)$ with $\mu^*(A) < \infty$.
	Show that $A \in \Sigma_\mu \iff \mu_*(A) = \mu^*(A)$.
    \end{parts}

  \item Let $f:X \to \R$ be measurable, and $g:\R \to \R$ be Lebesgue measurable. True or false: $g \circ f:X \to \R$ is measurable? Prove or find a counter example.

  \item Let $(X, \Sigma)$ be a measure space, and $f, g:X \to [-\infty, \infty]$ be measurable.
    Suppose whenever $g=0$, $f\neq 0$, and whenever $f = \pm \infty$, $g \in (-\infty, \infty)$.
    Show that $\frac{f}{g}:X \to [-\infty, \infty]$ is measurable.
    \noteb{
      Note that by the given data you will never get a `meaningless' quotient of the form $\frac{0}{0}$ or $\frac{\pm \infty}{\pm \infty}$.
      The remainder of the quotients (e.g. $\frac{1}{\infty}$) can be defined in the natural manner.}

  \item Let $f_n:X \to \R$ be a sequence of measurable functions such that $(f_n) \to f$ almost everywhere (a.e.). Let $g:\R \to \R$ be a Borel function.
    \begin{parts}
      \item If for a.e.\ $x \in X$, $g$ is continuous at $f(x)$, then show $(g \circ f_n) \to g\circ f$ a.e.
      \item Is the previous part true without the continuity assumption on $g$?
    \end{parts}



  \fullwidth{\noindent\emph{Optional problems, and details in class I left for you to check.}}
  \itemopt
    \emph{(An alternate approach to $\lambda$-systems.)}
    Let $\mc M \subset P(X)$.
    We say $\mc M$ is a \emph{Monotone Class}, if whenever $A_i, B_i \in \mc M$ with $A_i \subset A_{i+1}$ and $B_i \supset B_{i+1}$ then $\bigcup_1^\infty A_i \in \mc M$ and $\bigcap_1^\infty B_i \in \mc M$.
    If $\mc A \subset P(X)$ is an algebra, then show that the \emph{smallest} monotone class containing $\mc A$ is exactly $\sigma(A)$.
    \noteb{You should also address existence of a smallest monotone class containing $\mc A$.}

  \itemopt
    Prove that the completion $\Sigma_\mu$ we defined in class is the smallest $\mu$-complete $\sigma$-algebra that contains $\Sigma$.

  \itemopt Show that $f:X \to [-\infty, \infty]$ is measurable if and only if any of the following conditions hold
    \begin{mcparts}
      \item $\{f < a\} \in \Sigma$ for all $a \in \R$.
      \item $\{f > a\} \in \Sigma$ for all $a \in \R$.
      \item $\{f \leq a\} \in \Sigma$ for all $a \in \R$.
      \item $\{f \geq a\} \in \Sigma$ for all $a \in \R$.
    \end{mcparts}

  \itemopt Let $(f_n)$ is a sequence of real valued measurable functions.
    Define $f(x) = \lim f_n(x)$ if the limit exists, and $f(x) = \infty$ otherwise.
    Show that $f$ is measurable.



\end{assignment}%}}}
\newpage

\begin{assignment}{Wed 09/25}{Wed 10/02}%{{{ HW5
  \item
    Let $(X, \Sigma, \mu)$ be a measure space, and $(X, \Sigma_\mu, \bar\mu)$ it's completion.
    Show that $g:X \to [-\infty, \infty]$ is $\Sigma_\mu$-measurable if and only if there exists two $\Sigma$-measurable functions $f,h:X \to [-\infty, \infty]$ such that $f = h$ $\mu$-almost everywhere, and $f \leq g \leq h$ everywhere.

  \item
    Let $\mu$ be a regular (but not necessarily finite) Borel measure on a metric space $X$.
    \begin{parts}
      \item
	True or false: For any $f:X \to \R$ measurable and $\eps > 0$ there exists $g:X \to \R$ continuous such that $\mu\{ f \neq g \} < \eps$?
	Prove it or find a counter example.
      \item
	Do the previous subpart when $X = \R^d$.
    \end{parts}

  \item
    Let for $n \in \N$ define $A_n = \bigcup_{k \in \Z} [\frac{2k}{2^n}, \frac{2k+1}{2^n})$.
    If $E \in \mc B(\R)$ does $\dlim_{n \to \infty} \lambda( A_n \cap E )$ exist?
    Prove it.

  \item If $f \geq 0$ is measurable show that $\int_X f \, d\mu = 0 \iff f = 0$ almost everywhere.

  \item
    \begin{parts}
      \item Suppose $I\subset \R^d$ is a cell, and $f:I \to \R$ is Riemann integrable. Show that $f$ is measurable, Lebesgue integrable and that the Lebesgue integral of $f$ equals the Riemann integral.
      \item Is the previous subpart true if we only assume that an improper (Riemann) integral of $f$ exists? Prove or find a counter example.
    \end{parts}


  \fullwidth{\noindent\emph{Optional problems, and details in class I left for you to check.}}
  \itemopt
    Let $f:[0, 1] \to [0, 1]$ be the Cantor function, and $g (x) = \inf\{f = x\}$.
    Show that $f$ is (H\"older) continuous, and the range of $g$ is the Cantor set.
    What is the largest exponent $\alpha$ for which $f$ is H\"older-$\alpha$ continuous?

  \itemopt Let $\mu$ be the counting measure on $\N$, and $f:\N \to \R$ a function.
    \begin{parts}
      \item If $\sum_1^\infty \abs{f(n)} < \infty$, then show that $\sum_{n = 1}^\infty f(n) = \int_\N f \, d\mu$.
      \item If the series $\sum_{n=1}^\infty f(n)$ is conditionally convergent, show that $\int_\N f\, d\mu$ is not defined.
    \end{parts}



  \itemopt
    Let $X$ be a metric space $C \subset X$ be closed and $f:C \to \R$ be continuous.
    \begin{parts}
      \item
	If $0 \leq f \leq 1$, then show that there exists $F:X \to \R$ continuous such that $F(c) = f(c)$ for all  $c \in C$.
	\hintb{Let $F(x) = f(x)$ for all $x \in C$, and $F(x) = \inf\{ f(c) + \frac{d(x, c)}{d(x, C)} - 1 \st c \in C \}$ for $x \not \in C$.}
      \item
	\emph{(Tietze extension theorem in metric spaces)}
	Do the previous subpart without assuming $0 \leq f \leq 1$.
	\hintb{Put $g = \tan\inv(f)$, construct $G$ by the previous subpart and set $F = \tan(G)$.}
    \end{parts}

  \itemopt Finish the proof of Lusin's theorem. (I only proved it for bounded positive functions in class.)

  \itemopt
    Find a Borel measurable function $f:[0,1] \to \R$ which is not continuous almost everywhere.

  \itemopt Let $0 \leq s \leq t$ be two simple functions. Show $\int_X s \leq \int_X t$.

  \itemopt Show directly $\int_X \alpha f = \alpha \int_X f$ for any $\alpha \in \R$ and integrable function $f$.

\end{assignment}%}}}
\newpage

\begin{assignment}{Wed 10/02}{Never}%{{{ HW6
  \vspace{-.65\baselineskip}\fullwidth{\noindent\emph{In light of your \textbf{MIDTERM} this homework is optional.}}

  \item
    \begin{parts}
      \item If $f$ is a bounded measurable function and $\mu(X) < \infty$, then show $\int_X f d\mu = \inf\{ \int_X t\, d\mu \st t \geq f \text{ is simple}\}$.
      \item If $f, g$ are bounded measurable functions and $\mu(X) < \infty$ show directly that $\int_X (f + g) \, d\mu = \int_X f \, d\mu + \int_X g \, d\mu$.
    \end{parts}

  \item Let $f:[0, \infty) \to \R$ be a measurable function. We define the \emph{Laplace Transform} of $f$ to be the function $F(s) = \int_0^\infty \exp(-st) f(t) \, dt$ wherever defined.
    \begin{parts}
      \item If $\int_0^\infty \abs{f(t)} \, dt < \infty$, show that $F:[0,\infty) \to \R$ is continuous.
      \item If $\int_0^\infty t \abs{f(t)} \, dt < \infty$, show that $F:[0, \infty) \to \R$ is differentiable.
      \item If $f$ is continuous and bounded, compute $\lim_{s \to \infty} s F(s)$.
    \end{parts}

  \item
    For $p \in \R$ define define
    $\ds
      F(y) = \int_{0}^\infty \frac{\sin(xy)}{1 + x^p} \, dx.
    $
    \begin{parts}
      \item For what $p \in \R$ is $F$ defined? When defined, is $F$ continuous? Prove it.
      \item Show that $F$ is differentiable for $p > 2$, and not differentiable when $p = 2$.
    \end{parts}

  \item \emph{(Push forward measures)}
    Let $\mu$ be a measure on $(X, \Sigma)$, and $f:X \to Y$ be any function.
    Define $\tau \subset \mc P(Y)$ by $\tau = \{A \subset Y \st f\inv(A) \in \Sigma \}$.
    For $A \in \tau$ define $\nu(A) = \mu( f\inv (A))$.
    \begin{parts}
      \item Show that $\tau$ is a $\sigma$-algebra, and $\nu$ is a measure on $(Y, \tau)$. \noteb{The measure $\nu$ is called the push-forward of $\mu$ under $f$, and often denoted by $\mu_{f\inv}$.}
      \item If $g \in L^1(Y, \nu)$, then show that $g \circ f \in L^1(X, \mu)$ and $\int_X g \circ f \, d\mu = \int_Y g \, d\nu$.
    \end{parts}

  \item \emph{(Pull back measures)} Say $\nu$ is a measure on $(Y, \tau)$ and $f:X \to Y$ is surjective.
    \begin{parts}
      \item Show that $\Sigma = \{A \subset X \st f(A) \in \tau \}$ need not be a $\sigma$-algebra.
	If $\Sigma$ is a $\sigma$-algebra, show that $\mu(A) = \nu(f(A))$ need not be a measure on $(X, \Sigma)$.
      \item Define instead $\Sigma = \{ A \subset X \st f\inv( f(A) ) = A, \& f(A) \in \tau \}$, and $\mu(A) = \nu( f(A))$.
	Show that $\Sigma$ is a $\sigma$-algebra and $\mu$ is a measure.
      \item If $g \in L^1(Y, \nu)$, then show that $g \circ f \in L^1(X, \mu)$ and $\int_X g \circ f \, d\mu = \int_Y g \, d\nu$.
      \end{parts}

    \item \emph{(Linear change of variable)} Let $f: \R^d \to \R$ be integrable.
      \begin{parts}
	\item For any $y \in \R^d$ show that $\int_{\R^d} f(x + y) \, d\lambda(x) = \int_{\R^d} f(x) \, d\lambda(x)$.
	\item
	  If $T:\R^d \to \R^d$ an invertible linear transformation, and $E \in \mc L(\R^d)$.
	  Show that
	  $$
	    \int_{T\inv(E)} (f \circ T) \abs{\det T} \, d\lambda = \int_{E} f \, d\lambda.
	  $$
      \end{parts}



  \fullwidth{\noindent\emph{Details in class I left for you to check.}}
  \itemopt
    Check that if $s, t$ are non-negative simple functions then $\int_X (s + t) = \int_X s + \int_X t$.

  \itemopt
    Show that there exists $f:\R \to [0, \infty)$ Borel measurable such that $\int_a^b f \, d\lambda = \infty$ for all $a, b \in \R$ with $a < b \in \R$.
    \hintb{Let $g(x) = \Chi{\abs{x} < 1} \abs{x}^{-1/2}$, and define $h(x) = \sum_{m = -\infty}^\infty \sum_{n = 1}^\infty 2^{-m-n} g (x -m/n)$.}
\end{assignment}%}}}
\newpage

\begin{assignment}{Wed 10/09}{Wed 10/16}%{{{ HW7
  \item
    Do questions 3, 5, and 6 from HW6.

  \item
    \begin{parts}
      \item
	\emph{(Jensen's inequality)}
	Let $a, b \in [-\infty, \infty]$ with $a < b$ and $\varphi:(a, b) \to \R$ be a convex function.
	If $\mu(X) = 1$ and $f:X \to (a, b)$ is integrable then show
	$$
	  \varphi\paren[\Big]{ \int_X f \, d\mu } \leq \int_X \varphi \circ f \, d\mu.
	$$
      \item
	If $\varphi$ above is strictly convex, when can you have equality?
    \end{parts}

  \item
    \begin{parts}
      \item Suppose $p, q, r \in [1, \infty]$ with $p < q < r$. Prove that for all $f \in L^p \cap L^r$, $f \in L^q$. Further, find $\theta \in (0,1)$ such that $\norm{f}_q  \leq \norm{f}_p^\theta \norm{f}_r^{1-\theta}$.
      \item If for some $p \in [1, \infty)$, $f \in L^p(X) \cap L^\infty(X)$ show that $\lim_{q\to \infty} \norm{f}_q = \norm{f}_\infty$. \noteb{This sort of justifies the notation $\norm\cdot_\infty$.}
      \item
	 Let $p_0 \in (0, \infty]$, $\mu(X) = 1$ and $f \in L^{p_0}(X)$.
	 Prove $\lim_{p \to 0^+} \norm{f}_p = \exp\bigl( \int_X \ln \abs{f} \, d\mu \bigr)$.
  \end{parts}

  \fullwidth{\noindent\emph{Optional problems, and details in class I left for you to check.}}

  \itemopt
    Let $g \geq 0$ be measurable, and define $\nu(A) = \int_A g \, d\mu$.
    Show that $\nu$ is a measure, and $\int_E f \, d\nu = \int_E fg \, d\mu$.

  \itemopt
    Prove H\"older's inequality for $p=1$ and $q = \infty$.

  \itemopt
    If $p_i, q \in [1, \infty]$ with $\sum_1^N \frac{1}{p_i} = \frac{1}{q}$, show that $\norm{\prod_1^n f_i}_q \leq \prod \norm{f_i}_{p_i}$.

  \itemopt
    Show that $L^\infty$ is a Banach space.

  \itemopt
    For $p \in [0, 1)$ show that you need not have $\norm{f + g}_p \leq \norm{f}_p + \norm{g}_p$.

  \itemopt
    Let $p, q \in (1, \infty)$, $\frac{1}{p} + \frac{1}{q} = 1$, $f\in L^p$ and $g \in L^q$.
    Show that $\int_X \abs{f g} \, d\mu = \norm{f}_p \norm{g}_q$ if and only if there exists constants $\alpha, \beta \geq 0$ such that $\alpha f^p = \beta g^q$.

  \itemopt
    \begin{parts}
      \item If $X$ is $\sigma$-finite, then show
	$\norm{f}_\infty = \sup\limits_{g \in L^1 - \{0\} } \frac{1}{\norm{g}_1} \int_X fg \, d\mu$.
      \item Show that the previous subpart is false if $X$ is not $\sigma$-finite.
    \end{parts}



\end{assignment}%}}}
\newpage

\begin{assignment}{Wed 10/16}{Wed 10/23}%{{{HW8
  \item
    \begin{parts}
      \item If $\mu(X) <  \infty$, $1 \leq p <  q \leq \infty$, show $L^q(X) \subset L^p(X)$ and the inclusion map from $L^q(X) \to L^p(X)$ is continuous. Find an example where $L^q(X) \subsetneq L^p(X)$. \hintb{Show $\norm{f}_p \leq \mu(X)^{\frac{1}{p} - \frac{1}{q}} \norm{f}_q$.}
      \item Let $\ell^p = L^p(\N)$ with respect to the counting measure. If $1 \leq p < q$ show that $\ell^p \subsetneq \ell^q$.
	Is the inclusion map $\ell^p \hookrightarrow \ell^q$ continuous?
	Prove your answer.
    \end{parts}

  \item
    \begin{parts}
      \item Suppose $p \in [1, \infty)$, and $f \in L^p(\R^d, \lambda)$. For $y \in \R^d$, let $\tau_y f: \R^d \to \R$ be defined by $\tau_y f(x) = f(x - y)$.
	Show that $(\tau_y f) \to f$ in $L^p$ as $\abs{y} \to 0$.
      \item What happens for $p = \infty$?
    \end{parts}

  \item
    Suppose $\Sigma = \sigma(\mc C)$, where $\mc C \subset \mc P(X)$ is countable.
    If $\mu$ is a $\sigma$-finite measure and $1 \leq p < \infty$, show that $L^p(X)$ is seperable (i.e. has a countable dense subset).

  \item
    \begin{parts}
      \item
	Suppose $\lim_{\lambda \to \infty} \sup_n \int_{\abs{f_n} > \lambda} \abs{f_n} \, d\mu = 0$.
	Show that there exists an increasing funciton $\varphi$ with $\varphi(\lambda) / \lambda \to \infty$ as $\lambda \to \infty$, such that $\sup_n \int_X \varphi(\abs{f_n}) < \infty$.
      \item
	Suppose $\{f_n\}$ is uniformly integrable, and $\sup_n \int \abs{f_n} < \infty$.
	Show that $\lim_{\lambda \to \infty} \sup_n \int_{\abs{f_n} > \lambda} \abs{f_n} = 0$.
      \item
	Show that the previous part fails without the assumption $\sup_n \int \abs{f_n} < \infty$.
    \end{parts}

  \item
    Let $e_n(x) = e^{2\pi i n x}$, $X = [ 0, 1]$. For what $p \in [1, \infty]$ does $\{e_n\}$ have a convergent subsequence in $L^p(X, \lambda)$? Prove it.


  \fullwidth{\noindent\emph{Optional problems, and details in class I left for you to check.}}
  \itemopt
    In Vitali's convergence theorem prove that the assumption $f \in L^1$ is unnecessary.

  \itemopt
    If $(f_n) \to f$ in $L^1$, show that $\{ f_n \}$ is uniformly integrable.
    \noteb{This is part of Vitali's theorem which I didn't have time to prove in class.}

  \itemopt
    Show that if $(f_n) \to f$ in measure, then $(f_n)$ need not converge to $f$ in $L^p$.

  \itemopt
    Finish the proof that $C_c(X)$ is dense in $L^p$. \noteb{I only did the case when $X$ is compact in class.}

  \itemopt
      Show that simple functions are dense in $L^\infty$.

  \itemopt
    Show that $C_c(\R)$ is not dense in $L^\infty(\R)$.

  \itemopt
    Show that $L^\infty(\R)$ is not separable.

\end{assignment}%}}}
\newpage

\begin{assignment}{Wed 10/23}{Wed 10/30}%{{{ HW9
  \item Recall we defined the variation of $\mu$ by $\abs{\mu} = \mu^+ + \mu^-$, and the total variation by $\norm{\mu} = \abs{\mu}(X)$. (You should check that these are well defined.)
    \begin{parts}
      %\item
      %  If $\mu$, $\nu$ are two finite signed measures, show that $\abs{\mu + \nu}(A) \leq \abs{\mu}(A) + \abs{\nu}(A)$.
      \item
	Let $\mc M$ be the space of all finite signed measures on $(X, \Sigma)$.
    Show that $\mc M$ with total variation norm (i.e. with $\norm{\mu} = \abs{\mu}(X)$) is a Banach space.
      \item
	Show that $(\mu_n) \to \mu$ if and only if $(\mu_n(A)) \to \mu(A)$ uniformly in $A$, $\forall A \in \Sigma$.
    \end{parts}

  \item
    \begin{parts}
      \item
	For a signed measure, we define $\int_X f \, d\mu = \int_X f \, d\mu^+ - \int_X f \, d\mu^-$.
	Suppose $(f_n) \to f$, $(g_n) \to g$, and $\abs{f_n} \leq g_n$ almost everywhere with respect to $\abs{\mu}$.
	If $\lim \int_X g_n \, d\abs{\mu} = \int_X g \, d\abs{\mu} < \infty$, show that $\lim \int_X f_n \, d\mu = \int_X f \, d\mu$.

      \item
	Suppose $f, f_n \in L^1$, and $(f_n) \to f$ almost everywhere.
	Show that $\lim \int \abs{f_n - f} d\abs{\mu} = 0$ if and only if $\lim \int \abs{f_n} \, d\abs{\mu} = \int \abs{f} d\abs{\mu}$.
    \end{parts}

  \item
    \begin{parts}
      \item
	If $\mu$ is a positive $\sigma$-finite measure, and $\nu$ is a finite signed measure such that $\abs{\nu} \ll \mu$, show that there exists $f \in L^1(X, \mu)$ such that $d\nu = f \, d\mu$.
      \item
	Compute $\frac{d\nu}{d\abs{\nu}}$ in terms of the Hanh decomposition of $\nu$.
	\hintb[Notation]{We say $g = \frac{d\nu}{d\mu}$ if $d\nu = g \, d\mu$.}
    \end{parts}



  \item
    \begin{parts}
      \item
	Let $\nu_1$ and $\nu_2$ be two finite signed measures on $X$.
	%Show that there exists a ``smallest'' finite signed measure which is larger than both $\nu_1$ and $\nu_2$.
	Show that there exists a finite signed measure $\nu_1 \varmax \nu_2$ such that $\nu_1 \varmax \nu_2(A) \geq \nu_1(A) \varmax \nu_2(A)$, and for any other finite signed measure $\nu$ such that $\nu(A) \geq \nu_1(A) \varmax \nu_2(A)$ we ust have $\nu_1 \varmax \nu_2 \leq \nu$.

      \item If $\nu_1, \nu_2$ above are absolutely continuous with respect to a positive $\sigma$-finite measure $\mu$, prove $\nu_1 \varmax \nu_2 \ll \mu$ and express $\frac{d(\nu_1 \varmax \nu_2)}{d\mu}$ in terms of $\frac{d\nu_1}{d\mu}$ and $\frac{d\nu_2}{d\mu}$.
    \end{parts}

  \item
    Let $(\Omega, \mc F, P)$ be a measure space with $P(\Omega) = 1$, and $X \in L^1(\Omega, \mc F, P)$.
    \noteb{The probabilistic interpretation is that $\Omega$ is the sample space, $A \in \mc F$ is an event, $X$ is a random variable, and $P(X \in B)$ is the chance that $X \in B$, where $B \in \mc B(\R)$.}
  \begin{parts}
    \item
      Suppose $\mc G \subset \mc F$ is a $\sigma$-sub-algebra of $F$.
      Show that there exists a unique $\mc G$-measurable function $Y$ such that $\int_A Y \, dP = \int_A X \, dP$ for all $A \in \mc G$.
      \noteb{$Y$ is called the \emph{conditional expectation} of $X$ given $\mc G$, and denoted by $E(X \given \mc G)$.}

    \item
      \emph{(Tower property)}
      If $\mc H \subset \mc G$ is a $\sigma$-sub-algebra, show that $E(X \given \mc H) = E\bigl( E(X \given \mc G) \given \mc H)$ almost everywhere.

    \item \emph{(Conditional Jensen)}
      If $\varphi: \R \to \R$ is convex, show that $\varphi( E(X \given \mc G) ) \leq E( \varphi(X) \given G)$ almost everywhere.

    \item Suppose $X \in L^2(\Omega, \mc F, P)$. Show that $E(X \given \mc G)$ is the $L^2$-orthogonal projection of $X$ onto the subspace $L^2(\Omega, \mc G)$. \noteb{Namely show $E(X \given \mc G) \in L^2(\Omega, \mc G)$, and $\int_\Omega (X - E(X \given G)) Y \, dP = 0$ for all $Y \in L^2(\Omega, \mc G)$.}

  \end{parts}


  \fullwidth{\noindent\emph{Optional problems, and details in class I left for you to check.}}
  \itemopt
    In the proof of the Hanh decomposition, prove the following:
    Say $\mu(X) > -\infty$, and $\alpha = \inf\{\mu(B)\}$.
    Let $B_n'$ be a sequence of negative sets such that $\mu(B_n') \to \alpha$.
    Let $N = \cup B_n'$.
    Show $\mu(N) = \alpha$.

  \itemopt
    Prove the Hanh decomposition is unique up to null sets.
    %Namely if $X = P \cup N = P' \cup N'$ where $P, P'$ are positive, and $N, N'$ are negative, then show that $(P \Delta P') \cup (N \Delta N')$ is null. (Here $A \Delta B = (A - B) \cup (B - A)$ is the symmetric difference of $A$ and $B$.)

  \itemopt
    Prove uniqueness of the Jordan decomposition.

  \itemopt
    Show that the Radon-Nikodym theorem need not hold if $\mu, \nu$ are not $\sigma$-finite.
\end{assignment}%}}}
\newpage

\begin{assignment}{Wed 10/30}{Wed 11/06}%{{{ HW10
  \item
    \begin{parts}
      \item
	Let $p \in [1, \infty)$ and $q$ be conjugate H\"older exponent.
	If $X$ is $\sigma$-finite, show that there exists a bijective linear isometry between $(L^p)^*$ and  $L^q$.

      \item
	The above result is \emph{false} for $p = \infty$ even when $\mu(X) < \infty$.
	Find where our proof from class (when $\mu(X)<\infty$) fails when $p = \infty$.

      \item
	We can (partially) construct a counter example on $\ell^\infty$ as follows.
	The Hanh-Banach theorem shows that there exists exists $T \in (\ell^\infty)^*$ such that $Ta = \lim a_n$, for all $a = (a_n) \in \ell^\infty$ such that $\lim a_n$ exists and is finite.
	Show that there does not exist $b \in \ell^1$ such that $T a = \sum a_n b_n$ for all $a \in \ell^\infty$.
    \end{parts}

  \item
  \begin{parts}
    \item
      Suppose $\sum_{m=1}^\infty ( \sum_{n=1}^\infty |a_{m,n}| ) < \infty$.
      Show that $\sum_{m=1}^\infty (\sum_{n=1}^\infty a_{m,n}) = \sum_{n=1}^\infty (\sum_{m=1}^\infty a_{m,n})$.
    \item Give a counter example to (a) if we only assume $\sum_m \sum_n a_{m,n} < \infty$. Find a counter example where both iterated sums are finite.
  \end{parts}

  \item 
  \begin{parts}
    \item If $X$ and $Y$ are not $\sigma$-finite, show that Fubini's theorem need not hold.
    \item If $\int_{[-1,1]^2} f \, d\lambda$ is not assumed to exist (in the extended sense), show that both iterated integrals can exist, be finite, but need not be equal.
  \end{parts}

  \item \textit{(Fubini for completions.)} Suppose $(X, \Sigma, \mu)$ and $(Y, \tau, \nu)$ are two $\sigma$-finite, complete measure spaces.
      Let $\varpi = (\Sigma \otimes \tau)_{\pi}$ denote the completion of $\Sigma \otimes \tau$ with respect to the product measure $\pi = \mu \times \nu$.
  \begin{parts}
    \item
      Show that $\Sigma \otimes \tau$ need not be $\pi$-complete (i.e.\ $\varpi \supsetneq \Sigma \otimes \tau$ in general).
    \item
      Suppose $f:X\times Y \to [-\infty,\infty]$ is $\mc F$-measurable.
      Define as usual the slices $\varphi_{f,x}:Y \to [0, \infty]$ by $\varphi_{f,x}(y) = f(x,y)$, and similarly $\psi_{f,y}(x) = f(x,y)$.
      Show that for $\mu$-almost all $x\in X$, $\varphi_{f,x}$ is an $\tau$-measurable, and for $\nu$-almost all $y$, $\psi_{f,y}$ is an $\Sigma$-measurable.
    \item
      Suppose $f$ is integrable on $X\times Y$ in the extended sense.
      Define $F(x) = \int_Y f(x, y) \, d\nu(y)$ and $G(y) = \int_X f(x,y) \, d\mu(x)$.
      Show $F$ is defined $\mu$-a.e. and $\Sigma$-measurable.
      Similarly show $G$ is defined $\nu$-a.e., and $\tau$-measurable.
      Further, show and that $\int_X F \, d\mu = \int_Y G\, d\nu = \int_{X\times Y} f d(\mu \times \nu)$.
      %\hintb{Find a $\Sigma \otimes \tau$-measurable function $g$, such that $f = g$ $\mu \times \nu$-a.e., and apply Fubini to $g$.}
  \end{parts}

  \item
    Let $(X, \Sigma, \mu)$, $(Y, \tau, \nu)$ be two $\sigma$-finite measure spaces, $p \in [1, \infty]$, and $f:X\times Y \to \R$ is $\Sigma \otimes \tau$ measurable.
    Let $F(x) = \int_Y f(x,y) \, d\nu(y)$, and $\psi_{y,f}$ be the slice of $f$ defined by $\psi_{y,f}(x) = f(x,y)$.
    Show that $\norm{F}_{L^p(X)} \leq \int_Y \norm{\psi_{y,f}}_{L^p(X)} \, d\nu(y)$.
    \noteb{When $Y = \{1, 2\}$ with the counting measure, this is exactly Minkowski's triangle inequality.}

  \fullwidth{\noindent\emph{Optional problems, and details in class I left for you to check.}}

  \itemopt
    Let $\mu(X) < \infty$, $p \in [1, \infty)$ and $T \in (L^p)^*$.
    Let $\nu(A) = T( \Chi*{A} )$. We've seen in class that $\nu \ll \mu$ and so $d\nu = g \, d\mu$ for some $g \in L^1(\mu)$.
    \begin{parts}
      \item
	Show that $Tf = \int_X f g \, d\mu$ for all $f$ simple.
      \item
	If $\frac{1}{p} + \frac{1}{q} = 1$ show $\norm{g}_q = \sup \{ \int_X s g \}$, where the supremum runs over all \emph{simple} functions $s$ such that $\norm{s}_p \leq 1$.
	Conclude $g \in L^q$ and $\norm{g}_q \leq \norm{T}$.
      \item
	Show that $T f = \int_X f g \, d\mu$ for all $f \in L^p$, to conclude the proof.
    \end{parts}

  \itemopt
    Show that the Lebesgue measure on $\R^{m+n}$ is the product of the Lebesgue measures on $\R^m$ and $\R^n$ respectively.

  %\itemopt
  %  Verify that $\pi_1(E) \defeq \int_X \nu( S_x(E)) \, d\mu(x)$ is a measure.


\end{assignment}%}}}
\newpage

\begin{assignment}{Wed 11/06}{Wed 11/13}%{{{ HW11
  \item
    If $\frac{1}{p} + \frac{1}{q} = 1$, $f\in L^p$, $g \in L^q$ show that $f*g$ is bounded and continuous.
    If $p,q < \infty$, show further $f*g(x) \to 0$ as $\abs{x} \to \infty$.

  \item
    Let $\{\varphi_n\}$ be an approximate identity.
    \begin{parts}
      \item If $f \in C(\R^d) \cap L^\infty$, show $f * \varphi_n \to f$ pointwise.%
	%\hintb[Remark]{If $f$ is uniformly continuous, the convergence is uniform.}
      \item For $\alpha \in (0, 1)$ define
	$$
	  \norm{f}_{C^\alpha} = \norm{f}_\infty + \sup_{x \neq y} \frac{\abs{f(x) - f(y)} }{ \abs{x - y}^\alpha},
	  \text{ and }
	  C^{\alpha} = \{ f:\R^d \to \R \st \norm{f}_{C^\alpha} < \infty \}.
	$$
	If $f \in C^\alpha$, show that $f* \varphi_n \in C^\alpha$ \sout{and $f * \varphi_n \to f$ in $C^\alpha$}.

    \end{parts}

  \item
    Define $\mc S(\R^d) = \{ f \in C^\infty(\R^d) \st \forall m, \alpha, \; \sup_x (1 + \abs{x}^m) |D^\alpha f(x)| < \infty \}$.
    Here $m \in \N \cup \{0\}$, and $\alpha = (\alpha_1, \dots, \alpha_d) \in (\N\cup \{0\})^d$ is a multi-index, and $D^\alpha f = \del_1^{\alpha_1} \cdots \del_d^{\alpha_d} f$.
    The space $\mc S$ is called the \emph{Schwartz Space}.
    \begin{parts}
      \item If $p \in [1, \infty)$, $f \in L^p(\R^d)$, $g \in \mc S(\R^d)$, show that $f * g \in C^\infty(\R^d)$, and further $D^\alpha (f * g) = f * (D^\alpha g)$.
      \item For $p \in [1, \infty)$, show that $C^\infty_c$ and $\mc S$ are dense subsets of $L^p$ 
    \end{parts}

  \item Let $A, B \in \mc L(\R)$ be measurable, and define $A + B = \{a+b \st a \in A, b \in B\}$.
    %Show that $A + B \in \mc L(\R)$. Further,
    If $\lambda(A) > 0$ and $\lambda(B) > 0$ show $A + B$ contains an interval.

  \uplevel{%
    Though I encourage you to check the properties on the Dirichlet and Fej\'er kernels stated in the optional problems, you may assume them here without proof.

    Let $C\per = \{f \in C(\R) \st \tau_1 f = f \}$ denote all continuous functions with period $1$.
    Since the Fej\'er kernels are an approximate identity, it immediately follows that the Ces\`aro sums $\sigma_N f \to f$ uniformly, for any $f \in C\per$.
    For general $f \in C\per$, however, the partial sums $S_N f$ need not converge to $f$ even pointwise.
    (In fact, there exist many $f \in C\per$ such that $S_N f$ is divergent on a dense $G_\delta$.)
    If, however, $f$ is a little bit better than continuous, then the Fourier series of $f$ converges to $f$ pointwise.%
  }

  \item
    Let $\alpha \in (0, 1)$ and $f \in C^\alpha\per$. Show that $(S_N f) \to f$ uniformly, as $N \to \infty$.

  \fullwidth{\noindent\emph{Optional problems, and details in class I left for you to check.}}

  \itemopt If $f \in L^p$, $g \in L^q$ with $p,q \in [1,\infty]$ and $1/p + 1/q \geq 1$, show that $ f * g = g * f$.

  \itemopt If $f \in L^p$, $g \in L^q$, $h \in L^r$ with $p,q,r \in [1,\infty]$ and $1/p + 1/q + 1/r \geq 2$, show that $ (f * g) * h = f * (g * h)$.

  \itemopt
    Define the Derichlet kernel by $D_N(x) = \sum_{-N}^N \exp(2\pi i n x)$.
    \begin{parts}
      \item Show that $S_N f(x) = D_N * f(x) \defeq \int_0^1 f(y) D_N(x - y)  \, dy$.
	\noteb{Recall, $S_N f = \sum_{-N}^N \hat f(n) e_n$, where $e_n(x) = e^{2\pi i n x}$, and $\hat f(n) = \ip{f}{e_n} = \int_0^1 f(y) \bar e_n(y) \, dy$.}
      \item
	Show that $D_N(x) = \frac{\sin( (2N + 1) \pi x )}{\sin(\pi x)}$.
	Further show $\lim_{N \to \infty } \int_\eps^{1-\eps} \abs{D_N} = \infty$.
    \end{parts}

  \itemopt
    Define Fej\'er kernel by $F_N = \frac{1}{N} \sum_0^{N-1} D_n$.
    \begin{parts}
      \item Show that $\sigma_N f \defeq \frac{1}{N} \sum_0^{N-1} S_n f = F_N * f$.
      \item Show that $F_N(x) = \frac{\sin^2(N \pi x)}{ N \sin^2(\pi x)}$, and that $\{F_N\}$ is an approximate identity.
    \end{parts}

\end{assignment}%}}}
\newpage

\begin{assignment}{Wed 11/13}{Wed 11/20}%{{{ HW12
  \item
    Let $\mu$ be a finite signed Borel measure on $[0, 1]$.
    If $\forall n \in \Z \; \hat \mu(n) = 0$, show $\mu = 0$.

  \item
    Let $0 \leq r < s$.
    Show that any bounded sequence in $H^s\per$ has a subsequence that is convergent in $H^r\per$.

  \item
    Let $f \in L^2([0,1])$.
    Show that there exists a unique $u \in C^\infty( \R \times (0, \infty) )$ such that $u(x + 1, t) = u(x, t)$, $\lim_{t \to 0^+} \norm{u(\cdot, t) - f(\cdot)}_{L^2\per} = 0$, and $\delt u - \delx^2 u =0$.
    \hintb{You may assume the result of the optional problems.}

  \item
    Let $s \in (0, 1]$ and $f \in L^2\per$.
    Prove $f \in H^s\per \iff \sup\limits_{0 < h \leq 1} h^{-\alpha} \norm{\tau_h f - f}_{L^2} < \infty$.
    \hintb[Update]{The converse is false. A correction with solution will be posted.}

  \item
    \begin{parts}
      \item
	Let $n \in \N$ be even, $\frac{1}{n} + \frac{1}{n'} = 1$.
	If $\hat f \in \ell^{n'}(\Z)$, show that $f \in L^n\per([0,1])$ and $\norm{f}_{L^n} \leq \norm{\hat f}_{\ell^{n'}}$.
	\hintb{Let $n = 2m$. Then $\norm{f}_{L^n}^n = \norm{(f^m)^{\hatsym}}_{\ell^2}^2$.}
      \item
	Let $s > \frac{1}{2} - \frac{1}{p} \geq 0$, and $\frac{1}{p} + \frac{1}{q} = 1$.
	If $f \in H^s\per$ show $\hat f \in \ell^q(\Z)$.
	Further show that the map $f \mapsto \hat f$ is continuous from $H^s\per \to \ell^q$.
      \item If $n \in \N$ is even, $s > \frac{1}{2} - \frac{1}{n}$ then show that $H^s\per \subset L^n([0,1])$ and that the inclusion map is continuous.
	\noteb{This is one of the Sobolev embedding theorems.}
    \end{parts}


  \fullwidth{\noindent\emph{Optional problems, and details in class I left for you to check.}}
  \itemopt
    \begin{parts}
      \item
	If $f, g \in L^2\per([0,1])$, show that $(f*g)^{\hatsym}(n) = \hat f(n) \hat g (n)$.
	%\noteb{Here $L^2\per([0,1])$ denotes (equivalence classes of) all Lebesgue measurable functions $f:\R \to \C$ which have period $1$ (i.e. $\tau_1 f = f$), and $\int_0^1 \abs{f}^2 \, d\lambda < \infty$.}
      \item
	If $f, g \in L^2\per([0,1])$, show that $(fg)^{\hatsym}(n) = \hat f * \hat g (n) \defeq \sum_{m \in \Z} \hat f(m) \hat g(n-m)$.
    \end{parts}

  \itemopt
    \begin{parts}
      \item If $\alpha \in (0,1)$, $f \in C^\alpha\per([0,1])$, show that $\lim_{\abs{n} \to \infty} \abs{n}^\alpha \abs{\hat f(n)} = 0$.
      \item Show by example that the converse of the previous part is false.
    \end{parts}

  \itemopt
    For any $s \geq 0$ show that $H^s\per$ is a closed subspace of $L^2$.

  \itemopt
    Let $s \geq 1$, and $f \in H^s\per$.
    Show that $f$ has a weak derivative $Df$, and $Df \in H^{s-1}$.
    Further, show that the map $f \mapsto Df:H^{s} \to H^{s-1}$ is linear and continuous.

  \itemopt
    Let $s > 3/2$ and $f, g \in H^s\per$.
    Show that $fg \in H^1$, and further $D(fg) = (Df) g + f (Dg)$.

  \itemopt
    Find a function $f \in H^{1/2}\per - L^\infty$.
    \noteb{So the Sobolev embedding theorem is false for $s = 1/2$.}

  \itemopt
    Let $n \in \N \cup \{0\}$, $\alpha \in [0, 1)$ $s > 1/2 + n + \alpha$.
    Show that $H^s\per \subset C^{n, \alpha}\per[0, 1]$ and the inclusion map is continuous.
    \noteb{Recall $C^{n, \alpha}\per [0,1]$ is the set of all $C^n$ periodic functions on $\R$ (i.e. $\tau_1 f = f$) whose $n^\text{th}$ derivative is H\"older continuous with exponent $\alpha$.}
    %= \{ f \in C^n(\R) \st f(x + 1 ) = f(x), \text{and} \sup_{x\neq y} \frac{\abs{f^{(n)}(x) - 

  \itemopt
    Show that $f \in H^s$ for all $s \geq 0 \iff f \in C^\infty\per$.
\end{assignment}%}}}
\newpage

\begin{assignment}{Wed 11/20}{Wed 11/27}%{{{
  \item
    Let $s > 3/2$ and $f, g \in H^s\per$.
    Show that $fg \in H^1\per$, and further $D(fg) = (Df) g + f (Dg)$.
    \noteb{This was optional on last times homework.}

  \item
    \begin{parts}
      \item
	If $f \in L^1(\R^d)$ and $f$ is not identically $0$ (a.e.), then show that $Mf \not \in L^1(\R^d)$.

      \uplevel{\vspace{-1em}
	The next few subparts outline a proof that for any $p > 1$, the maximal function is an $L^p$ bounded sublinear operator.
	Let $p \in (1, \infty)$, $f \in L^p(\R^d)$ and $f \geq 0$.
      }

      \item
	Show that $\lambda\{Mf > \alpha \} \leq \frac{3^d}{(1 - \delta) \alpha } \int_{\{f > \delta \alpha\}} f$, for any $t > 0$, $\delta \in (0, 1)$ and $f \geq 0$ measurable.

      \item
	Let $p \in (1, \infty]$, and $d \in \N$.
	Show that there exists a constant $c = c(p, d)$ such that $\norm{Mf}_p \leq c \norm{f}_p$ for all $f \in L^p(\R^d)$.
	\hintb{For $p < \infty$, use the previous part, the identity $\norm{Mf}_p^p = \int_0^\infty p \alpha^{p - 1} \lambda\{Mf > \alpha\} \, d\alpha$ and optimise in $\delta$.}
    \end{parts}

  \item
    Let $\mu$ be a finite signed Borel measure on $\R^d$ such that $\mu \perp \lambda$.
    Show that $D\abs{\mu} = \infty$, $\mu$-almost everywhere.

  \item
    Let $\alpha \in [0, d]$, and $A \in \mc B(\R^d)$.
    If $H_\alpha(A) < \infty$, show $\dlim_{r \to 0} \frac{H_\alpha( A \cap B(x, r))}{c_\alpha r^\alpha} = 0$ for $H_\alpha$-almost all $x \not \in A$.

  \item
    \begin{parts}
      \item
	Suppose $f: [a, b] \to \R$ is a right continuous increasing function.
	Show that there exists a finite Borel measure $\mu$ such that $\mu( (x, y]) = f(y) - f(x)$ for every $x, y \in [a, b]$.
	Show further that $\mu = \mu_\text{ac} + \mu_\text{s} + \sum_i \alpha_i \delta_{a_i}$, where $\mu_\text{ac} \ll \lambda$, $\alpha_i > 0$, $a_i \in [a, b)$, $\sum_i \alpha_i < \infty$, and $\mu_\text{sc} \perp \lambda$ is such that  $\mu_\text{sc}( \{x\} ) = 0$ for all $x \in \R$.
	\hintb{If $f$ is strictly increasing and continuous, define $\mu(A) = \lambda( f(A))$, and consider its Lebesgue decomposition.}

      \item
	Let $f:[a, b] \to \R$ be monotone.
	Show that $f$ is differentiable almost everywhere, $f' \in L^1([a,b])$ and that $\abs{\int_a^b f'} \leq \abs{f(b) - f(a)}$.
    \end{parts}

  \fullwidth{\noindent\emph{Optional problems, and details in class I left for you to check.}}

  \itemopt
    Let $c_\alpha = \frac{ \pi^{\alpha/2} }{ \Gamma( 1 + \frac{\alpha}{2} ) }$ be the normalization constant from the definition of $H_\alpha$, the Hausdorff measure of dimension $\alpha$.
    \begin{parts}
      \item If $0 < H_\alpha(A) < \infty$, show $\dlimsup_{r \to 0} \frac{H_\alpha( A \cap B(x, r))}{c_\alpha r^\alpha} \in [2^{-\alpha}, 1]$ for $H_\alpha$-a.e.\ $x \in A$.
      \item Show that there exists $\alpha < d$ and $A \subset \R^d$ with $H_\alpha(A) \in (0, \infty)$ such that
	$$
	  \dliminf_{r \to 0} \frac{H_\alpha( A \cap B(x, r))}{c_\alpha r^\alpha} = 0
	  \quad\text{and}\quad
	  \dlimsup_{r \to 0} \frac{H_\alpha( A \cap B(x, r))}{c_\alpha r^\alpha} < 1,
	$$
	for $H^\alpha$-almost every $x \in \R^d$.
      \item
	If $C$ is the Cantor set, and $\alpha = \log 2 / \log 3$, compute $\dlimsup_{r \to 0} \frac{H_\alpha( C \cap B(x, r))}{c_\alpha r^\alpha}$.
    \end{parts}

  \itemopt
    \emph{(Infinite version of Vitali.)}
    Suppose $A \subset \cup B_\alpha$, where $\{B_\alpha\}_{\alpha \in \mc A}$ is an infinite collection of balls such that $\sup \lambda(B_\alpha) < \infty$.
    Show that there exists $\mc A' \subset \mc A$ such that the sub-collection $\{B_{\alpha'} \}_{\alpha' \in \mc A'}$ is disjoint and $A \subset \cup 5B_{\alpha '}$.

  \itemopt
    If $f \in L^1(\R^d)$, show that $Mf(x) \geq \abs{f(x)}$ at all Lebesgue points of $f$.
\end{assignment}%}}}
\newpage

\begin{assignment}{Wed 11/27}{Wed 12/04}%{{{

  \item
    Let $\mu$ be a positive finite Borel measure on $\R^d$, and $\alpha > 0$.
    Show that for every $A \subset \{ D\mu > \alpha \}$, we must have $\mu(A) \geq \alpha \lambda(A)$.

  \item
    \begin{parts}
      \item
	\emph{(Polar Coordinates.)}
	Let $f \in L^1(\R^2)$. Show that
	$$
	  \int_{\R^2} f(x, y) \, dx \, dy = \int_{[0,\infty)\times[0, 2\pi)} f(r \cos \theta, r \sin \theta) \, r \, dr \, d\theta
	$$
      \item
	\emph{(Higher dimensional version.)}
	Let $f \in L^1(\R^d)$.
	Let $S_1 = \{ y \in \R^d \st \abs{y} = 1 \}$ be the $d-1$ dimensional sphere of radius $1$.
	Show that there exists a unique measure $\sigma$ on $S_1$ such that
	$$
	  \int_{\R^d} f(x) \, dx = \int_{r \in [0, \infty)} \int_{y \in S_1} f(r y) \, r^{d-1} \, d\sigma(y) \, d\lambda(r).
	$$
	\hintb{
	  For $A \in \mc B(S_1)$ define $\sigma(A) = \lambda(A^*)$ where $A^* = \{ r x \st x \in A, r \in [0, 1] \}$.
	  Now for any $B \in \mc B(S_1)$ prove the desired equality when $f = \Chi*{A}$ where $A = \{ r x \st a < r < b, x \in B \}$.
	}
      \end{parts}

  \fullwidth{\noindent\emph{Optional problems, and details in class I left for you to check.}}

  \itemopt
    Show that the arbitrary union of closed (non-degenerate) cells is Lebesgue measurable.

  \itemopt
    Find an example of $E \in \mc L(\R^d)$ and $x \in \R^d$ such that $\lim_{r \to 0} \frac{\lambda(E \cap B(x, r))}{\lambda(B(x,r))}$ does not exist.

  \itemopt
    Suppose $f : \R \to \R$ is measurable.
    Let $\alpha, \beta > 0$ with $\alpha / \beta \not\in \Q$.
    If $f$ has period $\alpha$, and also has period $\beta$ (i.e. for all $x \in \R$, $f(x) = f(x + \alpha) = f(x + \beta)$), then show that $f$ is constant almost everywhere. (But $f$ need not be constant everywhere!)

  \itemopt
    We say the family $\{E_r\}$ \emph{shrinks nicely} to $x \in \R^d$ if there exists $\delta > 0$ such that for all $r$, $E_r \subset B(x, r)$ and $\lambda(E_r) > \delta \lambda(B(x,r))$.
    If $\{E_r\}$ shrinks nicely to $x$, show that $\lim \frac{1}{\lambda(E_r)} \int_{E_r} f = f(x)$ for all Lebesgue points of $f$.

  \itemopt
    If $f \in L^1(\R^d)$, show that $Mf(x) \geq \abs{f(x)}$ at all Lebesgue points of $f$.

  \itemopt
    If $f:[a,b] \to \R$ is absolutely continuous, then show that $f$ is of bounded variation, and that the variation is absolutely continuous.
    Conclude $f$ can be written as the difference of two monotone absolutely continuous functions.

  \itemopt
    Let $U, V \subset \R^d$ be open and $\varphi: U \to V$ be $C^1$ and injective.
    If $x_0 \in U$ and $\grad \varphi(x_0)$ is not invertible, show that
    $$
      \lim_{r \to 0} \frac{\lambda( \varphi( B(x_0, r) ) ) }{\lambda(B(x_0, r)) } = 0.
    $$


\end{assignment}%}}}
\clearpage

\begin{assignment}{Wed 12/04}{never}%{{{ HW15
  \item
    \begin{parts}
      \item
	If $\int_{\R^d} (1 + \abs{x}) \abs{f(x)} \, dx < \infty$, show that $\hat f$ is differentiable and $\del_j \hat f(\xi) = -2\pi i ( x_j f(x) )^{\hatsym}(\xi)$.
	\hintb[Note]{$(x_j f(x))^{\hatsym}(\xi)$ means $\hat g(\xi)$, where $g(x) = x_j f(x)$.}
      \item
	If $f \in C^1_0(\R^d)$, and $\grad f \in L^1$ show that $(\del_j f)^{\hatsym}(\xi) = +2 \pi i \xi_j \hat f(\xi)$.
      \item
	Show that the mapping $f \mapsto \hat f$ is a bijection in the Schwartz space.
    \end{parts}

  \item
    If $\mu$ is a finite Borel measure on $\R^d$ define $\hat \mu(\xi) = \int e^{-2\pi i \ip{x}{\xi}} \, d\mu(x)$.
    If $\hat \mu(\xi) = 0$ for all $\xi$, show that $\mu = 0$.
    \hintb{Show that $\int f \, d\mu = 0$ for all $f \in \mc S$.}


  \item
    For $f \in L^1$, the formula $\hat f(\xi) = \int f(x) e^{-2\pi i \ip{x}{\xi}}$ allows us to prove many identities: E.g. $(\delta_\lambda f)^\wedge (\xi) = \hat f(\lambda \xi)$, etc.
    For $f \in L^2$, the formula  $\hat f(\xi) = \int f(x) e^{-2\pi i \ip{x}{\xi}}$ is no longer valid, as the integral is not defined (in the Lebesgue sense).
    However, most identites remain valid, and can be proved using an approximation argument.
    I list a couple here.
    \begin{parts}
      \item
	For $f \in L^1$ we know $(\tau_x f)^\wedge(\xi) = e^{-2\pi i \ip{x}{\xi}} \hat f(\xi)$.
	Prove it for $f \in L^2$.

      \item
	Similarly, show that $(\delta_\lambda f)^\wedge (\xi) = \hat f(\lambda \xi)$ for all $f \in L^2$.

      \item
	Let $F$ denote the Fourier transform operator (i.e.\ $Ff = \hat f$), and $R$ denote the reflection operator (i.e.\ $Rf(x) = f(-x)$).
	Note that our Fourier inversion formula (for $f \in L^1$, $\hat f \in L^1$) is exactly equivalent to saying $F^2f = Rf$.
	Prove $F^2 f= Rf$ for all $f \in L^2$.
    \end{parts}

  %\item Here's another method of computing the Fourier transform of the Gaussian:  Let $\varphi(x) = \frac{1}{\sqrt{2\pi}} e^{-x^2/2}$ for $x \in \R$. Use the identity $\varphi'(x) = -x \varphi(x)$ to obtain an ODE for $\hat\varphi$. Solve this ODE and show $\hat\varphi = \sqrt{2\pi}\varphi$. Now use Fubini to compute the Fourier transform of $\frac{1}{(2\pi)^{n/2}} e^{-\abs{x}^2/2}$ for $x \in \R^n$.

  \item
    \emph{(Uncertainty principle)} Suppose $f \in \mathcal S(\R)$.
    Show that
    $$
      \Bigl( \int_{\R} \abs{x f(x)}^2 \, dx \Bigr) \Bigl(\int_{\R} \abs{\xi \hat f(\xi)}^2 \, d\xi\Bigr) \geq \frac{1}{16\pi^2} \norm{f}_{L^2}^2 \norm{\hat f}_{L^2}^2
    $$
    \noteb{%
      This illustrates a nice localisation principle about the Fourier transform.
      The first integral measures the spread of the function $f$.
      The second the spread of the Fourier transform $\hat f$.
      Here you show that this product is bounded below!
      The proof, once you know enough Physics, reduces to the above inequality.
      
      Hint: Consider $\int_\R x f(x) f'(x) \, dx$.
      }

  \item
    \emph{(Central limit theorem)}
    Let $f \in L^1(\R)$ be such that $f \geq 0$ and $\int x^2 f(x) \, dx < \infty$.
    Define $g_n = (f * \cdots * f)$ ($n$-times), and $h_n(x) = \delta_{1/\sqrt{n}} g_n (x) = \sqrt{n} g_n( \sqrt{n} x )$.
    Show
    $$
      \hat h_n(\xi) \xrightarrow{n \to \infty}
	% \frac{1}{\sqrt{2\pi} \sigma} \exp\bigl(-\frac{(x - \mu)^2}{2 \sigma^2} \bigr),
	\exp\bigl(-2\pi i \mu \xi - 2 \pi^2 i \sigma^2 \xi^2 \bigr),
    $$
    where $\mu = \int x f(x) \, dx$ and $\sigma^2 = \int (x - \mu)^2 f(x) \, dx$.
    \noteb{The central limit theorem says that tabulating results of a large number of independent trials of any experiment produces a ``bell curve''.
    The key step in the proof, which you will no doubt see next semester, is showing that any function convoved with itself often enough looks like a Gaussian.}

  \item
    \emph{(Sobolev spaces)}
    For $f \in L^2(\R^d)$ and $s \geq 0$ define
    $$
      \norm{f}_{H^{s}}^2 = \int (1 + \abs{\xi}^s)^2 \abs{\hat f(\xi)}^2 \, d\xi,
      \qquad\text{and}\qquad
      H^s = \{ f \in L^2 \st \norm{f}_{H^{s}} < \infty \}.
    $$
    Intuitively, we think of $H^s$ as the space of functions with ``$s$'' ``weak-derivatives'' in $L^2$. (This will be formalized in your functional analysis course.)
    \begin{parts}
      \item
	If $f\in C^n_0(\R^d)$ and $D^\alpha f \in L^2$ for all $\abs{\alpha} < n$, then show that $f \in H^n(\R^d)$.

      \item
	Let $s \in (0, 1)$ and $f \in L^2(\R^d)$.
	Show that $f \in H^s(\R^d)$ if and only if
	$$
	  \int_0^\infty \paren[\Big]{ \frac{ \norm{ \tau_h f - f }_{L^2} }{h^s} }^2 \, \frac{dh}{h} < \infty.
	$$
    \end{parts}

  \item
    \emph{(Sobolev embedding)}
    If $n \in \N$ and $f \in H^s(\R^d)$ for $s > n + \frac{d}{2}$ then show that $f \in C^n$, and further the inclusion map $H^s \to C^n$ is continuous.

  \item
  \begin{parts}
    \item
      \emph{(Elliptic regularity)}
      Let $Lu = \sum a_{ij} \del_i \del_j u + \sum b_i \del_i u + c u$, where $a_{ij}, b_i, c$ are constants.
      Suppose $\exists \lambda >  0$ such that $a_{ij} = a_{ji}$ and $\abs{\sum a_{ij} \xi_i \xi_j} \geq \lambda \abs{\xi}^2$ for all $\xi \in \R^n$ (this assumption is called ellipticity).
      If $f \mc S$ and $u, \del_i u, \del_i\del_j u \in L^2 \cap C^0$ are such that $Lu = f$, show that $u \in C^\infty$.
      \noteb{%
	To emphasize why this is surprising, choose for example $L = \lap$.
	Then $\lap u = f$ makes no mention of a mixed derivative $\del_1 \del_2 u$.
	Yet, all such mixed derivatives exist and are smooth.
	Hint: If $f \in H^s$ show that $u \in H^{s + 2}$.}
    \item
      Show by example that the previous subpart is false without the ellipticity assumption.
  \end{parts}


  \item
    \emph{(Trace theorems)}
    Let $p \in \R^m$ be fixed.
    Given $f:\R^{m + n} \to \R$ define $S_p f:\R^n \to \R$ by $S_p f(y) = f(p, y)$.
    \begin{parts}
      \item
	Let $s > m/2$, and $s' = s - m/2$.
	Show that there exists a constant $c$ such that $\norm{S_p f}_{H^{s'}(\R^n)} \leq c \norm{f}_{H^s(\R^{m+n})}$.
      \item
	Show that the section operator $S_p$ extends to a continuous linear operator from $H^s(\R^{m+n})$ to $H^{s'}(\R^n)$.
      \noteb{%
	Given an arbitrary $L^2$ function on $\R^{m+n}$ it is of course impossible to restrict it to an $m$-dimensional hyper-plane.
	However, if your function has more than $n/2$ ``Sobolev derivatives'', then you can make sense of this restriction, and the restriction still has $s - n/2$ ``Sobolev derivatives''.}
    \end{parts}

  \item
    \emph{(Reliech Lemma)}
    Let $K \subset \R^d$ be compact, $0 \leq s_1 < s_2$, and suppose $\{f_n\}$ are a sequence of functions supported in $K$.
    If the sequence $\{f_n\}$ is bounded in $H^{s_2}$, then show that it has a convergent subsequence in $H^{s_1}$.
    \noteb{This is the generalization of the Arzella-Ascolli theorem in this context.}
\end{assignment}%}}}
\end{document}
