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\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 09/05}{Wed 09/12}%{{{HW1
  \uplevel{\note{Following the notation of Cohn, I use $\lambda$ to denote the Lebesgue measure.}}
  \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
    \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 an orthogonal linear transformation, and $A \in \mc L$.
	Show that $T(A) \in \mc L$ and $\lambda(T(A)) = \lambda(A)$.
	\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, $\delta > 0$ and define $\mc E_\delta = \{B(x, r) \st x \in X, r \in (0, \delta) \}$.
	Given $\alpha > 0$ define
	$
	  \rho (B(x, r)) = c_\alpha r^\alpha,
	$
	where $c_\alpha = \pi^{\alpha/2} / \Gamma( 1 + \alpha /2 )$ is a normalization constant.
	Let $H_{\alpha,\delta}^*$ be the outer measure obtained with this choice of $\rho$ 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 $\alpha$.}
	\noteb{Don't reprove Caratheodory.}

      \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 $H_d = \lambda$, 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}



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

  \itemopt We saw in class $\ell(I) = I$ for closed cells. Show it for arbitrary cells.

  \itemopt Show that $m^*(a + E)  = m^*(E)$ for all $a \in \R^d$, $E \subset \R^d$.

  \itemopt Show that the arbitrary intersection of $\sigma$-algebras on $X$ is also a $\sigma$-algebra.

  \itemopt Verify that the counting measures and delta measures are measures.
  \itemopt When proving Caratheodory, we proved in class $\Sigma$ is a $\sigma$-algebra, and that $\mu^*|_\Sigma$ is \emph{finitely} additive. Show that $\mu^*|_{\Sigma}$ is countably additive. 
\end{assignment}%}}}
\newpage
\begin{assignment}{Wed 09/12}{Wed 09/19}%{{{ HW2
  \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 Here's an alternate (cleaner) approach to proving $\mc L = \mc B_\lambda$. We do it by proving a stronger statement than necessary.
    \begin{parts}
      \iffalse%{{{
      \item Suppose $G_\delta^1, G_\delta^2, \dots$  are a countable collection of $G_\delta$'s in $\R^d$ such that $G_\delta^n \subset \overline{B(0, n+1)} - B(0, n-1)$. Show that $\bigcap_1^\infty G^n_\delta$ is also a $G_\delta$.
      \item If $A \in \mc L(\R^d)$ show that there exists $F, G$ such that $F$ is an $F_\sigma$, $G$ is a $G_\delta$, $F \subset A \subset G$ and $\lambda(G - F) = 0$. Conclude $\mc L = \mc B_\lambda$. \noteb{This is a stronger statement than what is required to show $\mc L = \mc B_\lambda$.}
      \fi%}}}
      \item
	If $A \in \mc L(\R^d)$ show that for any $\eps > 0$ there exists two sets $C, U$ such that $C \subset A \subset U$, $C$ is closed, $U$ is open and $\lambda( U - C ) < \epsilon$.
      \item
	For $A \in \mc L(\R^d)$, show that that there exists an $F_\sigma$, $F$ and a $G_\delta$, $G$ such that $F \subset A \subset G$ and $\lambda(G - F) = 0$. Conclude $\mc B_\lambda = \mc L$.
    \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}

  \item
    Find $E \in \mc B(\R)$ so that for all $a < b$, we have $0 < \lambda(E \cap (a, b)) < b - a$.

  \fullwidth{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}.}
  \item \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 $\mu_0$.

  \fullwidth{\noindent\emph{Optional problems, and details in class I left for you to check.}}
  \itemopt
    Prove any open subset of $\R^d$ is a countable union of cells. Conclude $\mc L \supset \mc B$.

  \itemopt
    Show that the cardinality $\mc B(\R)$ is the same as that of $\R$, however, the cardinality of $\mc L(\R)$ is the same as that of $\mc P(\R)$.
    Conclude $\mc B(\R) \subsetneq \mc L(\R)$. \noteb{There are of course other ways to prove this.}

  \itemopt
    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 $\mu(A_1) < \infty$. Given an example to show this is not true if $\mu(A_1) = \infty$.

  \itemopt
    We saw in class $\lambda(A) = \sup \{ \lambda(K) \st K \subseteq A \mathop \& K \text{ is compact} \}$ for all bounded sets $A \in \mc L$. Prove it for arbitrary $A \in \mc L$.


  \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 $\lambda(B) = 0$.

\end{assignment}%}}}
\newpage

\begin{assignment}{Wed 09/19}{Wed 09/26}%{{{ HW3
  \item
    Let $X$ be a topological space, and $\mu$ be a regular Borel measure on $X$.
    Show that $X$ has a \emph{maximal} open set of measure $0$.
    Namely, show that there exists $U \subset X$, such that $U$ open set, $\mu(U) = 0$ and further for any open set $V \subset X$ with  $\mu(V) = 0$, we must have $V \subset U$.
    \noteb{The complement of $U$ is defined to be the \emph{support} of the measure $\mu$, and denoted by $\supp(\mu)$.}

  \item
    Let $\Sigma \supset \mc B(\R^d)$, and $\mu$ be a regular measure on $(\R^d, \Sigma)$.
    Suppose $A \in \Sigma$ is $\sigma$-finite (i.e. $A = \cup_1^\infty A_n$, and $\mu(A_n) < \infty$).
    Show that $\mu(A) = \sup\{ \mu(K) \st K \subset A \text{ is compact}\}$.
    \noteb{This remains true if we replace $\R^d$ with any Hausdorff space.}

  \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
    Let $\kappa \in (0, 1)$.
    Does there exist $E \in \mc L(\R)$ 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 or find a counter example.
    \noteb{I'm aware that this looks suspiciously like a homework problem you already did.
    Also, this problem has a short, elegant solution using only what we've seen in class so far.}

  \item
    For $i \in \{1, 2\}$, let $(X_i, \Sigma_i, \mu_i)$ be two measure spaces with $\mu_i(X_i) < \infty$.
    Define $\Sigma_1 \otimes \Sigma_2 = \sigma\{ A_1 \times A_2 \st A_i \in \Sigma_i \}$.
    \begin{parts}
      \item
	Let $x_1 \in X_1$ and $A \in \Sigma_1 \otimes \Sigma_2$. Let $S_{x_1}(A) = \{ x_2 \in X_2 \st (x_1, x_2) \in A \}$, and $T_{x_2}(A) = \{ x_1 \in X_1 \st (x_1, x_2) \in A \}$. Show that $S_{x_1}(A) \in \Sigma_2$ and $T_{x_2}(A) \in \Sigma_1$.
      \item
	If $A \in \mc P(X_1 \times X_2)$ is such that for all $x_i \in X_i$, $S_{x_1}(A) \in \Sigma_2$ and $S_{x_2}(A) \in \Sigma_1$.
	Must $A \in \Sigma_1 \otimes \Sigma_2$?
      \item
	Show that there exists a measure $\nu$ on $(X_1 \times X_2, \Sigma_1 \otimes \Sigma_2)$ such that for all $A_i \in \Sigma_i$ we have $\nu(A_1 \times A_2) = \mu_1(A_1) \mu_2(A_2)$.
    \end{parts}

  \item
    \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 $\mc A$.}

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

  \itemopt
    Let $X$ be a second countable locally compact Hausdorff space, and $\mu$ be a Borel measure on $X$ that is finite on compact sets.
    Show that $\mu$ is regular.

  \itemopt Is any $\sigma$-finite Borel measure on $\R^d$ regular?
  \itemopt Show that any $\lambda$-system that is also a $\pi$-system is a $\sigma$-algebra.
  \itemopt If $\Pi$ is a $\pi$-system, then $\lambda(\Pi) = \sigma(\Pi)$. (We only proved $\lambda(\Pi) \subseteq \sigma(\Pi)$.)
\end{assignment}%}}}
\newpage

\begin{assignment}{Wed 09/26}{Wed 10/03}%{{{ HW4
  \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}

  \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, 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 $X$ be a metric space, $\Sigma \supset \mc B(X)$ a $\sigma$-algebra on $X$, and  $\mu$ a regular finite measure on $(X, \Sigma)$.
    Let $f:X \to \R$ be measurable.
    \begin{parts}
      \item For any $\eps > 0$ and $i \in \N$, show that there exists finitely many disjoint compact sets $\{K_{i,j} \st \abs{j} \leq N_i \}$ such that
	$$
	  \mu\Bigl(X - \union_{j=-N_i}^{N_i} K_{i, j}\Bigr) < \frac{\eps }{ 2^i},
	  \quad\text{and}\quad
	  f(K_{i,j}) \subset \bigl[\frac{j}{2^i}, \frac{j+1}{2^i}\bigr)
	$$

      \item
	\emph{(Lusin's Theorem)}
	For any $\eps > 0$ show that there exists $K_\epsilon \subset X$ compact such that $f:K_\eps \to \R$ is \emph{continuous}, and $\mu(X - K_\eps) < \eps$.
	\hintb{Let $K_\eps = \intersect_{i = 1}^\infty \union_{\abs{j} \leq N_i} K_{i,j}$. Define $g_i:K_\eps \to \R$ by $g_i(x) = j / 2^i$ if $x \in K_{i,j}$ and $\abs{j} \leq N_i$. Show $g_i:K \to \R$ is continuous and $(g_i) \to f$ uniformly on $K_\eps$.}
      \uplevel{A standard extension theorem now shows that for any $f:X \to \R$ measurable and $\epsilon > 0$, there exists $g_\epsilon :X \to \R$ \emph{continuous} such that $\mu\{f \neq g_\eps \} < \epsilon$.}
    \end{parts}


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

\end{assignment}%}}}
\newpage

\begin{assignment}{Wed 10/03}{Wed 10/10}%{{{ HW5
  \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}

  \item
    \begin{parts}
      \item Let $(X, \Sigma, \mu)$ be a complete measure space, $f:X \to [-\infty,\infty]$ be measurable and suppose $\int_X f \, d\mu$ is defined. If $g:X \to [-\infty,\infty]$ is such that $f = g$ a.e., then show $\int_X f \, d\mu = \int_X g \, d\mu$.
    \uplevel{\vspace{-.8\baselineskip}All the convergence theorems we've seen so far hold if we replace pointwise convergence with a.e.\ convergence. I ask you to prove one below; you should verify the others on your own.}
      \item Suppose $(f_n)$ is a sequence of measurable functions, $f_n \geq 0$ a.e., and $(f_n) \to f$ a.e.\ on $E$. Show that $\liminf \int_E f_n \, d\mu \geq \int_E f \, d\mu$.
    \end{parts}

  \item Let $f:\R^d \to [-\infty, \infty]$ be an integrable function such that $\int_I f \, d\lambda = 0$ for all cells $I$. Must $f = 0$ a.e.? Prove or find a counter example.

  \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\label{qnLinChangeOfVar}
    \begin{parts}
      \item Let $T:\R^d \to \R^d$ be linear, and $A \in \mc L$. Show that $\lambda(T(A)) = \abs{\det(T)} \lambda(A)$.
	\hintb{Check it separately for $\det(T) = 0$. For $\det(T) \neq 0$, write $T$ as a product of elementary transformations, and check the result for cells. (This should have been on HW1, but I `inadvertently' added the assumption that $T$ was orthogonal.)}
      \item\label{ptLinChangeOfVar} \emph{(Linear change of variable)} Let $f: \R^d \to \R$ be integrable, $T:\R^d \to \R^d$ 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{Optional problems, and details in class I left for you to check.}}
  \itemopt For simple functions, check that $\int_E s$ is well defined.
  \itemopt For positive functions check $f \leq g \implies \int_E f \leq \int_E g$.
  \itemopt For arbitrary integrable functions, check $\int_E \alpha f \, d\mu = \alpha \int_E f \, d\mu$.
  \itemopt If $\int_X f \, d\mu < \infty$, then show $f < \infty$ a.e.
  \itemopt If $\int_X \abs{f} \, d\mu = 0$, then show that $f = 0$ a.e.
  \itemopt Prove the following generalization of Fatou's Lemma: If $f_n \geq 0$ are measurable, then $\liminf \int_E f_n \, d\mu \geq \int_E \liminf f \, d\mu$.
  \itemopt Finish the proof of showing $\int_X g \, d\mu = \int_Y g \circ f \, d\mu_{f\inv}$. Use this to give a quick proof that $\int_{\R^d} f(x + y) \, dx = \int_{\R^d} f(x) \, dx$. (This trick also helps with~Q\ref{qnLinChangeOfVar}(\ref{ptLinChangeOfVar}).)
\end{assignment}%}}}
\newpage

\begin{assignment}{Wed 10/10}{Never}%{{{
  \fullwidth{In view of your Midterm on 10/17, this homework is optional.}
  \itemopt If $\mu(E) = 0$, and $f:E \to [-\infty, \infty]$ is any measurable function, then show directly from the definition that $\int_E f \, d\mu = 0$.

  \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 $(X, \Sigma, \mu)$ be a measure space and $f:X \to Y$ some function.
    Define $\tau = \{A \subset Y \st f(f\inv(A)) = A, \& f\inv(A) \in \Sigma\}$.
    For $A \in \tau$, define $\mu_f(A) = \mu( f(A))$.
    Show that $(Y, \tau, \mu_f)$ is a measure space.
    If $g : Y \to [-\infty, \infty]$ is integrable, can you write $\int_Y g \, d\mu_f$ in terms of an integral over $X$ with respect to $\mu$?

  \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
    Let $f \sim g$ if $\mu\{f \neq g\} = 0$. For $p \in [1, \infty)$, define
    $$
      \mc L^p = \bigl\{ f: X \to \R \text{ measurable, such that } \int_X \abs{f}^p \, d\mu < \infty \bigr\}
      \quad\text{and}\quad
      L^p = \mc L^p / \sim.
    $$
    For $f \in L^p$, pick any $f' \in f$, and define $\norm{f}_p = (\int_X \abs{f'}^p \, d\mu)^{1/p}$.
    Show that this is well defined and satisfies all the axioms of a Banach space except completeness and the triangle inequality.
    \noteb{Completeness and the triangle inequality are of course true but are harder to prove. I will prove them in class.}

  \itemopt Show that $f \leq \esssup_X f$ almost everywhere.

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

  \itemopt Prove H\"older's inequality if $p=1$ or $p = \infty$.

  \itemopt
    \begin{parts}
      \item Prove $\norm{f}_1 = \sup_{\norm{g}_\infty = 1} \int_X fg \, d\mu$.
      \item If $X$ is $\sigma$-finite, then show $\norm{f}_\infty = \sup_{\norm{g}_{1} = 1 } \int_X fg \, d\mu$.
    \end{parts}

  \itemopt
    \begin{parts}
      \item \emph{(Young's inequality)} Let $x, y \in \R$, $p, q \in (1, \infty)$ with $\frac{1}{p} + \frac{1}{q} = 1$.
	Show that $\abs{xy} \leq \frac{\abs{x}^p}{p} + \frac{\abs{y}^q}{q}$, and equality holds if and only if $\abs{x}^p = \abs{y}^q$.
      \item Use Young's inequality to give an alternate proof of H\"older's inequality.
    \end{parts}

  \itemopt
    \begin{parts}
      \item Suppose $\varphi$ is a strictly convex function and $\mu(X) = 1$. For what functions can you have equality in Jensen's inequality. Namely, when is $\varphi(\int_X f \, d\mu) = \int_X \varphi \circ f \, d\mu$?
      \item For what functions $f, g$ can you have equality in H\"older's inequality?
    \end{parts}
\end{assignment}%}}}
\newpage

\begin{assignment}{Wed 10/17}{Wed 10/24}%{{{ HW7
  \item
    \begin{parts}
      \item If $\mu(X) <  \infty$, $1 \leq p <  q$, 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 $p, q, r \in [1, \infty]$ 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}

  \item
    For any $p \in [1, \infty]$, show that simple functions are dense in $L^p(X)$.
    That is, for any $\epsilon > 0$, $f \in L^p(X)$ show that there exists a simple function $s \in L^p(X)$ such that $\norm{f - s}_p < \epsilon$.

  \item
    Let $X$ be a metric space and $\mu$ be a regular Borel measure on $(X, \mc B(X))$. 
    Assume further and $X = \bigcup_1^\infty U_n$, where  $U_n$ is open, $\overline{U}_n$ is compact, and $\overline{U}_{n} \subset U_{n+1}$.
    \begin{parts}
      \item
	For any $p \in [1, \infty)$, show that continuous compactly supported functions are dense in $L^p(X)$.
	\noteb{You may assume the \emph{Tizete extension theorem} from topology, which guarantees (in a more general situation) that if $C \subset X$ is closed and $f:C\to \R$ is continuous, then there exists a continuous function $F:X \to \R$ such that $F = f$ on $C$.}
      \item Is the previous part true for $p=\infty$? Prove or find a counter example.
    \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 happnes for $p = \infty$?
    \end{parts}



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

  \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 Let $0 < p < q < \infty$. Then $L^p \not\subset L^q$ iff $X$ contains sets of arbitarily small, positive, measure. Also, $L^q \not\subset L^p$ iff $X$ contains sets of arbitarily large (but finite) measure.

  \itemopt \emph{(Vitali's convergence theorem.)}
    Let $f_n, f \in L^1$. Show that $(f_n) \to f$ in $L^1$ if and only if
    \begin{inparaenum}[(1)]
      \item $(f_n) \to f$ in measure,
      \item $\{f_n\}$ is uniformly integrable, and
      \item For all $\eps > 0$ there exists $F \in \Sigma$ with $\mu(F) < \infty$ such that $\int_{F^c} \abs{f_n} < \eps$.
    \end{inparaenum}
    \noteb{I proved the forward direction in class, and sketched the reverse. Fill in the details of the reverse.}

\end{assignment}%}}}
\newpage

\begin{assignment}{Wed 10/24}{Wed 10/31}%{{{HW8
  \item
    Suppose $\Sigma = \sigma(\mc C)$, where $C \subset \mc P(X)$ 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 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.
  \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 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 signed measurs on $X$, 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}



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

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

  \itemopt
    Say $\mu$ is a signed measure, and $A_i \in \Sigma$ are pariwise disjoint.
    If $\abs{\mu(\union A_i)} < \infty$, then must $\sum_1^\infty \abs{\mu(A_i)} < \infty$?
    Prove, or find a counter example.

  \itemopt If $g \in L^1(X, \mu)$, let $\nu(A) = \int_A g$.
    Show that $\nu$ is a signed measure on $X$, and $\int f \, d\nu = \int fg \, d\mu$.



  \itemopt
  \begin{parts}
    \item
      Prove the Hanh decomposition is unique, up to sets of measure $0$.
      \noteb{That is show $X = P_1 \cup N_1$ and $X = P_2 \cup N_2$, then $P_2 = P_1 - A \cup B$, where all subsets of $A, B$ have measure $0$, and a similar statement for $N$.}

    \item
      Show that the measures $\mu^+$ and $\mu^-$ we defined in class are independent of the Hanh decomposition used to define them.

    \item
      We say $\mu$ and $\nu$ are \emph{mutually singular} if $X = A \cup B$ where $A, B \in \Sigma$ with $A \cap B = \emptyset$, and for all measurable $A' \subset A$, $B' \subset B$ we have $\mu(A') = 0$ and $\nu(B') = 0$.
      Show that the Jordan decomposition is unique if the measures are assumed to be mutually singular.
  \end{parts}

  \itemopt If $\mu = \mu_1 - \mu_2$ where $\mu_1$ and $\mu_2$ are positive, show that $\mu_1 \geq \mu^+$ and $\mu_2 \geq \mu^-$.


\end{assignment}%}}}
\newpage

\begin{assignment}{Wed 10/31}{Wed 11/07}% HW9 {{{

  \item
    \begin{parts}
      \item Let $\nu$ be a finite (positive) measure. Prove $\nu \ll \mu \iff \forall \eps > 0, \exists \delta > 0 \suchthat \mu(A) < \delta \implies \nu(A) < \eps$. \noteb{This sort of justifies the name ``absolutely continuous''.}
      \item
	Is the previous part true if $\nu$ is not finite?
	Prove or find a counter example.
    \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 expection} 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}

  \item
    Let $\mu$ be a positive measure and $\nu$ a finite signed measure.
    Let $\nu = \nu_{\text{ac}} + \nu_\text{s}$ be the Lebesgue decomposition of $\nu$.
    Show that $\norm{\nu} = \norm{\nu_{\text{ac} }} + \norm{\nu_\text{s}}$.

  \item Let $\mu$ be $\sigma$-finite, and define $\varphi:L^\infty \to (L^1)^*$ by $\varphi_g(f) = \int_X f g \, d\mu$.
	Show that $\varphi$ is a bijective linear isometry.
	\noteb{In this sense we say $L^\infty$ is the dual of $L^1$.
	The reverse identification is not true in general: $L^1$ can be identified with an \emph{subspace} of $(L^\infty)^*$, but need not be all of it.
	The proof of this requires the \emph{Hanh-Banach} theorem.}

  \fullwidth{\noindent\emph{Optional problems, and details in class I left for you to check.}}
  \itemopt
    Show that the Radon Nicodym theorem is not true if $\nu$ is $\sigma$-finite, but $\mu$ is not.
    Where does the proof we had in class break down if $\mu$ is not $\sigma$-finite?
  \itemopt Finish the proof of the Lebesgue decomposition (existence and uniqueness) when $\nu$ is $\sigma$-finite.

  \itemopt If $X, Y$ are Banach spaces show that $B(X,Y)$ with operator norm is a Banach space.

  \itemopt
    Let $p \in (1, \infty]$, $1/p + 1/q = 1$, and $c < \infty$.
    If $g$ is a measurable function such that $\sup \{ \int_X s g \st s \text{ is simple, and } \norm{s}_p \leq 1 \} \leq c$, show that $g \in L^q$ and $\norm{g}_q \leq c$.

  \itemopt
    If $\mu$ is a finite signed measure, show that $\abs{\int f \, d\mu} \leq \int \abs{f} d\abs{\mu}$.
\end{assignment}%}}}
\newpage

\begin{assignment}{Wed 11/07}{Wed 11/14}%{{{ HW10
  \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 $\int_{X\times Y} f \, d(\mu \times \nu)$ 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 $\mc \pi = (\Sigma \otimes \tau)_{\mu \times \nu}$ denote the completion of $\Sigma \otimes \tau$ with respect to $\mu \times \nu$.
  \begin{parts}
    \item
      Show that $\Sigma \otimes \tau$ need not be $\mu \times \nu$-complete (i.e.\ $\pi \supsetneq \Sigma \otimes \tau$ 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{You should verify that when $Y = \{1, 2\}$ with the counting measure, the above is exactly Minkowski's triangle inequality.}

  \item
    For $p \in [1, \infty)$ define $\norm{f}_{L^{p, \infty}} = \sup\{ \lambda \mu\{ \abs{f} > \lambda\}^{1/p} \st \lambda > 0 \} \}$, and the weak $L^p$ space (denoted by $L^{p, \infty}$) by $L^{p, \infty} = \{ f \st \norm{f}_{L^{p, \infty}} < \infty \}$.
    \noteb{As usual, we use the convention that functions that are equal almost everywhere are identified with each other.}
  \begin{parts}
    \item
      If $f \in L^p$, show $f \in L^{p, \infty}$ and $\norm{f}_{L^{p, \infty}} \leq \norm{f}_{p}$.
      Is the converse true?

    \item
      If $f, g \in L^{p, \infty}$, show that $f + g \in L^{p, \infty}$.
      Show further that $\norm{f + g}_{L^{p, \infty}} \leq c (\norm{f}_{L^{p, \infty}} + \norm{g}_{L^{p, \infty}})$ for some constant $c$ independent of $f, g$.
      \noteb{Thus $\norm{\cdot}_{L^{p, \infty}}$ is called a quasi-norm, and $L^{p, \infty}$ is called a quasi-Banach space.}

    \item If $\mu$ is $\sigma$-finite, $1 \leq p < q < r < \infty$ and $f \in L^{p,\infty} \cap L^{r, \infty}$ then show $f \in L^q$.
  \end{parts}


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

  \itemopt
    Show that the Lebesgue measure on $\R^{m+n}$ is the product of the Lebesgue measurs on $\R^m$ and $\R^n$ respectively.
    \noteb{Note, you've previously seen that $\mc L(\R^{m+n}) \supsetneq \mc L(\R^m) \otimes \mc L(\R^n)$; however $\mc B(\R^{m+n}) = \mc B(\R^m) \otimes \mc B(\R^n)$.}

  \itemopt
    For $E \in \Sigma \otimes \tau$, define $f_E(x) = \nu(S_x(E))$ and $g_E(y) = \mu(T_y(E))$. Show that $f:X \to \R$ and $g:Y \to \R$ are measurable.
    \hintb{First assume $\mu, \nu$ are finite. Let $\Lambda = \{E \st f_E, g_E \text{ are measurable} \}$. Show that $\Lambda$ is a $\lambda$-system, and $\Lambda$ contains all rectangles.}

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

\end{assignment}%}}}
\newpage

\begin{assignment}{Wed 11/14}{Wed 11/21}%{{{ HW 11
  \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
    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 Show that $\mc S$ is dense subset of $L^p$ for $p \in [1, \infty)$.
      \item Show that $C^\infty_c$ is a dense subset of $L^p$ for $p \in [1, \infty)$.
    \end{parts}

  \item
    \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 $f, g \in L^2([0,1])$, show that $(fg)^{\hatsym}(n) = \hat f * \hat g (n) \defeq \sum_{m \in \Z} \hat f(m) \hat g(n-m)$.
    \end{parts}
	

  \item
    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.
    \begin{parts}
      \item
	If $f \in C\per[0, 1]$, show that $(\sigma_N f) \to f$ uniformly.
	\noteb{Here $C\per[0, 1] = \{f \in C(\R) \st \tau_1 f = f \}$ denotes all continuous functions with period $1$.}

      \uplevel{\vspace{-1em}If $f \in C\per[0, 1]$, it turns out that the partial sums $S_N f$ need not converge to $f$ even pointwise.
      (In fact, there exist many $f \in C\per([0,1])$ such that $S_N f$ is divergent on a dense $G_\delta$ in $[0,1]$.)
      If, however, $f$ is a little bit better than continuous, then the Fourier series of $f$ converges to $f$ pointwise.
      }
      \item
	Let $f \in C\per([0, 1])$ and $\alpha > 0$. If $\sup_{x,y} \frac{\abs{f(x) - f(y)} } {\abs{x - y}^\alpha} < \infty$, then show that $(S_N f) \to f$ pointwise, as $N \to \infty$. \noteb{In fact, $(S_N f) \to f$ uniformly.}

    \end{parts}


  \item Let $\mu$ be a finite signed Borel measure on $[0, 1]$.
    If $\forall n \in \Z \; \hat \mu(n) = 0$, show $\mu = 0$.

  \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 11/21}{Wed 11/28}%{{{ HW 12
  \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 part of the Sobolev embedding theorem.}
    \end{parts}

  \item
    Let $f \in L^2([0,1])$.
    Show that there exists $u \in C^\infty( [0,1] \times (0, \infty) )$ such that $u(0, t) = u(1, t)$, $\lim_{t \to 0^+} \norm{u(\cdot, t) - f(\cdot)}_2 = 0$, and $\delt u - \delx^2 u =0$.
    \hintb{You may assume the result of the optional problems.}

  \item
    Finish the change of variable proof using the following approach.
    Recall $U, V \subset \R^d$ are open connected sets, and $\varphi:U \to V$ is a $C^1$ bijection whose inverse is also $C^1$.
    Our aim is to show $\lambda(\varphi(A)) = \int_A \abs{\det \grad \varphi} \, d\lambda$ for all $A \subset U$ Borel.

    Assume first that $\varphi, \varphi\inv$ are both \emph{uniformly} $C^1$, and $U, V$ are bounded.
    In this case we showed in class that  $\lambda(\varphi(A)) \leq \int_A \abs{\det \grad \varphi} \, d\lambda$ for all Borel $A \subset U$.
    \begin{parts}
      \item 
	If $f:V \to [0, \infty]$ is Borel, show that $\int_V f \leq \int_U f \circ \varphi \, \abs{\det \grad \varphi} \, d\lambda$.
      \item
	Show that $\lambda(A) = \int_A \abs{\det \grad \varphi} \, d\lambda$.
	\hintb{This follows very quickly previous part.}
      \item
	Prove the previous subpart \emph{without} the additional assumptions that $\varphi, \varphi\inv$ are \emph{uniformly} $C^1$, and $U, V$ are bounded.
    \end{parts}

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

  \itemopt
    \begin{parts}
      \item If $f \in L^1\per([0,1])$, show that $2 \abs{\hat f(n)} \leq \int_0^1 \abs{ f(y) - f(y - \frac{1}{2n})  } \, dy$.
      \item Use the previous subpart to give an alternate (perhaps more illuminating) proof of the Riemann Lebesgue lemma.
      \item If $\alpha \in (0,1)$, $f \in C^\alpha\per([0,1])$, show that $\sup_n \abs{n}^\alpha \abs{f(n)} < \infty$.
      \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 $0 \leq r \leq s$. Show that any bounded sequence in $H^s\per$ has a subsequence that is convergent subsequence in $H^r\per$.

  \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 If $\norm{\grad \varphi - I}_{L^\infty} < \eps$, and $\varphi(0) = 0$, then show that $\varphi( (-1, 1)^d ) \subset (-1 - d\eps, 1 + d \eps)^d$.

  \itemopt \emph{(Polar Coordinates.)}
    Let $f \in L^1(\R^2)$. Show that
    $$
      \int_{\R^2} f(x, y) \, dx \, dy = \int_{[0,\infty)\times[0, 2\pi)} f(r \cos \theta, r \sin \theta) \, r \, dr \, d\theta
    $$



\end{assignment}%}}}
\newpage

\begin{assignment}{Wed 11/28}{Wed 12/05}%{{{ HW13
  \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 $\R^d$. Define $D\mu(x) = \lim_{r \to 0^+} \frac{\mu(B(x, r))}{\lambda(B(x,r))}$.
    \begin{parts}
      \item
	If $\mu \perp \lambda$, show that $D\mu = 0$ almost everywhere with respect to $\lambda$.
	\hintb{Write $\mu = \mu_1 + \mu_2$ where $\supp(\mu_1)$ is compact with Lebesgue measure $0$, and $\norm{\mu_2} < \eps$.}
      \item
	If $\mu \perp \lambda$, show that $D \abs{\mu} = \infty$ almost everywhere with respect to $\mu$!
      \item 
	Show that $D \mu = \frac{d \mu_\text{ac}}{d \lambda}$ almost everywhere with respect to $\lambda$.
	\noteb{Here $\mu = \mu_\text{s} + \mu_\text{ac}$ is the Lebesgue decomposition of $\mu$ with respect to $\lambda$.}
    \end{parts}

  \item
    Suppose $f: \R \to \R$ is a monotone function.
    Show that $f$ is differentiable almost everywhere.
    \hintb{Suppose first $f$ is monotone, injective and bounded.
    Show that $\mu(A) = \lambda(f(A))$ defines a finite Borel measure.
    How does this help?}

  \item
    If $f, g:[0,1] \to \R$ are absolutely continuous, then show that $f g$ is absolutely continuous.
    Conclude $[fg]_0^1 = \int_0^1 f' g + \int_0^1 f g'$.

  \item
    Show that $f: \R \to \R$ is Lipshitz if and only if $f$ is absolutely continuous and $f' \in L^\infty(\R)$.


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

  \itemopt
    Show that the arbitary 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.


\end{assignment}%}}}
\clearpage

\begin{assignment}{Wed 12/05}{never}%{{{ HW14
  \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!
      Thus, if one is locallized the other is forced to be spread out.

      In quantum mechanics, Haysenberg's uncertainty principle says the product of errors in measuring the position and momentum (respectively) of a particle 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
	For $s \in (0, 1]$ show that there exists a constant $c$ such that for all $x \in \R^d$, and $f \in H^s$ we have $\norm{f - \tau_x f}_{L^2} \leq c \abs{x}^s \norm{f}_{H^{s}}$.
    \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 $f \in C^\infty$, $D^\alpha f \in L^1$ for all $\alpha$, and $u, \del_i u, \del_i\del_j u \in L^1 \cap C^0$ 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}
