\documentclass[10pt]{article}

\usepackage{amsmath,amssymb}
\usepackage{dsfont}

%\setlength{\textwidth}{160mm} \setlength{\textheight}{200mm} \setlength{\parindent}{8mm} \setlength{\oddsidemargin}{0pt} \setlength{\evensidemargin}{0pt}

\renewcommand{\labelenumi}{(\alph{enumi})}
\renewcommand{\labelenumii}{\Roman{enumi}.\arabic{enumii}}
\renewcommand{\labelenumiii}{\alph{enumiii}.}

\newcommand{\norm}[1]{\|#1\|}
\newcommand{\RR}{\mathbb{R}}
\newcommand{\NN}{\mathbb{N}}
\newcommand{\CC}{\mathbb{C}}
\newcommand{\ZZ}{\mathbb{Z}}

\newcommand{\R}{\mathds{R}}
\newcommand{\Z}{\mathds{Z}}
\newcommand{\N}{\mathds{N}}
\newcommand{\Q}{\mathds{Q}}
\newcommand{\E}{\mathds{E}}
\newcommand{\C}{\mathds{C}}
\newcommand{\F}{\mathds{F}}
\newcommand{\B}{\mathds{B}}
\newcommand{\M}{\mathbf{M}}
\newcommand{\lub}{\mathop{\textrm{lub}}}
\newcommand{\glb}{\mathop{\textrm{glb}}}
\newcommand{\romancounter}{\renewcommand{\labelenumi}{\roman{enumi}.}}
\def\signum{\mathop{\textrm{sgn}}}

\newcounter{exer}[section]
\newenvironment{exer}
{\stepcounter{exer}\hskip -.5in \makebox[.5in][l]{\Roman{section}.\arabic{exer}}\begin{minipage}[t]{\textwidth}}
{\end{minipage}\newline}

\newcommand{\alphcounter}{\renewcommand{\labelenumi}{\alph{enumi}.}}
\newcommand{\Alphcounter}{\renewcommand{\labelenumi}{\Alph{enumi}.}}
\newcommand{\Romancounter}{\renewcommand{\labelenumi}{\Roman{enumi}.}}

\newcommand{\comment}[1]{}


\begin{document}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\thispagestyle{empty}

\begin{center} \Large\textbf{Analysis Test Bank} \\
\begin{small}June 10, 2010\end{small} \end{center}

\vskip.5 true in

\noindent This document is the \textbf{Syllabus} for the \emph{Common Core Analysis Comprehensive Exam}. It is followed by the \emph{Test Bank} of sample problems for review. In this document, functions are understood to be defined on an abstract measure space, or else on either the real line or real $n$--dimensional Euclidean space.  Functions are real valued unless stated to the contrary.

\vskip.1in

 \centerline{\textbf{Prerequisites}}

 \noindent \emph{Graduate students are presumed to have preparation in advanced calculus, such as Math 4031, 4032, and 4035 at LSU. Not every student will have learned each of the topics below, so it is a good idea to round out your understanding as needed by means of reading or auditing one or more of these advanced calculus courses.}

    \begin{itemize}
    \item From Math 4031: Continuous functions, uniform continuity, pointwise versus uniform convergence, compact sets, and the Riemann integral.
     \item From Math 4032: Differentiable functions, infinite series of constants and functions, Taylor's theorem, $\mathcal{C}^{\infty}$--functions, and the Weierstrass approximation theorem.
     \item From Math 4035: The topology of Euclidean space, derivatives as linear transformations, Jacobians, the inverse-- and implicit--function theorems. The topics in this list are particularly helpful as preparation for Math 7550---differential geometry.
     \end{itemize}


\centerline{\large \textbf{Major Test Topics}}
\begin{enumerate}

\item Measure spaces: Fields, Borel Fields and Measures
    \begin{itemize}
    \item Monotone Classes and Borel Fields
    \item Additive Measures, Outer Measures
    \item E. Hopf’s Extension Theorem
    \end{itemize}

\item Lebesgue Measure
    \begin{itemize}
    \item  Measurable Sets, Borel Sets, and the Real Line
    \item Measure Spaces and Completions, Semimetric Space of Measurable Sets
    \item Lebesgue Measure in $\textbf{R}^n$; comparison with Jordan Measure in $\textbf{R}^n$
    \end{itemize}

\item Measurable Functions (on measure spaces)
    \begin{itemize}
    \item Measurable Functions, Limits of Measurable Functions
    \item Simple Functions and Egoroff’s Theorem; Convergence in Measure; Lusin's Theorem
    \end{itemize}

\item The Integral (on measure spaces)
    \begin{itemize}
    \item Simple Functions; the Class of nonnegative Measurable Functions, and the Class L of Lebesgue Integrable Functions
    \item Lebesgue Dominated Convergence Theorem; the Monotone Convergence theorem, and Fatou’s lemma
    \item $L^1(X,\mathfrak{A},\mu)$ as a Complete Normed Linear Space; Complex Valued Functions
    \end{itemize}

\item Product Measures and Fubini’s Theorem (for products of measure spaces)
    \begin{itemize}
    \item Product Measures, Fubini’s Theorem, Comparison of Lebesgue and Riemann Integrals
    \item Applications of Fubini's theorem to $L^1$ convolution, Fourier transform, etc.
    \end{itemize}

\item Functions of a Real Variable
    \begin{itemize}
    \item Functions of Bounded Variation; Fundamental Theorem for the Lebesgue Integral; Lebesgue's Theorem and Vitali's Covering Theorem. The Cantor function.
    \item Absolutely Continuous \& Singular Functions
    \item Convex functions
    \end{itemize}

\item General Countably Additive Set Functions
    \begin{itemize}
    \item Hahn Decomposition Theorem
    \item Radon--Nikodym Theorem
    \item Lebesgue Decomposition Theorem
    \end{itemize}

\end{enumerate}



\newpage

\thispagestyle{empty}

\centerline{\textbf{Suggested Reading}}
\smallskip
\emph{For Graduate Real Analysis: }
\medskip
\begin{itemize}
\item D.S. Bridges, \emph{Foundations of Real and Abstract Analysis}, Springer, 1997
\item P. Halmos, \emph{Measure Theory}, D. Van Nostrand Company, Inc., 1950.
\item A. Browder: \emph{Mathematical Analysis}, Springer, 1996.
\item I.P. Natanson, \emph{Theory of Functions of a Real Variable}, Vol. 1, 1955.
\item L. Richardson, \emph{Measure and Integration: a Concise Introduction to Real Analysis}, John Wiley \& Sons, 2009.
\item H.L. Royden, \emph{Real Analysis}, Macmillan, 1988.
\item W. Rudin, \emph{Real and Complex Analysis}, McGraw-Hill.
\item G.F. Simmons, \emph{Introduction to Topology and Modern Analysis}, McGraw-Hill, 1963.
\item E.M. Stein and R. Shakarchi, \emph{Real Analysis-Measure Theory, Integration, \& Hilbert Spaces}, Princeton U. Press, 2005.
\item R.L. Wheeden and A. Zygmund, \emph{Measure and Integral: An Introduction to Real Analysis}. Marcel Dekker, 1977.
\end{itemize}

\medskip

For Advanced Calculus/Undergraduate Analysis Review:

 \begin{itemize}
 \item L. Richardson, \emph{Advanced Calculus: An Introduction to Linear Analysis}, John Wiley \& Sons, 2008.
 \item Walter Rudin, \emph{Principles of Mathematical Analysis}, McGraw-Hill, 1964.
 \end{itemize}

\newpage



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\pagestyle{plain}
\setcounter{page}{1}
%\setcounter{section}{0}

\begin{center}
\Large \textbf{Analysis Test Bank}
\end{center}
{\bf Problems representing the core topics of math 7311 (Analysis I) are found in sections II, III, and IV, below.}

%\begin{enumerate}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\smallskip

\section{Preliminaries: Advanced Calculus}
{\bf These topics are found in Math 4031, 4032, and 4035.} \emph{Unless otherwise stated, functions are real valued and defined on the real line and/or real $n$-space. In this section only, the integral is meant exclusively in the sense of Riemann. Students who have not studied these topics in undergraduate analysis may need to learn them independently as graduate students.  It is possible also to audit Math 4031, 4032, and 4035.}

\smallskip


\begin{exer}
%\alphcounter
\begin{enumerate}
  \item Let $f_n : \RR\to\RR$ be given by $f_n (x) = {x \over n}$
($n\in\NN$). Show that the sequence $(f_n)_{n\in\NN}$ is  pointwise
convergent on $\RR$, uniformly convergent on $[0,1]$,  but not
uniformly convergent on $\RR$.
  \item Let $f_n : [0,1]\to\RR$ be given by $f_n (x) = {1 \over {1 + nx^2}}$ ($n\in\NN$).  Show that $(f_n)_{n\in\NN}$ is a bounded subset of $C [0,1]$ and that no subsequence
of $(f_n)_{n\in\NN}$ converges in $C[0,1]$.
\end{enumerate}
\end{exer}

%\item Let $f_n : \RR\to\RR$ be given by $f_n (x) = {x \over {1 + nx^2}}$ ($n\in\NN$).
%\smallskip
%\begin{enumerate}
%\item  Show that $\sup_{x\in\RR} | f_n(x) |
% = {1 \over {2 \sqrt{n}}}$.  Conclude that the sequence $(f_n)_{n\in\NN}$ converges uniformly on
%$\RR$ to a function $f$.  What is $f$?
%\item  Show that the equation $f^{\prime} (x) = \lim_{n~\rightarrow~\infty}
%f_n^{\prime} (x)$ is true if $x \neq 0$, but false if $x = 0$.
%\end{enumerate}

\begin{exer}  Identify all subsets of $[0,1]$ on which $\sum_{n = 0}^{\infty} x^n$ converges uniformly.  Explain.
\end{exer}

\begin{exer} Recall that a step function is finite linear combination of
characteristic functions of intervals.
 \begin{enumerate}
\item  Show that every continuous function on $[0,1]$ is a uniform
limit of step functions.
\item  Is the converse true?
\end{enumerate}
\end{exer}

\begin{exer}  Prove or disprove:
\begin{enumerate}
\item The product of two uniformly continuous functions
on $\RR$ is also uniformly continuous.
\item The product of two uniformly continuous functions
on $[0,1]$ is also uniformly continuous.
\end{enumerate}
\end{exer}

\begin{exer}  Prove or disprove the following two statements:
\begin{enumerate}
\item  If $\sum_{n = 1}^{\infty} a_n$ converges, then
$\sum_{n = 1}^{\infty} a_n \cos n x$ converges pointwise everywhere
on $\RR$.
\item  If $\sum_{n = 1}^{\infty} a_n$ converges absolutely, then
$\sum_{n = 1}^{\infty} a_n \cos n x$ converges to a continuous
function on $\RR$.
\end{enumerate}
\end{exer}

%\item Let $$f(x)=
%\begin{cases}
%x^2&\;\text{ if }\;$x$ \text{ is irrational}\\
%0&\; \text{ if }\;$x$ \text{ is rational.}\
%\end{cases}$$  Show that $f$
%is continuous at only one point, and that $f$ is differentiable
%there.

\begin{exer}   Let $C_{0,0}[0,1]$ be the space of all
continuous real functions $f$ on the interval $[0,1]$ satisfying
$f(0) = f (1) = 0$. Let ${\cal P}_{0,0}$ be the subspace of
polynomials in $C_{0,0}[0,1]$.  Show that ${\cal P}_{0,0}$ is dense
in $C_{0,0}[0,1]$ in the sup norm.
\end{exer}

\begin{exer}  Let $(x_n)_{n\in\NN}$ be a sequence of real numbers.  Prove that
the following are equivalent.
\begin{enumerate}
\item  $\lim_{n\to\infty} x_n = a$.
\item  Every subsequence of $(x_n)_{n\in\NN}$ contains a subsequence that
converges to $a$.
\end{enumerate}
\end{exer}

%\item  Show that the function
%$f(x) = 2^{-x} + 3 \cdot 2^{-3 x} + \ldots + (2n + 1)2^{-(2n + 1)x}
%+ \ldots$
% is continuous on $(0, \infty)$.

%\item
%
%\begin{enumerate}
%  \item  Prove or disprove:  A continuous function on the
%interval $[0, \infty)$ can be approximated uniformly by polynomials.
%  \item  Prove or disprove: If $f$ and $g$ are both functions from
%$\RR$ into $\RR$, and $\lim_{t \to a} g(t) = b$ and  $\lim_{t \to b}
%f(t) = c$, then $\lim_{t\to a} f(g(t)) = c$.
%\end{enumerate}

%\item  Show: if $f \in C [0,1]$ and $\int_{0}^{1} f(x) x^n dx = 0$ for all
%$n\in \NN_0$, then $f = 0$.

\begin{exer}  Prove: If $f \in C[0,1]$ and $\int_{0}^{1} f(x) e^{-nx} \,
dx = 0$ for all $n \in \NN_0$, then $f = 0$.
\end{exer}

\begin{exer}  Let $f_n : [1,\infty) \to \RR$ be defined by $f_n (x) := {{n + 1} \over n} e^{-nx}$  ($n\in\NN$).  Show
that the series $\sum_{k = 1}^{\infty} f_k$ converges uniformly to a
continuous function.
\end{exer}

\begin{exer}  $f_n : \RR \to \RR$ be defined by $f_n (x):= (\sin x)^n $  ($n\in\NN$).  Does $(f_n)_{n\in\NN}$ converge uniformly?
\end{exer}

\begin{exer}  Let $f_n : [0,\infty) \to \RR$ be defined by $f_n(x) := (x \slash n)e^{-(x \slash n)}$ ($n\in\NN$).
\begin{enumerate}
\item  Determine $f(x) = \lim_{n \rightarrow \infty} f_n(x)$.  Show that the sequence
$(f_n)_{n\in\NN}$ converges uniformly to $f$ on $[0,a]$ for any
nonnegative real number $a$. Does the sequence converge uniformly
to $f$ on $[0, \infty)$?  Justify your answer.
\item  Show that $$\lim_{n \to \infty} \int_0^a f_n (x) \, dx  =
\int_{0}^{a} f(x) \, dx,$$ but that $$\lim_{n \to \infty}
\int_{0}^{\infty}
 f_n(x) \, dx \neq \int_{0}^{\infty} f(x) \, dx.$$
\end{enumerate}
\end{exer}

\begin{exer}   Let $\displaystyle f_n(x) = \sum_{i = 0}^{n - 1} {1 \over n} f \left(x + {i \over n}\right)$,
where $f$ is a continuous function on $I\hskip-.13cmR$.  Show that
the sequence of functions $(f_n)_{n\in\NN}$ converges uniformly on
every finite segment $[a,b]$ to the function $F(x) = \int_{x}^{x +
1} f(s)\, ds$.
\end{exer}

\begin{exer}  Does $e^z (x^2 + y^2 + z^2 ) - \sqrt{1 + z^2} + y = 0$ have
a solution $z = f(x,y)$, where $f$ is continuous at $(x = 1, y = 0)$ and
$f(1,0)=0$? Explain carefully.
\end{exer}

\begin{exer}  Let $f : \RR \rightarrow \RR$ be an infinitely differentiable function.

\begin{enumerate}
\item
Use Taylor's formula with remainder to show that, given $x$ and $h$,
$f^{\prime} (x) = (f(x + 2h) - f(x)) \slash 2h - hf^{\prime
\prime} (\xi)$ for some $\xi$.
\item  Assume $f(x) \rightarrow 0$ as $x \rightarrow \infty$, and that
$f^{\prime \prime}$ is bounded.  Show that $f^{\prime} (x)
\rightarrow 0$ as $x \rightarrow \infty$.
\end{enumerate}
\end{exer}

%\item  Let $f(x,y) = xy - \cos y + x^2 + 1$.  At what points on the set
%$\{(x,y) : f(x,y) = 0 \}$ does the condition $f(x,y) = 0$ fail to
%define either $x$ as a function of $y$ or $y$ as a function of $x$?

\begin{exer}  Prove or give a counterexample: If $f$ is a uniform limit of polynomials
on $[-1,1]$, then the Maclaurin series of $f$ converges to $f$ in
some neighborhood of $0$.
\end{exer}

%\item  Can the surface whose equation is
%$x y - z \log y + e^{x z} = 1$
% be represented in the form $z = f(x,y)$ in a neighborhood of $(0,1,1)$?
%In the form $y = g(x, z)$ in a neighborhood of $(0,1,1)$?

\begin{exer}  Let $f(x) = x^k \sin (1 \slash x)$ if $x \neq 0$ and $f(0) = 0$.
\begin{enumerate}
\item  If $k = 2$, show that $f$ is differentiable everywhere but $f^{\prime}$
fails to be continuous at some point.
\item  If $k = 3$, does $f$ have a second derivative at all points?  If so, is
$f^{\prime \prime}$ a continuous function?  Give your reasons.
\end{enumerate}
\end{exer}

\begin{exer}   Let $f$ be defined on $\RR^3$ by
$f(x,y,z) = x^2 + 4y^2 - 2yz - z^2$.
Show that $f(2,1,-4) = 0$ and $f_z(2,1,-4) \neq 0$, and that
there exists therefore a differentiable function $g$ in a
neighborhood of $(2,1)$ in $\RR^2$, such that $f(x,y,g (x,y)) = 0$.
Find $g_x(2,1)$ and $g_y(2,1)$. What is the value of $g(2,1)$?
\end{exer}

\begin{exer}  Suppose that a function $f$ is defined on $(0,1]$ and has a finite
derivative with $| f^{\prime} (x) | < 1$.  Define $a_{n} := f(1
\slash n)$ for $n = 1,2,3,\ldots $.  Show that $\lim_{n \to \infty}
a_n$ exists.
\end{exer}

%\item  Prove or disprove: The series $\sum_{n = 1}^{\infty} (-1)^n
%{{n^2 + n} \over {n^2}}$ is uniformly convergent on $[-1,1]$.

\begin{exer}  Define a function $f$ on $\RR$ by
$$f(x)=\left\{\aligned
e^{-1/x^2}\text{\rm , if $x>0$}\\
0\quad \text{\rm ,  if $x\le 0$     }\endaligned \right.$$
\begin{enumerate}
\item Check whether $f$ is infinitely differentiable at 0, and, if
so, find $f^{(n)}(0),$ $n=1,2,3,\cdots$.  Show details.
\item  Does $f$ have a power series expansion at $0$?
\item Let $g(x)=f(x)f(1-x)$.  Show that $g$ is a nontrivial infinitely
differentiable function on $\RR$ which vanishes outside $(0,1)$.
\end{enumerate}
\end{exer}

\begin{exer}  Prove that a function $f : \RR^{n} \to \RR$ is continuous at
$x = (x_1, \cdots, x_n) \in \RR^{n}$ if the partial derivatives
$f_{x_1} , \ldots, f_{x_n} $ exist and are bounded in a neighborhood
of $x$.
\end{exer}

\begin{exer}  Let $f_N(x)~=~\sum_{n = 1}^{N} a_n \sin (nx)$ for $a_n,x \in \RR$.
If $\sum_{n = 1}^{\infty} na_{n}$ converges absolutely, show that
$(f_{N})_{N\in\NN}$ converges uniformly to a function $f$ on $\RR$,
and that $(f_{N}^{\prime})_{N\in\NN}$ converges uniformly to
$f^{\prime}$ on $\RR$.
\end{exer}

\begin{exer}  Let $f$ be a twice continuously differentiable real-valued function on $\RR^n$.  A point $x \in \RR^n$ is a critical point of $f$
if all partial derivatives of $f$ vanish at $x$ (i.e.,
$\bigtriangledown f(x) = 0$), a critical point $x$ is nondegenerate
if the $n\times n$ matrix
$$\left [{{\partial^2 f} \over {\partial x_i \partial x_j}} (x) \right ]$$
is nonsingular.  Let $x$ be a nondegenerate critical point of $
f$.  Prove that there is an open neighborhood of $x$ which contains
no other critical points. (i.e., the nondegenerate critical points
are isolated.)
\end{exer}

%\item  Show that the power series in $x$ for the function $f(x) = e^{a x}
%\cos (bx)$ ($a,b\in\RR$) has either no zero coefficients or
%infinitely many zero coefficients.

%\item  Let $f$ be continuous function on $[0,1]$.  Define
%$$I_{n} = \prod_{j = 1}^{n} \left [ {1 + {1 \over n} f \left ({j \over n} \right )} \right ]$$
%  for every integer $n \geq 1$.  Determine the limit
%$\lim_{n \rightarrow \infty} I_n$.

\begin{exer}  Show that a function $f(x) = e^{-x} + 2e^{-2x} + \ldots
+ ne^{-nx} + \ldots$ is continuous on $(0,\infty)$.
\end{exer}

%\item   Let $f(x) = e^{-1 \slash x^2}$ if $x > 0$ and $f(x) = 0$ if
%$x \leq 0$.  Verify, using induction, that for each positive integer
%$k$, there is a polynomial $p_k$ such that $f^{(k)} (x) = f(x) p_k
%(x) x^{-3k}$ for all $x > 0$.  Show that the function $\phi : x
%\rightarrow f (1 + x) f(1 - x)$ has the following properties:
%
%\begin{enumerate}
%\item $\phi$ is smooth (i.e. has derivatives of all orders);
%\item $\phi \geq 0$;
%\item $\phi (x) = 0$ if $| x | \geq 1$;
%\item $\int_{-\infty}^{\infty} \phi (x) dx > 0$.
%\end{enumerate}

\begin{exer}  Let $f : \RR \to \RR$ be a twice differentiable function with
bounded first and second derivatives.  By considering the Taylor
expansion of $f$, show that: $ \| f^{\prime}\|_\infty \leq {1 \over
h} \| f\|_\infty + h \| f^{\prime \prime} \|_\infty$ for every $h
> 0$.  By minimizing over $h$, show that $\| f^{\prime} \|_{\infty}
\leq 2 \sqrt{\| f \|_{\infty} \| f^{\prime \prime} \|_{\infty}}$,
where $\| g \|_{\infty}$ denotes $\sup_{x \in \RR} | g(x) |$.
\end{exer}

%\item  Define the Hermite polynomial of degree $n$ by
%$$H_n (x) = (-1)^n
%e^{{{x^2} \over {2}}} ({{d} \over {dx}} )^n e^{- {{x^2} \over
%{2}}}\quad ( n \geq 0, x\in\RR).$$
%
%Use Taylor's theorem to prove the identity
%$$\sum_{n = 0}^{\infty} {{t^n} \over {n!}} H_n (x) = e^{t x - {{1} \over{2}} t^2}.$$

\begin{exer}  Let $f$ be a real-valued differentiable function on an interval $(a,b)$. Show that $f$ is Lipschitz continuous if and only if $f$ has bounded derivative.
\end{exer}

\begin{exer}
Let $E$ be a nonempty subset of $\mathbb R$. Let $c\in \mathbb R$.
Show $\sup E=c$ if and only if $x\le c$ for all $x\in E$ and for
each $\epsilon >0$, there is an $x\in E$ with $x+\epsilon >c$.
\end{exer}

\begin{exer}
Let $\langle a_n\rangle$ and $\langle b_n\rangle$ be sequences in
$ [-\infty ,\infty ]$. Show
\begin{itemize}
\item[(a)]  $\lim\sup a_n=-\lim\inf(-a_n)$

\item[(b)] $\lim\inf(a_n+b_n)\ge\lim\inf a_n+\lim\inf b_n$
\end{itemize}
\end{exer}

\begin{exer}
Show every open subset $U$ of $\mathbb R$ can be written has a
countable disjoint union $\cup_{i\in I}(a_i,b_i)$ of disjoint open
intervals.
\end{exer}

\begin{exer}
Let $E$ be a subset of $\mathbb R^d$. Recall a subset $S$ of $E$
is said to be dense in $E$ if for each $x\in E$ and each $\epsilon
>0$, one has $B_ {\epsilon}(x)\cap S\ne\emptyset$. Using the
countability of the set of points in $\mathbb R^d$ with rational
entries, show every subset $E$ of $\mathbb R^ d$ has a countable
dense subset.
\end{exer}

\begin{exer}
Let $F_1$ and $F_2$ be disjoint compact subsets of $\mathbb R^d$.
Show $d(F_1,F_2)>0$.
\end{exer}

\begin{exer}
Show if $F_1$ and $F_2$ are nonempty compact subsets of $\mathbb
R^d$, there are points $x_1\in F_1$ and $x_2\in F_2$ with
$d(x_1,x_2)=d (F_1,F_2)$.
\end{exer}

\begin{exer}
Show if $F\subseteq O\subseteq \R^d$ and $F$ is compact and
$O$ is open, then there is a continuous function $f$ such that
$0\le f(x)\le 1$ for all $x\in \R^d$ and $f(x)=1$ on $F$
and $f(x)=0$ for $x\notin O$. {\bf Hint:} Consider
$f(x)=\min \left\{\frac {d(x,O^c)}{d(F,O^c)},1\right\}$.
\end{exer}

\begin{exer}   Assume that $A \geq 0$, $B > 0$, and $f$ is continuous and
nonnegative on $[a,b]$.  Assume that $f(t) \leq A + B \int_{a}^{t} f(s)\,ds$ for $a \leq t \leq b$.  Prove that $f(t) \leq A e^{B (t-a)}$ for $a \leq t \leq b$.
\end{exer}

\begin{exer}
Let $f:[a,b]\to \R$ be monotonically increasing. Show the set of points in $[a,b]$ where $f$ is discontinuous is countable.
\end{exer}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



\section{Lebesgue measure, $\sigma$-algebras, Borel sets, measurable
functions}

\nopagebreak

\emph{Problems concerning $\R$, on $\R^n$ refer to Lebesgue measure, $l$. Other problems are about arbitrary measure spaces $(X,\mathfrak{A},\mu)$.}
\nopagebreak
\medskip

\begin{exer} Let $\mathcal A$ be a $\sigma$-algebra of subsets of $\mathbb
X$, and let $f:\mathbb{X}\to \mathbb{X}$ be arbitrary.  Prove that
the set $$\left\{f^{-1}[S] \,\big|\, S\in{\mathcal A}\right\}$$ is a $\sigma$-algebra.
\end{exer}

\begin{exer} Suppose that $g:X\to \R$ is $\mu$-measurable
and that $f:\R \to \R$ is Borel measurable, meaning that
$f^{-1}[(a,\infty)]$ is a Borel set for each real number $a$. Prove
that $f\circ g$ is
$\mu$-measurable.
\end{exer}

\begin{exer} Prove that every Lebesgue measurable subset of $\R$ is sandwiched between an $F_{\sigma}$-set and a $G_{\delta}$-set of the same Lebesgue measure.
\end{exer}

\begin{exer} Prove that the Cantor set is a Borel set.
\end{exer}

\begin{exer}
\begin{enumerate}

\item Show the set of irrational numbers is a $G_{\delta}$ set but is not an $F_{\sigma}$ set. {\bf Hint:} Show $\mathbb Q$ is not a
$G_{\delta}$, for otherwise you could obtain a decreasing sequence
$G_n$ of dense open sets that have empty intersection. Then use
the decomposition of each $G_n$ into a disjoint countable union of
open intervals.

\item Using the fact that the set of rational numbers in any closed interval $a\le x\le b$ where $a<b$ is not a $G_{\delta}$ set, give an example of a Borel subset of $\mathbb R$ which is neither an $F_{\sigma}$ or a $G_{\delta}$ set.

\item Let $f$ be any function from $\RR$
to $\RR$.  Prove that the set of points of discontinuity of $f$ is
of type $F_\sigma$.

\item   Can a function from $\R$ to $\R$ be
continuous on the rationals and discontinuous on the irrationals?
What if the roles of the rationals and irrationals are interchanged?
\end{enumerate}
\end{exer}

\begin{exer} Let $f:\R\to \R$ be a measurable function that is not almost everywhere infinite. Prove that there exists a subset $S\subset\R$ of
positive measure such that $f$ is bounded on $S$.
\end{exer}

\begin{exer}  Let $f:X\to \R^*$ be an extended real-valued measurable function on a finite measure space $(X,\mathfrak{A},\mu)$. Suppose that $f(x)$ is finite for almost all $x$. Prove that for each $\epsilon >0$ there exists a set $A\in \mathfrak{A}$, with $\mu(X\setminus A)<\epsilon$, such that $f$ is bounded on $A$.  Give an example for which it is impossible to require that $X\setminus A$ be a $\mu$-null set.
\end{exer}

\begin{exer}   If $E$ is a measurable subset of $[0,1]$ prove that there is a measurable subset $A\subset E$ such that $l(A)=\frac{1}{2}l(E).$
\end{exer}

\begin{exer} Let $E\subset {\mathbb R}$ be a measurable set with the property that $$l(E\cap
I)\le \frac{l(I)}{2},$$ for every  open interval $I$. Prove that
$l(E)=0$.
\end{exer}

\begin{exer} Let $E$ be a measurable set in $[0,1]$ and let $c>0$. If $l(E\cap
I)\ge c l(I)$, for all open intervals $ I \subset [0,1]$ show that
$l(E)=1$.
\end{exer}

\begin{exer} Let $A \subset (0,1)$ be a measurable set and $l(A)=0$. Show that
$$l\{x^2: x\in A\}=0 \quad \text{and} \quad l\{\sqrt{x}: x\in A\}=0.$$
\end{exer}

\begin{exer}
Show the Cantor ternary set is totally disconnected; that is show
it contains no nonempty open interval.
\end{exer}

\begin{exer}
Show the Cantor ternary set consists of all $x$ in $[0,1]$ that
can be written in form $$x=\sum_{k=1}^{\infty}\frac {a_k}{3^k}$$
where each $a_k$ is either $0$ or $2$. Use the digits $a_n$ to define the value of  Cantor function $\phi$ at $x$.
\end{exer}

\begin{exer}
 Let $\delta_1,\delta_2,\ldots ,\delta_d$ be nonzero real numbers. Let $E$ be a subset of $\mathbb R^d$.  Define $$\phi (x_1,x_2,\ldots , x_d) = (\delta_1x_1, \delta_2x_2,\ldots ,\delta_dx_d).$$ Show that the \emph{outer measure}
$$l^*(\phi (E))=|\delta_1\delta_2\cdots\delta_d|\,l^*(E).$$ Then show $E$ is Lebesgue measurable if and only if $\phi (E)$ is Lebesgue measurable and then
$$l(\phi (E))=|\delta_1\delta_2\cdots\delta_d|\,l(E).$$
\end{exer}

\begin{exer}
An extended real-valued function $f:X \to [-\infty , \infty ]$ is \emph{measurable} if $f^{-1}[-\infty ,\alpha )$ is measurable for each $\alpha\in \R$. Show $f^{-1}(\alpha)=\{x\mid f(x)=\alpha \}$ is measurable for $\alpha\in [-\infty ,\infty ]$.
\end{exer}

\begin{exer} Let $(X,\mathfrak{A},\mu)$ be a complete measure space, and define a Carath\'{e}odory outer measure, \[\mu^*(E)=\inf \left\{\sum_{n\in\N} \mu(A_n) \ \Bigg|\ E\subseteq \bigcup_{n\in \N}A_n,\ A_n\in \mathfrak{A}\ \forall n \right\},\] for all $E\in \mathfrak{P}(X)$.
\begin{enumerate}
    \item If $\mu(X)<\infty$, show that a set $E$ in the power set $\mathfrak{P}(X)$ is measurable if, and only if, for each $\epsilon >0$ there is a measurable set $W$ with outer measure $\mu^{*}(E\vartriangle W) < \epsilon$. (Suggestion: Consider the complete semimetric space structure that can be placed on $\mathfrak{A}$.)
    \item Extend the preceding part to the case of a $\sigma$-finite measure space.
\end{enumerate}
\end{exer}

\begin{exer}
Show that there is a sequence of Lebesgue measurable functions $f_n:[0,1] \to \R$ that converges pointwise everywhere to a function $f$ having the property that there is not a subset $A$ of $[0,1]$ of Lebesgue measure $0$ such that $f_n$ converges uniformly to $f$ on the complementary set $[0,1]\setminus A$. Compare this exercise with the statement of Egoroff's theorem.
\end{exer}

\begin{exer}
Let $Q=[0,1]^d$ be the unit cube in $\R^d$ and suppose $f:Q\to \R$ is a continuous function. Show the Lebesgue measure of the compact set \[\Gamma =\{(x,f(x))\mid x\in Q\}\] in $\mathbb R^{d+1}$ is zero. (\emph{Hint:} Use uniform continuity.)
\end{exer}

\begin{exer}
Let $\mu$ be a \emph{finite} and \emph{finitely additive} measure on a field $\mathfrak{A}\subseteq \mathfrak{P}(X)$. Prove that $\mu$ is countably additive \emph{on} $\mathfrak{A}$ if and only if for each decreasing sequence $$A_1\supseteq A_2\supseteq \ldots\supseteq A_n\supseteq \ldots$$ with $\bigcap_{n=1}^{\infty}A_n=\emptyset$ we have $\lim_{n\to \infty}\mu(A_n)=0$.
\end{exer}

\begin{exer}
Prove that Lebesgue measure $l$ is translation-invariant on the real line.
\end{exer}

\begin{exer}
In the semimetric space $(\mathfrak{L},d)$ formed from  $([0,1],\mathfrak{L},l)$, where $l$ is Lebesgue measure, prove that each of the following sets is dense.
    \begin{enumerate}
    \item $\mathcal{O}$, the set of all \emph{open} sets in $[0,1]$.
    \item $\mathcal{K}$, the set of all \emph{closed} sets in $[0,1]$.
    \end{enumerate}
\end{exer}

\begin{exer}
Suppose $A$ and $B$ are measurable subsets of $\R^n$, each one of strictly positive but finite measure. Prove that there exists a vector $c\in \R^n$ such that \[l((A+c)\cap B)>0.\] (Hints: Consider the outer measure of $A$ and $B$. Or, alternatively, consider the convolution $1_{-A}*1_B$.)
\end{exer}

\begin{exer}
 Let $f:X\to \R$. Prove that $f$ is measurable (meaning that $f^{-1}(-\infty,a]\in \mathcal{L}$ for all $a\in \R$) if and only if $f^{-1}(B)\in \mathcal{L}$ for each Borel set $B$. (Hint: Show that the family $$\mathfrak{S} =\left\{A\in \mathfrak{P}(\R)\,\big|\, f^{-1}(A)\in \mathcal{L}\right\}$$ is a $\sigma$-field.)
\end{exer}

\begin{exer}
Suppose $f_n:X\to \R$ is a measurable function for each $n\in \N$, where $(X,\mathfrak{A},\mu)$ is a measure space. Prove that the set $$S=\left\{x\,\Big|\, \lim_{n\to \infty}f_n(x) \ \mathrm{exists} \right\}$$ is a measurable set.
\end{exer}

\begin{exer}
Suppose $E_n$ is a sequence of measurable sets in $X$ such that $\sum_{n\in \N} \mu (E_n)<\infty$. Set $F_N=\bigcup_{n\ge N}E_n$.
\begin{enumerate}

\item Show that $$\lim_ {N\to\infty}\mu (F_N)=0.$$
\item One defines $\limsup E_n$ by $$\limsup_{n\to\infty}E_n=\bigcap_{N\in \N} F_N.$$ Show that $\limsup E_n$ consists of all points $p$ which are in infinitely many of the sets $E_n$.
\item Part (a) shows that $\mu(\limsup E_n)=0$ if $\sum_{n=1}^{\infty}\mu (E_n)<\infty$. Give an example of a sequence of measurable sets $E_n$ such that $\mu (E_ n)\to 0$ as $n\to\infty$, but $\mu (\limsup E_n)\ne 0$.
\end{enumerate}
\end{exer}

\begin{exer}
Suppose $f_n$ is a sequence of real-valued Lebesgue measurable
functions on $X$. Suppose that for each $\epsilon > 0$, $\mu\{x\mid |f_n(x)|\ge\epsilon \}\to 0$ as $n\to\infty$.  Prove the following conclusions. \begin{enumerate}
\item There is a subsequence $f_{n_k}$ of $f_n$ such that $f_{n_k}(x)\to 0$ as $k\to\infty$ almost everywhere.
\item Show for each $\epsilon >0$, there is a measurable subset $E_{\epsilon}$ of $X$ and a subsequence $f_{n_k}$ such that $\mu (E_{\epsilon}) <\epsilon$ and $f_{n_k}$converges uniformly to $0$ off $E_{\epsilon}$.
\end{enumerate}
\end{exer}

\begin{exer}
Prove or give a counterexample:  $E\times F$ is $\mu\otimes \nu$-measurable in  $(X,\mathfrak{A},\mu)\bigotimes (Y,\mathfrak{B},\nu)$ if and only if both $E$ and $F$ are measurable with respect to $\mu$ and $\nu$ respectively.
\end{exer}

\begin{exer}
Let $f\ge 0$ be a Lebesgue measurable function on $\mathbb R^d$. Define $\mu (E)=\int_Ef\,dl$.
\begin{enumerate}
\item Show that $\mu$ is a measure on the Lebesgue measurable sets.
\item Show that $\mu \prec l$.
\item Show that $\mu$ is purely infinite on the set $E=\{x\mid f(x)=\infty \}$. (Note $\mu$ is purely infinite on $E$ if and only if $\mu (F)=0$ or $\mu (F)=\infty$ on any measurable subset $F$ of $E$.)
\item Show that $\mu$ is $\sigma$-finite on the set $E=\{x\mid f(x)<\infty \}$.  Note this means there is a sequence $E_n$ of measurable sets with $E=\bigcup_{n=1}^{\infty}E_n$ and $\mu (E_n)<\infty$ for all $n$.
\end{enumerate}
\end{exer}

\begin{exer}
It is known the ball of radius $1$ in $\mathbb R^{2m}$ has volume
(i.e., Lebesgue measure) $$\frac {\pi^m}{m!}.$$ Find the volume of
any ball in $\mathbb R^{2m}$ which has radius $r>0$.
\end{exer}

\comment{
\begin{exer} Let $r>0$. The sphere $S_r$ of radius $r$ in
$\mathbb R^{2m}$ is the collection of all points $x\in \mathbb
R^{2m}$ satisfying $|x|=r$. The surface area of this sphere is
defined by $$A_r=\lim_{h\to 0}\frac {m(B_ {r+h}(0))-m(B_r(0))}h.$$
Find $A_r$.
\end{exer}}

\begin{exer}  Let $f:\R^n\to \R$ be Lebesgue measurable. Prove that $f$ is equal almost everywhere to a \emph{Borel} measurable function $h$. Hint: The function $f$ is the pointwise limit of a sequence $\phi_n\in \mathfrak{S}$, the set of simple functions. Modify the functions $\phi_n$ suitably.
\end{exer}

%Show that if $f$ is a Lebesgue measurable function on $\mathbb R^d$, then there exists a Borel measurable function $h$ on $\mathbb R^d$ such that $f=h$ a.e. {\bf Hint:} Let $q\in \Q$ and let $E_q=\{x\mid f(x)<q\}$. The set $E_q$ is Lebesgue measurable. Choose a Borel set $B_q\supseteq E_q$ with $l (B_q-E_q) =0$. Set $F_q=\bigcup_{q'<q}B_q$. Note $F_q\subseteq F_{q'}$ if $q<q'$. Define $h(x)=\inf\{q\mid x\in F_q\}$. Show $h$ is a Borel function and equals $f$ a.e. {\bf Hint:} Show $f=h$ on the complement of $$\bigcup_{q\in \Q} (B_q-E_q)$$ and show $h$ is a Borel function.


\begin{exer}
 Let $F$ be a measurable subset of $\mathbb R$.  Let
$E$ be the set of $x$ in $F$ such that there is a $\delta >0$ with
$m((x-\delta ,x+\delta )\cap F)=0$. Show that $E$ has measure $0$.
\end{exer}

\begin{exer}
 Show there are no countably infinite $\sigma$-algebras.
\end{exer}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{The Integral: Convergence Theorems, Product Measures \& Fubini's Theorem.}
\nopagebreak

\nopagebreak
\emph{Integrals on $\R$ or $\R^n$ refer to Lebesgue measure unless stated explicitly to the contrary. Other problems concern integrals on an arbitrary measure space $(X,\mathfrak{A},\mu)$.}
\nopagebreak

\medskip

\begin{exer}
Let $ \chi_{[-n,n]} (\cdot)$ denote the characteristic function of
the interval $[-n,n]$ ($n\in\NN$). Consider the sequence of
functions $f_{n}(x) := \chi_{[-n,n]} (x) \sin ({\pi x \over n})$ ($x
\in \RR$).
\begin{enumerate}
\item  Determine $f(x) = \lim_{n \rightarrow \infty} f_n(x)$ and  show that the sequence $(f_n)_{n\in\NN}$ converges uniformly on compact subsets of $\RR$.  Does the sequence converge uniformly on $\RR$?
\item  Show that
$$\int_{- \infty}^{\infty} f(x) \, dx = \lim_{n \rightarrow \infty} \int_{- \infty}^{\infty}
f_n (x) \, dx.$$ Are the assumptions of Lebesgue's dominated
convergence theorem satisfied?
\end{enumerate}
\end{exer}

\begin{exer}  Let $f$ be a positive function on $(0,1]$ such that $f$ is Riemann
integrable on $[t,1]$ for all $t\in (0,1)$, but $\lim_{x \to 0^{+}}
f(x) = \infty$.  Assume that the improper (Riemann) integral $(R)
\int_{0}^{1} f(x) dx$ exists.  Show that $f$ is a measurable
Lebesgue integrable function and that
$$\int_{[0,1]} f(x)\, dx~=~(R) \int_{0}^{1} f(x)\, dx.$$
\end{exer}

\begin{exer}   For each of the following problems, check whether the limit exists.
If so, find its value.
\begin{enumerate}
  \item $ \lim_{n \rightarrow \infty} \int_{1}^{n} ({1 - {x \over n}})^n\,
dx,$
  \item $ \lim_{n \rightarrow \infty} \int_{1}^{2 n} ({1 - {x
\over n}} )^n\, dx.$
\end{enumerate}
\end{exer}

\begin{exer}
\begin{enumerate}
  \item  Characterize in terms of the set of points of continuity those bounded functions on $[0,1]$ that are Riemann integrable.
  \item   Define $g$ on $[0,1]$ by $$g(x)= \begin{cases} 0&\;\text{ if }\;$x$ \text{ is irrational or 0},\\
      1/q&\; \text{ if }\; x = p/q \text{ in lowest terms}.
\end{cases}$$  Is $g$ a Riemann integrable function? Give a proof of your assertion.
\end{enumerate}
\end{exer}

%  \item Let $({r_n})$ be an enumeration of the rationals in $[0,1]$.
%Define $f$ on $[0,1]$ by
%$$f(x)= \begin{cases}
%\frac 1n &\;\text{ if }\; x=r_n\\
%0&\; \text{ if }\;\text{ if $x$ is irrational}\
%\end{cases}$$  Is $f$
%Riemann integrable on [0,1]?  Explain!

\begin{exer} Show that if
$$f(x) =\begin{cases}
1&\;\text{ if }\;\text{ if $x$ is rational},\\
-1&\; \text{ if }\;\text{ if $x$ is irrational},\
\end{cases}$$ then $f$ is
not Riemann integrable on the interval $[0,1]$.  Is $ f$ Lebesgue
integrable?  Explain.
\end{exer}

\begin{exer}   Show there are no bounded sequences $(a_n)_{n\in\NN}$ and $(b_n)_{n\in\NN}$ for
which $$f_n (x) = a_n \sin (2 \pi n x) + b_n \cos (2 \pi n x )$$
converges to 1 almost everywhere on $[0,1]$.
\end{exer}

\begin{exer}   Let $f(x)$ be a real-valued measurable function on a finite measure space $(X,\mathfrak{A},\mu)$.  Show that
$$\lim_{n \rightarrow \infty} \int_{X} \cos^{2n} (\pi f(x))\, d\mu = \mu \{x \mid f(x)\in \Z \},$$  where $\Z$ is the set of integers.
\end{exer}

\begin{exer}
\begin{enumerate}
  \item Show that $f(x) = x^{-r}$ is a Lebesgue integrable function on
$[0,1]$ if $0 \leq r < 1$.
  \item If $0 \leq r < 1$ let $$\displaystyle{a_n = \int_{0}^{1} {{dx} \over {n^{-1}
+ x^{r}}}}.$$  Compute $\lim_{n \rightarrow \infty} a_n$.  Be sure to quote the appropriate integration theorems to justify your calculations.
 \end{enumerate}
\end{exer}

\begin{exer}   Let $f_n : \R\to \R$ be defined by $$f_n(x)=\begin{cases}
\frac 1n\ &\textrm{ if }\ |x| \le n, \\ 0 &\textrm{ if }\ |x| >n.
\end{cases}$$
\begin{enumerate}
\item  Show that $f_n$ converges to $0$ uniformly on $\R$, and that $$\lim_{n\rightarrow\infty}\int_{\RR}f_n(x)\,dx=2$$ while $$\int_{\RR}\left(\lim_{n\rightarrow\infty}f_n\right)(x)\,dx
\neq 2.$$
\item  Explain why this example  does not contradict the Lebesgue dominated convergence theorem.
\end{enumerate}
\end{exer}

\begin{exer}
\begin{enumerate}
  \item Show that $f(x) = \frac{1}{\sqrt{x}}$ is Lebesgue integrable on $(0,1)$.
  \item Find $$\inf \left\{\int_{0}^{1} \psi(x) \, dx\,
  \Big|\, \psi\in \mathfrak{S},\, \psi (x) \geq \frac{1}{\sqrt{x}}\,\forall\, x\in (0,1) \right\}.$$
(Here $\mathfrak{S}$ is the set of simple functions, which are finite linear combinations of characteristic functions of measurable sets with extended real-valued coefficients.)
  \end{enumerate}
\end{exer}

\begin{exer} Give an example of a sequence $(f_n)_{n\in\N}$ of bounded, measurable functions on $[0,1)$ such that $$\lim_{n \rightarrow \infty} \int_{0}^{1} |f_n (x)|\,dx=0$$  but such that $f_n$ converges pointwise  nowhere.
\end{exer}

\begin{exer}  Prove that $$\lim_{n \rightarrow \infty} \int_{- \infty}^{\infty} (\sin nt) f(t) dt=0$$ for every Lebesgue integrable function $f$ on $\R$. (Hint:  Do the problem first for step functions.)
\end{exer}

\begin{exer} Let $${f_n(x)=\frac{n}{x(\ln x)^n}}$$ for $x \geq e$ and
$n \in {\N}$.
\begin{enumerate}
\item  For which  $n \in  {\N}$ does the Lebesgue integral $\int_{e}^{\infty} f_n(x)\,dx$ exist?
\item  Determine $\lim_{n\to \infty}f_n(x)$ for $x > e$.
\item  Does the sequence $( f_n )_{n\in\N}$ satisfy the assumptions of Lebesgue's dominated convergence theorem?
\end{enumerate}
\end{exer}



\begin{exer}  Define $f(x) = \int_{\R} \cos (xy)\, g(y) \, dy$ for $x \in
\R$ where $g$ is an integrable function on $\R$.  Show that $f$ is continuous.
\end{exer}

%\item  Give an example of a Lebesgue integrable function $f$ on
%$[0,1]$ such that $$\int_{0}^{1} f(x) \, dx = 1,$$ but $f$ is not
%Riemann integrable.

\begin{exer}   Let $f$ be a differentiable function on $[-1,1]$.  Prove that
$\lim_{\epsilon \to 0} \int_{\epsilon < | x | \leq 1} {1 \over x}
f(x)\, dx$ exists.
\end{exer}



\begin{exer}   Let $\displaystyle{f_n(x)=\frac {x^n}{n!} e^{-x}}$ for all integers $n\ge 0$.
\begin{enumerate}
\item  Show that $\lim_{n \rightarrow \infty} f_n (x) = 0$ for
all $x > 0$.
\item  Show that $f_n \in L^1 (0, \infty)$ with $\|
f_n \|_1 = 1$ for all $n\ge 0$.
\item  Show that $\lim_{k\rightarrow\infty}\int_0^k\frac {
x^n}{n!}(1-\frac xk)^k\,dx=1$ for all $n\ge 0$.
\end{enumerate}
\end{exer}

\begin{exer} Prove that $\displaystyle \lim_{b\to \infty} \int_{0}^{b} \frac {\sin x}{x} \, dx$ exists but that the function $\displaystyle \frac{\sin x}{x}$ is not integrable over $(0, \infty)$.
\end{exer}

\begin{exer}  Compute the following limit and justify your calculations:
$$\lim_{n \rightarrow \infty} \int_{0}^{\infty} \left (1 + {{x} \over {n}}
\right )^{-n} \sin \left ( {x \over n} \right )\, dx$$
\end{exer}

\begin{exer}   Find and justify the limits:
\begin{enumerate}
  \item $\displaystyle{\lim_{n \rightarrow \infty}
\int_{0}^{n} {{\sin x} \over {1 + nx^2}} \, dx}$
  \item $\displaystyle{\lim_{n \rightarrow \infty}
    \int_{0}^{e^n} {x \over {1 + nx^2}} \, dx}$.
\end{enumerate}
\end{exer}

\begin{exer} Let $f \in L^1 (0,\infty)$, and suppose that $\int_0^{\infty}
x | f(x) | \, dx < \infty$. Prove that the function $$g(y) =\int_0^{\infty} e^{-xy} f(x) dx$$  is differentiable at every $y \in (0,\infty)$.
\end{exer}

\begin{exer} Let $f \in L^1 (\R)$  with respect to Lebesgue measure, and suppose that $$\int_{\R} | x | | f(x) | \, dx < \infty.$$ Show that the function $$g(y) = \int_{\R} e^{ixy} f(x) dx$$  is differentiable at every $y \in \R$.
\end{exer}

\begin{exer}  Prove that, if $f$ is a real-valued Lebesgue integrable function on $\R$, then $$\lim_{x \rightarrow 0} \int_{\R} |f(x+t)-f(t)|\,dt=0.$$
\end{exer}

\begin{exer}   Let $f\in L^1(X,\mathfrak{A},\mu)$, a $\sigma$-finite measure space with $X=\dot{\bigcup}_{n\in \N}X_n$ such that $\mu(X_n)<\infty$ for each $n$.  Let $A_n\in \mathfrak{A}$ be such that $\mu(A_n)\to 0$.
\begin{enumerate}
\item  Prove:  $\displaystyle \lim_{n\to \infty} \int_{A_n}f\,d\mu=0.$
\item  Prove or disprove: $\displaystyle \lim_{N \to \infty} \int_{\bigcup_{n\ge N}X_n}f(x)\,dx=0$.
\end{enumerate}
\end{exer}

\begin{exer}  Give an example of a sequence of uniformly bounded measurable
functions $f_n$ on $[0,1]$ that converges in measure to $0$ as $n \to \infty$, yet the sequence $f_n(x)$ does not converge for any $x \in [0,1].$
\end{exer}



\begin{exer}  For each $n\ge 3$ let
$$f_n(x)=\begin{cases}
n^2x,\ &0\le x<\frac 1n\\
2n-n^2x,\ &\frac{1}{n} \le x\le \frac{2}{n}\\
0,\ &\frac{2}{n}<x\le 1.
\end{cases}$$
Sketch the graphs of $f_3$ and $f_4$.  Prove that if $g$ is a continuous real-valued function on $[0,1]$, then $$\lim_{n \rightarrow \infty} \int_{0}^{1} f_n (x) g (x) dx = g (0).$$ (Hint:  First show that $\int_0^1f_n(x)\,dx=1$.)
\end{exer}



\begin{exer} Assume that the real-valued measurable function $f(t,x)$ and its partial derivative ${{\partial} \over {\partial t}} f (t,x)$ are bounded on $[0,1]^2$. Show that for $t \in (0,1)$ $$\frac{d}{dt} \left [{\int_{0}^{1} f(t,x) \, dx} \right ] = \int_{0}^{1} \frac{\partial}{\partial t} f (t,x) \, dx .$$ (Hint:  Consider the difference quotient for the derivative on the left.)
\end{exer}




\begin{exer}   Prove that if $f_n$ is Lebesgue integrable on $[0,1]$ for each $n\in\N$, and $$\sum_{n\in \N} \int_{0}^{1}  |f_n (x)|\, dx < \infty,$$ then $\sum_{n\in \N} f_n(x)$ is convergent almost everywhere, and
$$\int_{0}^{1} \sum_{n\in \N} f_n(x)\,dx=\sum_{n\in \N} \int_{0}^{1}f_n(x)dx.$$
\end{exer}

\begin{exer}   Let $f \in L^1(\R)$.  Prove that $\lim_{n\to \infty} \int_{0}^{n} e^{-nx}f(x)\,dx=0.$
\end{exer}

\begin{exer} Let $f$ be a nonnegative integrable function on $[0,1]$. Suppose that for every $n\in {\N}$ $$\int_{0}^{1}(f(x))^n\, dx=\int_{0}^{1}f(x)\, dx.$$ Show that $f$ is almost everywhere the characteristic function for some measurable set.
\end{exer}

\begin{exer} Provide an example of a sequence $\{f_n\}$ of measurable functions on $[0,1]$ such that $f_n\to f$ almost everywhere, and $f_n\geq 0$, yet $\liminf \int_0^1 f_n\,dl \neq \int_0^1 f\,dl$.
\end{exer}

\begin{exer}
Determine $$\lim_{n\to\infty}\int_1^{\infty}\sin\left(\frac xn\right)\frac {
n^3}{1+n^2x^3}\,dx.$$
\end{exer}

\begin{exer}
Give an example of a sequence of measurable real-valued functions $f_n$ such that $$\liminf\int f_n(x)\,dx<\int\liminf f_n(x)\,dx.$$ Also give a sequence where $f_n\ge 0$ and $$\int\liminf f_n(x)\,dx<\liminf\int f_n(x)\,dx.$$
\end{exer}

\begin{exer}
Prove the function $$\phi (x)=\int_0^{\infty}\frac {t^{x-1}}{1+t}\,dt$$
is continuous for $0<x<1$.  Is $\phi (x)$ differentiable on $0<x<1$?
\end{exer}

\begin{exer} Let $f$ be a real-valued function on $\R\times \R$. Suppose that $$x\mapsto f(x,y)$$ is continuous for each $y$, and that $$y\mapsto f(x,y)$$ is Lebesgue measurable for each $x$.  Show $f$ is Lebesgue measurable as a function on the plane. (Hint: Express $f$ as a pointwise limit of measurable functions on the plane.)
\end{exer}

\begin{exer}
Let $p(x_1,x_2)$ be a polynomial in two variables, but not the zero polynomial. Prove that the set of points $x\in \R^2$ with $p(x)=0$ has 2-dimensional Lebesgue measure $(l\times l)\left(p^{-1}(0)\right)=0$.
\end{exer}

\begin{exer}
Let $(X,\mathfrak{A},\mu)$ be a measure space, where $\mathfrak{A}$ is the $\sigma$-field generated by a field $\mathfrak{E}$ and $\mu$ is the countably additive measure arising via the E. Hopf Extension theorem from a finitely additive measure on $\mathfrak{E}$.  Suppose $f$ is integrable on this measure space and that $\int_Ef\,d\mu=0$ for each $E\in \mathfrak{E}$.  Show that  $f(x)=0$ $\mu$-almost everywhere.
\end{exer}

\begin{exer}
Prove that the family $\mathcal{S}$ of step-functions having finite carrier on the real line is dense in $L^1(\R)$.
\end{exer}

\begin{exer}
Suppose $f\in L^1(X,\mathfrak{A},\mu)$. Show if $\int_Af\,d\mu\ge 0$ for all $\mu$-measurable sets $A$, then $f(x)\ge 0$ for almost all $x$.
\end{exer}

\begin{exer}
Show if $f$ is an integrable function on $\R^d$ and one has $$\int_ {\R^d}fh\,dl=0$$ for every bounded continuous function $h$ on $\R^d$, then $f=0$ almost everywhere.
\end{exer}

\begin{exer}
Let $f$ be a Lebesgue integrable function on $\R$ and let $E$ be a bounded measurable set. Let $x_n$ be a sequence in $\R$ with $|x_n|\to\infty$ as $n\to\infty$. Show $$\int_{x_n+E}f\,dl\to 0$$ as $n\to\infty $.
\end{exer}

\begin{exer}
Let $f$ be a real-valued measurable  function on a complete measure space $(X,\mathfrak{A},\mu)$.  If $g=f$ almost everywhere, prove that $g$ is measurable.
\end{exer}

\smallskip
\hrulefill
\smallskip

\emph{The following three problems are related, but each has sufficient substance to stand alone as a problem.}

\smallskip

\begin{exer}  Suppose $h:\R\times \R\to \R$ is given by $h(x,y)=x-y$. Let $N\subset \R$ be \emph{both} a Borel set and a Lebesgue null set. Prove that  $h^{-1}(N)$ is a null set in $\R^2$. (Hint: Use Fubini's Theorem. Be sure to justify that your integrand for Fubini's Theorem is measurable.)
\end{exer}

\begin{exer} Let $g:\R^2 \to \R$ be \emph{Borel measurable} and $\phi:\R\to \R$ be \emph{Lebesgue measurable}.  Show that $\phi\circ g$ is \emph{Lebesgue measurable} on $\R^2$, \emph{provided} that $g^{-1}$ carries Lebesgue null sets to Lebesgue null sets.
\end{exer}

 \begin{exer} Suppose $f$ and $h$ are in $L^1\left(\R\right)$, and let $H(x,y)=f (x-y)h(y)$. Show that $H\in L^1\left(\R^2\right)$.  Use this to show the function $$f*h(x)=\int_{\R}f(x-y)h(y)\,dy$$ is defined almost everywhere and is an integrable function on $\R$. Then show that $\|f*h\|_1\le \|f\|_1\|h\|_1$. (Hint: Use Fubini's Theorem.  You may \emph{assume} the measurability of the integrand of the double integral over the plane.)
 \end{exer}

\smallskip
\hrulefill
\smallskip

\begin{exer}
Let $f:[0,1]\to \R$ be Lebesgue measurable and suppose $H(x,y)=f(x) -f(y)$ is integrable on $[0,1]^2$. Show $f$ is integrable on $[0,1]$ and determine the integral of $H$.
\end{exer}

\begin{exer}
Suppose $f$ is integrable on $\R^d$. Prove $$H(\alpha)=m\{x\mid f(x)>\alpha \}-m\{x\mid f(x)<-\alpha \}$$ is integrable for $\alpha\ge 0$ and show $$\int_{[0,\infty )}H\, dl_1=\int_{\mathbb R^d}f\,dl.$$
\end{exer}

\begin{exer}
Let $(X,\mathfrak{A},\mu)$ be a measure space for which $\mu(X)<\infty$. Show $$L^q(X,\mathfrak{A},\mu)\subseteq L^p(X,\mathfrak{A},\mu)$$ whenever $1\le p\le q\le\infty$.
\end{exer}

\begin{exer}
Let $(X,\mathfrak{A},\mu)$ be a probability space, meaning that the nonnegative measure $\mu(X)=1$. Suppose that $f\in L^p(X,\mathfrak{A},\mu)$.
    \begin{enumerate}
    \item Use Jensen's inequality to prove that $$\left( \int_X |f|^p\,d\mu\right)^\frac{1}{p} \le \left(\int_X|f|^q\,d\mu\right)^\frac{1}{q},$$ provided that $1\le p<q< \infty$.
    \item Prove that $\|f\|_p\le \|f\|_q$ for all $1\le p\le q\le \infty$.
    \end{enumerate}
\end{exer}

\begin{exer}
Let $f\in L^1(\R^2)$. Show $\int_0^1f(x,y+n)\,dl(y)$ exists a.e. in $x$ and defines a function $H_n$ in $L^1(\R)$. Then determine if the sequence $ H_n$ has a limit in $L^1(\R)$.
\end{exer}

\begin{exer}
Let $f$ be defined on a measure space, $(X,\mathfrak{A},\mu)$. Suppose that for each $\epsilon>0$, there is a set $W\in \mathfrak{A}$ such that $\mu(W) <\epsilon$ and $f\big|_{W^c}$ is measurable on $W^c$. Prove that $f$ is Lebesgue measurable.
\end{exer}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



\section{Bounded Variation, Absolutely Continuous Functions and Measures, Convex Functions, the Fundamental Theorem of
Calculus, and the Radon-Nikodym Derivative.}

\nopagebreak

\nopagebreak

\emph{Integrals on $\R$ or $\R^n$ refer to Lebesgue measure. Other problems concern integrals on an arbitrary measure space $(X,\mathfrak{A},\mu)$.}
\nopagebreak

\nopagebreak

\begin{exer}
Let $F$ and $G$ be functions on $[a,b]$ into $\R$ and let $c$ be a constant. If $F$ and $G$ have bounded variation, prove that $$V_a^b(F+G)\le V_a^bF+V_a^bG,$$ and that $V_a^b cF=|c|V_a^bF$.
\end{exer}

\begin{exer}
Show $f$ has bounded variation if and only if both $f^{+}$ and $f^{-}$ have bounded variation. {\bf Hint:} Show that $|f|$ has bounded variation.
\end{exer}

\begin{exer}
Show if $f$ is absolutely continuous on $[a,b]$, then the total variation $V_a^xf$ where $a\le x\le b$ is given by $$V_a^xf=\int_a^x|f'(t)|\,dl(t).$$ {\bf Hint:} First show $F'(x)\ge |f'(x)|$ a.e by using $V_x^{x+h}f\ge |f
(x+h)-f(x)|$ and then conclude $\int_a^c|f'(x)|\,dx\le\int_a^cF'(x)\,dx\le V_a^cf$. Then show $V_a^cf\le\int_a^c|f'(x)|\,dx$.
\end{exer}

\begin{exer}
Show $f:[a,b]\to \R$ is absolutely continuous if and only if $f^{+}$ and $f^{-}$ are absolutely continuous on $[a,b]$.
\end{exer}

\begin{exer}
Let $V_a^xg$ denote the total variation of $g$ from $a$ to $x$. Show if $f$ is absolutely continuous on $[a,b]$, then $V_a^xf=V_a^x|f|$ for $a\le x\le b$ and then give an example of a function $f$ of bounded variation where $V_a^xf$  and $V_a^x|f|$ are different. You may use the fact that $V_a^xf=\int_a^x|f'|\,dl$ if $f$ is absolutely continuous.
\end{exer}





\comment{
\begin{exer}
Show if $f$ is absolutely continuous on $[a,b]$, then
$V_{a}^{x}f=V_{a}^{x}|f|$ and then give an example of a function
$f$ of bounded variation where $V_a^xf$ and $V_a^x|f|$ are
different.
\end{exer}
}


\begin{exer}  If $f$ is continuous on an interval $[a,b]$ and has
a bounded derivative in $(a,b)$,
show that $f$ is of bounded variation on $[a,b]$.  Is the
boundedness of $f^{\prime}$ necessary for $f$ to be of bounded
variation?  Justify your answer.
\end{exer}




%\item
%\begin{enumerate}
%  \item Prove that if a real-valued function $f$ is of bounded variation
%on an interval $[a,b]$, then $f$ has right and left-hand limits at
%all $x \in (a,b)$.
%  \item Prove that a function $f : [a,b] \rightarrow \RR$
%of bounded variation has at most countably many points of
%discontinuity.
%\end{enumerate}

\begin{exer}  Let $f (x) = x^2 \sin ({{1} \over {x}})$, $g(x) = x^2 \sin ({{1} \over {x^2}})$
for $x \neq 0$, and $f(x) = g(x) = 0$ for $x = 0$.

\begin{enumerate}
\item  Show that $f$ and $g$ are differentiable everywhere (including at $x = 0$).
\item  Show that $f$ is bounded variation on the interval $[-1,1]$, but $g$ is not.
\item  Let $f(x) = x \sin(1 \slash x)$ for $x \neq 0$ and $f(x) = 0$ for $x = 0$. Is $f$ of bounded variation on $[-1,1]$?
\end{enumerate}
\end{exer}

\begin{exer}  Show that  the product of two absolutely continuous functions on
a closed finite interval $[a,b]$ is absolutely continuous.
\end{exer}

\begin{exer} A real-valued function $f$ on an interval $I$ for which there
exists a constant $C$ such that
 $$|f(x)-f(y)|\le C|x-y|$$ for all $x$ and $y$ in $I$ is
called a \emph{Lipschitz function}.
    \begin{enumerate}
  \item Show that a Lipschitz function is absolutely continuous.
  \item  Show that an absolutely continuous function $f$ on an interval is Lipschitz if
and only if $f^{\prime}$ is essentially bounded.
  \end{enumerate}
\end{exer}

%\item  Let $f$ be absolutely continuous in the interval $[\epsilon, 1]$
%for each $\epsilon > 0$.  Does the continuity of $f$ at $0$ imply
%that $f$ is absolutely continuous on $[0,1]$?  What if $f$ is also
%of bounded variation on $[0,1]$?

\begin{exer} Let $f$ be a nonnegative integrable function on $[0,1]$ and let
$I=\int_{0}^{1 }f(x)\, dx$. Show that
$$\sqrt{1+I^2}\le
\int_{0}^{1}\sqrt{1+f^2(x)}\, dx\le 1+I.$$
\end{exer}

\begin{exer} Suppose $f$ is a nonnegative integrable function on $[0,1]$.
Prove that $$ \sqrt{ \int_{0}^{1}f(t)\, dt}\ge \int_{0}^{1}\sqrt{f(t)}\, dt.$$
\end{exer}

\begin{exer}
\begin{enumerate}
  \item Provide an example of a function on $[0,1]$ that is not
absolutely continuous but is of bounded variation.
  \item Provide examples of two different continuous functions on
$[0,1]$ that have the same derivative {\it a.e.} and that are both
equal to zero at 0.
\end{enumerate}
\end{exer}





%
%\item How is the total variation of an absolutely continuous function
%$f$ on $[0,1]$ related to \nolinebreak $f'$?
%


%\item Prove that, if $F$ and $G$ are absolutely continuous real-valued
%functions on $[0,1]$, then $FG$ is absolutely continuous on $[0,1]$.



\begin{exer} Suppose $F$ is absolutely continuous on $[0,1]$ and that $g\in
L^1([0,1])$, with $\int_0^1 g=0$.  Prove the ``integration by
parts'' law:
\[
\int_0^1 F(x)\,g(x)\,dx \,\,\,=\,\,\,
   -\int_0^1\left[F'(x){\,\int_0^x\!g}\right]\,dx
\]
\end{exer}


\begin{exer}
\begin{enumerate}
  \item Provide an example of a function of unbounded variation on $[0,1]$
that has a derivative equal to zero at almost all $x\in[0,1]$.
  \item Provide an example of a function that is absolutely continuous on
$[0,1]$ but has an unbounded derivative.
\end{enumerate}
\end{exer}

\begin{exer} Prove that, if $f$ is differentiable a.e. on $[0,1]$ and $f'$ is
not in $L^1[0,1]$, then $f$ is not of bounded variation on $[0,1]$.
\end{exer}

\begin{exer} Suppose that $f$ is a real-valued function of bounded variation
on $[0,1]$. Prove that
\begin{enumerate}
  \item $f$ has a right- and left-hand limit at each point in $(0,1)$;
  \item $f$ can have only countable many points of discontinuity;
  \item If, in addition to being of bounded variation on $[0,1]$, $f$ is
absolutely continuous on $[0,T]$ for each $T<1$, then there exists an absolutely continuous function $g$ on $[0,1]$ that coincides with $f$ on $[0,1)$.
\end{enumerate}
\end{exer}

\begin{exer}
Suppose that both $f$ and $\frac{\partial f}{\partial y}$ lie in $L^1\left([a,b]\times [c,d]\right)$. Suppose also that $f(x,y)$ is absolutely continuous as a function of $y$ for almost all fixed values of $x$. Prove that $$\frac{\partial}{\partial y} \int_a^bf(x,y)\,dl(x) = \int_a^b \frac{\partial f}{\partial y} (x,y)\,dl(x)$$ for almost all $y$. Take care to establish that both sides exist. Hint: Use Fubini's Theorem to prove that $$g(y)=\int_a^bf(x,y)\,dl(x) -\int_c^y \left( \int_a^b \frac{\partial f}{\partial t}(x,t)\,dl(x) \right) dl(t)$$ is a constant function of $y$.
\end{exer}

\begin{exer}
Suppose that we have two decompositions of the measure space for a \emph{signed measure} $\mu$, as in the Hahn Decomposition Theorem. That is $$X=P\cup N=P'\cup N',$$ where each measurable subset of $P$ or $P'$ has nonnegative measure, and each measurable subset of $N$ or $N'$ has non-positive measure. Prove that every measurable subset of  $$(P\vartriangle P')\cup (N\vartriangle N')$$ is a $\mu$-null set.
\end{exer}

\begin{exer}
Suppose that the (nonnegative) measures $\lambda,\, \mu,\, \nu$ on a $\sigma$-field $\mathfrak{A}\subset \mathfrak{P}(X)$ have the relationship $$\lambda\prec \mu \prec \nu$$ where $\lambda$ and $\mu$ are finite and $\nu$ is $\sigma$-finite. Prove that $\lambda\prec \nu$ and that $$\frac{d\lambda}{d\nu} = \frac{d\lambda}{d\mu} \frac{d\mu}{d\nu}.$$ (Hints:  Let $f=\frac{d\lambda}{d\mu},\ g=\frac{d\mu}{d\nu},\  h=\frac{d\lambda}{d\nu}$ and show that there is a sequence $f_n$ of special simple functions such that $f_n\nearrow f$ pointwise almost everywhere.  Show that $\left| \lambda(A) -\int_Af_n\,d\mu\right|\to 0$ for all $A\in \mathfrak{A}$ as $n\to \infty$.  Show that $ \int_A f_n\,d\mu = \int_Af_n\,g\,d\nu$ for all $A\in \mathfrak{A}$. Use the essential uniqueness of the Radon-Nikodym derivative to complete the proof that $f_ng_n\to h$.)
\end{exer}

\begin{exer}
Let $E_1\supset E_2\supset \ldots\supset E_n\supset \ldots$ be a decreasing nest of measurable sets in the complete measure space $(X,\mathfrak{A},\mu)$. Let $f$ be integrable on $(X,\mathfrak{A},\mu)$ and suppose $$\mu\left(\bigcap_{n\in \N} E_n\right) = 0.$$ Use the Radon-Nikodym derivative to prove that $\int_{E_n}f\,d\mu\to 0$ as $n\to \infty$ by letting $\nu$ be defined by $\frac{d\nu}{d\mu}=f$.
\end{exer}

\begin{exer}
We consider the relationship between the concept of absolute continuity of functions on the real line with that of absolute continuity of measures.

    \begin{enumerate}
    \item If $\lambda$ and $\mu$ are any two finite, positive measures on a $\sigma$-field $\mathfrak{A}\subset \mathfrak{P}(X)$, prove that $\lambda \prec \mu$ if and only if they satisfy the following condition: for each $\epsilon>0$ there exists a $\delta >0$ such that $\mu(A)<\delta$ implies that $\lambda(A)< \epsilon$. (Hint: For one direction, use the Radon-Nikodym theorem.)

    \item Suppose now that the finite measure $\lambda$ is defined on the Lebesgue measurable sets of $([a,b),\mathfrak{L})$. Define $f(x)=\lambda[a,x)$ for all $x\in [a,b]$. Prove that $f$ is an absolutely continuous function on $[a,b]$ if and only if $\lambda\prec l$. (Hint: From right to left is easy by part (a).  For the other direction, prove that $\lambda(A)=\int_A f'\,dl$ for all $A\in \mathfrak{L}$.)
    \end{enumerate}
\end{exer}

\begin{exer}
Let $\mu$ and $\nu$ be nonnegative finite measures on $(X,\mathfrak{A})$.  If $\nu \perp \mu$ and $\nu \prec \mu$, prove that $\nu=0$, the identically zero measure on $\mathfrak{A}$.
\end{exer}

\begin{exer}
Let $f$ be a monotone increasing function on $[a,b]$ and define a measure $\mu$ on the Borel sets of $[a,b]$ by letting it assign to each interval $[a,x)$ the measure $\mu[a,x)=f(x)-f(a)$. (This definition extends easily to elementary sets and then to Borel sets.)
    \begin{enumerate}
     \item Let $\mu_1$ be the absolutely continuous part of $\mu$ with respect to Lebesgue measure, and find the Radon-Nikodym derivative $$\frac{d\mu_1}{dl}.$$
    \item  Show that the singular part $\mu_{0}$ and the absolutely continuous part $\mu_1$ of $\mu_f$ can be used to define absolutely continuous and singular parts of the function $f$.
    \end{enumerate}
\end{exer}




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



\section{$L^p$, $L^2$, Dual Spaces, Topics from either 7311 or 7330.}
\nopagebreak

\emph{Integrals on $\R$ or $\R^n$ refer to Lebesgue measure. Other problems concern integrals on an arbitrary measure space $(X,\mathfrak{A},\mu)$.}
\nopagebreak
\medskip

\begin{exer}   Let $C([a,b])$ be the space of real continuous
functions on a closed interval $[a,b]$ equipped with the $\sup$
norm. Let ${\cal{M}} = \{f \in C([a,b])$  : $f (x) > 0$ for all $x
\in [a,b] \}$. Show that $\cal{M}$ is an open subset of $C([a,b])$.
\end{exer}

\begin{exer}   Let $X$ be the normed linear space obtained by putting the norm
$\| f \|_1 = \int_{0}^{1} | f(t)|\, dt$ on the set of real
continuous functions on $[0,1]$.

\begin{enumerate}
\item   Show that $X$ is not a Banach space.
\item   Show that the linear functional $\Lambda f = f (1 \slash 2)$ is
not bounded.
\end{enumerate}
\end{exer}

\begin{exer}  Show that $L^p [0,1]$ is separable for $1 \leq p < \infty$, but not separable for $p = \infty$.
\end{exer}

\begin{exer}  Consider a Lebesgue measurable function $f$ on $\R$ with
$\int_\R f(t)^2 \, dt < \infty$.  Show that the function $$g(x)=\int_{\R}f(t- x) f(t) \, dt$$ is continuous.
\end{exer}

\begin{exer}  Show that $L^p (0,1) \subset L^q (0,1)$ for any $p > q \geq 1$.
Here the integrability is with respect to the Lebesgue measure.  Is
the inclusion map for $L^p (0,1)$ into $L^q (0,1)$ continuous?
\end{exer}

\begin{exer}  Prove or disprove the equality $\displaystyle{L^{\infty} [0,1] = \bigcap_{1 \leq
p < \infty} L^p [0,1]}$.
\end{exer}

\begin{exer}  Let $f \in L_p (\R)$ with $1 \leq p < \infty$.  Show
that $\displaystyle{\int_{| x | > n} | f (x) |^p \, dx \rightarrow
0}$  for $n \rightarrow \infty.$
\end{exer}

\begin{exer}  Let $g_n = n \chi_{[0,n^{-3}]}$.  Show that
$\int_0^1 f(x) g_n(x)\, dx \rightarrow 0$ as $n\to\infty$ for all $f
\in L^2 [0,1]$,
  but not all $f \in  L^1 [0,1]$.
\end{exer}

\begin{exer}  Provide an example of the following:
\begin{enumerate}
   \item A nonzero bounded linear functional on $L^p[0,1]$, $1<p<\infty$.
  \item A nonzero bounded linear functional on $\ell^\infty$.
\end{enumerate}
\end{exer}

\begin{exer} Describe precisely how the dual of $\ell^1$ is represented
concretely.
\end{exer}

\begin{exer} Why is the dual of $L^\infty$ not equal to $L^1$, in other
words, why is $L^1$ not reflexive?
\end{exer}

\begin{exer} What is the completion of the space of continuous functions on
$[0,1]$ in the $p$-norm ($1\leq p<\infty$)?  In the $\infty$-norm?
\end{exer}

\begin{exer} Let $p+q=pq$. For $g\in L^q(E)$, define $\hat g\in(L^p(E))^*$ as
$\hat g(f) = \int_Egf$.  Prove that $\|\hat g\| = \|g\|_{L^q(E)}$.
\end{exer}

\begin{exer} Prove that the linear space of finite sequences is dense in $\ell^p$ for
$1\leq p<\infty$, but it is not dense in $\ell^\infty$.
\end{exer}

\begin{exer} Prove that $L^\infty({\mathbb R}) \cap L^p({\R})$ is dense in
$L^p({\R})$.
\end{exer}

\begin{exer} Let $1<p<q<r<\infty$.  If $f\in L^p({\R})$ and $f\in
L^r({\R})$, then $f\in L^q({\R})$.
\end{exer}

\begin{exer} Prove that $\ell^\infty$ is not separable, that is, it has no
countable dense set.
\end{exer}

\begin{exer}   Let $f \in L^{\infty} [0,1]$ and $\int_{0}^{1} x^n
f(x)\, dx = 0$ for $n \in \NN$.  Show that $f = 0$ a.e.
\end{exer}

\begin{exer}  Let $f$ be a nonnegative Lebesgue measurable function on
$(0, \infty)$ such that $f^2$ is integrable.  Let $F(x)=\int_{0}^{x}f(t)\, dt$ where $x > 0$.  Show that $\lim_{x \rightarrow 0^+}\frac{F(x)}{\sqrt{x}} =0.$
\end{exer}

\begin{exer}  Let $f$ be a measurable nonnegative function on $(X,\mathfrak{A},\mu)$ $0<\mu(X)<\infty$.  Let $$\|f\|_{\infty} = \sup \left\{M \mid \mu(f^{-1}(M-\delta,M))>0\, \forall \delta>0\right\}.$$  Show that $$\lim_{n \rightarrow \infty} \left (\int_{X}  f(x)^n \, d\mu(x) \right )^{\frac{1}{n}}
 = \|f\|_{\infty}.$$
\end{exer}

\begin{exer} Let $1\le p<\infty$. Suppose $f_n\in L^p([0,1])$, $\norm{f_n}_p\le 1$, and $f_n\rightarrow f$ almost everywhere.
    \begin{enumerate}
    \item Show that $f\in L^p([0,1])$ and $\norm{f}_p\le 1$.
    \item Let $1<p<\infty$ and $g\in L^q([0,1])$ where $\frac 1p+\frac 1q=1$. Prove that $$\int_0^1f_ng\to \int_0^1fg.$$
    \item Give an example to show that the conclusion in item \emph{b} would be false if $p=1$.
    \end{enumerate}
\end{exer}

\begin{exer}
Let $1<p<\infty$. Suppose $1\le q\le\infty$ where $q\ne p$. Give
an example of a measurable function $f$ on $\R$ such that
$f\in L^p(\R)$ but $f\notin L^q(\R)$.
\end{exer}

\begin{exer}
Suppose $1\le p\le q\le\infty$ and $\frac 1p+\frac 1q=1$.
\begin{itemize}
\item[(a)]
Let $f\in L^p(\R)$ where $1\le p<\infty$. Find a function
$g\in L^q (\R)$ so that $|g|_q=1$ and $\int fg\,dm=|f|_p$.
\item[(b)]
Give an example of an $f\in L^{\infty}(\R)$ for which there
is no $g\in L^1(\R)$ such that $|g|_1=1$ and $\int
fg\,dm=|f|_{\infty}$.
\end{itemize}
\end{exer}

\begin{exer}
Let $f$ and $g$ be functions in $L^2(\R^d)$. Suppose $\|x_n\|\to\infty$ as $n\to\infty$. Show $$\int f(x)g(x+x_n)\,dl(x)\to 0\text{\ as } n\to\infty.$$
\end{exer}

\begin{exer}
Let $1\le p<\infty$. Suppose $f_n\in L^p(\R)$ and $f\in L^p(\R)$ and $$\|f_n-f\|_p\to 0\text{\ as }n\to\infty .$$ Show that there is a subsequence $f_{n(k)}$ such that $f_{n(k)}(x)$ converges a.e.\ to $f(x)$ as $n\to\infty$.
\end{exer}

\begin{exer}  A function $f : [0,1] \to L^1  [0,1]$ is called Lipschitz
continuous if there exists $M > 0$ such that $\| f(t) - f (s)\|_1
\leq M |t - s|$ for all $t, s \in [0,1]$.  It is called
differentiable at a point $s \in (0,1)$ if the quotients $\dfrac{f (t) - f (s)}{t - s}$ converge in $L^1 [0,1]$ as $t \to s$.  Let $f : [0,1] \to  L^1 [0,1]$ be given by $f (t) = {\chi}_{[0,t]}$, where ${\chi}_{[0,t]}$ denotes the characteristic function of the interval $[0,t]$.  Show that $f$ is Lipschitz
continuous and nowhere differentiable.
\end{exer}


\end{document}


\bye



















%\if@twoside
%    \def\ps@headings{
%        \def\@evenhead{{\thepage}\hfil{\leftmark}\hfil}
%        \def\@oddhead{\hfil{\rightmark} \hfil{\thepage}}
%        \def\chaptermark##1{\markboth{\MakeUppercase{%
%                    \ifnum \c@secnumdepth>\m@ne
%                    \if@mainmatter
%\thechapter.\enspace\fi \fi ##1}}{} }
%        \def\sectionmark##1{\markright {\MakeUppercase{%
%            {\ifnum\c@secnumdepth>\z@
%\thesection.\enspace\fi ##1}}}} }
%\fi

\newtheorem{thm}{Theorem}[section]
\newtheorem{prop}[thm]{Proposition}
\newtheorem{lem}[thm]{Lemma}
\newtheorem{cor}[thm]{Corollary}
\newtheorem{defi}[thm]{Definition}
\newtheorem{ex}[thm]{Example}
\newtheorem{rem}[thm]{Remark}

\section{\bf  Riemann integral, Lebesgue integral (via
completion), convergence theorems} \pni
\begin{exer}{6.1}  Let $f$ be a positive function on $(0,1]$ such that $f$ is Riemann
integrable on $[t,1]$ for all $t~\in~(0,1)$, but $\lim_{x \rightarrow 0^{+}} f(x) = \infty$.  Assume that the improper (Riemann) integral $(R)
\int_{0}^{1} f(x) dx$ exists.  Show that $f$ is a measurable function and is
Lebesgue integrable and
$$\int_{[0,1]} f(x) dx~=~(R) \int_{0}^{1} f(x) dx.$$

\begin{exer}{6.2}  For each of the following problems, check whether the limit exists.
If so, find its value.
$${\rm (a)} \lim_{n \rightarrow \infty} \int_{1}^{n} ({1 - {x \over n}})^n
dx,~~~~{\rm (b)} \lim_{n \rightarrow \infty} \int_{1}^{2 n} ({1 - {x \over n}}
)^n dx.$$

\begin{exer}{6.3}    \hfill
\itemitem{a}  Give a necessary and sufficient condition for a bounded function
on $[0,1]$ to be Riemann integrable.
\itemitem{b}  Let $({r_n})$ be an enumeration of the rationals in $[0,1]$.
Define $f$ on $[0,1]$
by
$$f(x)~=~\cases 1\slash n&if\endcases $$
$$f(x)~=~\cases 1\slash n&if\endcases $$
$$f(x)~=~\cases 1\slash n&if\endcases $$
$$f(x)~=~\cases 1\slash n&if\endcases $$
$$f(x)~=~\cases 1\slash n&if\endcases $$
$$f(x)~=~\cases 1\slash n&if\endcases $$
$$f(x)~=~\cases 1\slash n&if\endcases $$
$$f(x)~=~\cases{
1 \slash n & if $x = r_n$ \cr
0 & if $x$ is irrational
}
$$\itemitem{}Is$$

f$Riemannintegrableon$[0,1]$?Explain!\itemitem{c}Showthatif$$f(x)~
=~\left\{\eqalign{1&x\rm rational\cr-1&x\rm irrational\cr}\right\}$$\itemitem{}
then$f$isnotRiemannintegrableonanyinterval$[0,1]$.Is$f$Lebesguein
tegrable?Explain$
f$Riemannintegrableon$[0,1]$?Explain!\itemitem{c}Showthatif$$f(x)~
=~\left\{\eqalign{1&x\rm rational\cr-1&x\rm irrational\cr}\right\}$$\itemitem{}
then$f$isnotRiemannintegrableonanyinterval$[0,1]$.Is$f$Lebesguein
tegrable?Explain$
f$Riemannintegrableon$[0,1]$?Explain!\itemitem{c}Showthatif$$f(x)~
=~\left\{\eqalign{1&x\rm rational\cr-1&x\rm irrational\cr}\right\}$$\itemitem{}
then$f$isnotRiemannintegrableonanyinterval$[0,1]$.Is$f$Lebesguein
tegrable?Explain$
f$Riemannintegrableon$[0,1]$?Explain!\itemitem{c}Showthatif$$f(x)~
=~\left\{\eqalign{1&x\rm rational\cr-1&x\rm irrational\cr}\right\}$$\itemitem{}
then$f$isnotRiemannintegrableonanyinterval$[0,1]$.Is$f$Lebesguein
tegrable?Explain$
f$Riemannintegrableon$[0,1]$?Explain!\itemitem{c}Showthatif$$f(x)~
=~\left\{\eqalign{1&x\rm rational\cr-1&x\rm irrational\cr}\right\}$$\itemitem{}
then$f$isnotRiemannintegrableonanyinterval$[0,1]$.Is$f$Lebesguein
tegrable?Explain$
f$Riemannintegrableon$[0,1]$?Explain!\itemitem{c}Showthatif$$f(x)~
=~\left\{\eqalign{1&x\rm rational\cr-1&x\rm irrational\cr}\right\}$$\itemitem{}
then$f$isnotRiemannintegrableonanyinterval$[0,1]$.Is$f$Lebesguein
tegrable?Explain$
f$ Riemann integrable on $[0,1]$?  Explain!
\itemitem{c}  Show that if
$$f(x)~=~
\left \{
\eqalign{1& x {\rm rational} \cr
                 -1& x {\rm irrational} \cr}
\right \}
$$
\itemitem{}  then $f$ is not Riemann integrable on any interval $[0,1]$.  Is $f$
Lebesgue integrable?  Explain!

\begin{exer}{6.4}  Show there are no bounded sequences $\{a_n \}$ and $\{b_n \}$ for
which $f_n (x) = a_n \sin (2 \pi n x) + B_n \cos (2 \pi n x )$ converges to 1
almost everywhere on $[0,1]$.

\begin{exer}{6.5}  Let $f(x)$ be a real-valued measurable function on $[0,1]$.  Show
$\lim_{n \rightarrow \infty} \int_{0}^{1} ( \cos (\pi f(x)))^{2 n} dx = m
\{x: f(x) {\rm is~a n~interger} \}$ where $m$ denotes Lebesgue measure.

\begin{exer}{6.6}    \hfill
\itemitem{a}  Show that $f(x) = x^{-r}$ is a Lebesgue integrable function on
$[0,1]$ if $0 < r < 1$.
\itemitem{b}  If $0 < r < 1$ let $a_n = \int_{0}^{1} {{dx} \over {n^{-1}
+ x^{r}}}$ (Lebesgue integral).  Compute $\lim_{n \rightarrow \infty} a_n$.  Be
sure to quote the appropriate integration theorems to justify your calculations.

\begin{exer}{6.7}  Let $f_n : \RR \rightarrow \RR$ be defined
by
$$f_n(x)=\cases \endcases $$
$$f_n(x)=\cases \endcases $$
$$f_n(x)=\cases \endcases $$
$$f_n(x)=\cases \endcases $$
$$f_n(x)=\cases \endcases $$
$$f_n(x)=\cases \endcases $$
$$f_n(x)=\cases \endcases $$
$$f_n(x) = \cases{
$1 \over n$ & if $| x | \leq n$ \cr
0 & if $| x | > n$. \cr
}
$$\itemitem{a}Showthat$$

f
f
f
f
f
f
f_n$ converges to $0$ uniformly on $\RR$
and that $\lim_{n \rightarrow \infty} \int_{\RR} f_n (x) dx = 2$
while $\int_{\RR} \left (\lim_{n \rightarrow \infty} f_n \right )
(x) dx \neq 2$.
\itemitem{b}  Explain why the example in part (a) does not contradict the
Lebesgue dominated convergence theorem.

\begin{exer}{6.8}     \hfill
\itemitem{a}  Show that $f(x) = 1 \slash \sqrt{x}$ is Lebesgue integrable on
$(0,1)$.
\itemitem{b}  Find $I n f \{\int_{0}^{1} \psi | \psi~{\rm is~a~simple~function,~
a n d} \psi (x) \geq 1 \slash \sqrt{x}~{\rm o n}~(0,1) \}$.  (Simple functions are
finite linear combinations of characteristic functions of measurable sets with
extended real-valued coefficients.)

\begin{exer}{6.9}  Give an example of a sequence $\{f_n \}$ of bounded, measurable
functions on $[0,1)$ such that
$$\lim_{n \rightarrow \infty} \int_{0}^{1} | f_n | = 0$$
\item{}  but such that $f_n$ converges pointwise {\it nowhere}.

\begin{exer}{6.10}  Consider a Lebesgue-measurable function $f$ or {\bf R} with
$\int \limits_{\bf R} f(t)^2 dt < \infty$.  show that the function $g(x) =
 \int \limits_{\bf R} f(t - x) f(t) dt$ is continuous.
\begin{exer}{6.11}  Prove that $\lim_{n \rightarrow \infty} \int_{- \infty}^{\infty}
(\sin nt) f(t) dt~=~0$ for every Lebesgue integrable function $f$ or {\bf R}.
(Hint:  Do the problem first for step functions.)

\begin{exer}{6.12}  Let $f_n (x) = {{n} \over {x (ln x)^n}}$ for $x \geq e$ and
$n \in N$.
\itemitem{a}  For which  $n \in N$ does the Lebesgue integral $\int
\limits_{e}^{\infty} f_n (x) dx$ exist?
\itemitem{b}  Determine $\lim_{n \rightarrow \infty} f_n (x)$ for $x > e$.
\itemitem{c}  Does the sequence $\{ f_n \}$ satisfy the assumptions of
Lebesgue's dominated convergence theorem?

\begin{exer}{6.13}  Define $f(x) = \int_{\bf R} \cos (xy) g(y) dy$ for $x \in$
{\bf R} where $g$ is an integrable function on {\bf R}.  Show that $f$ is
continuous.

\begin{exer}{6.14}  Define $g : \RR \rightarrow \RR$ by
$g(x) = 0$ if $x$ is irrational and $g(x) = {1 \over q}$ if $x = p \slash q$
in lowest terms.  Is $g$ a Riemann integrable function?  Give a proof of your
assertion.

\begin{exer}{6.15}  Give an example of a Lebesgue integrable function $f$ on
$[0,1]$ such that $\int_{0}^{1} f(x) dx = 1$, but $f$ is not Riemann integrable.

\begin{exer}{6.16}      \hfill
\itemitem{a}  Let $f$ be in $L_{\infty} [0,1]$ such that $\int_{0}^{1} x^n
f(x) dx = 0$ for $n = 1, 2, 3, \ldots$.  Show that $f \equiv 0$.
\itemitem{b}  If, in part (a), $f$ is not continuous but just a bounded
Lebesgue measurable function, show that $f = 0$ {\it a.e.}

\begin{exer}{6.17}  Let $f$ be a nonnegative Lebesgue measurable function on
$(0, \infty)$ such that $f^2$ is integrable.  Let $F(x) = \int_{0}^{x} f(t)
dt$ where $x > 0$.  Show that
$$\lim_{x \rightarrow 0+} {{F(x)} \over {\sqrt{x}}} = 0.$$

\begin{exer}{6.18}  Let $f$ be a differentiable function on $[-1,1]$.  Prove that
the following limit exists
$$\lim_{\in \rightarrow 0} \int_{\in < | x | \leq 1} {1 \over x}
f(x) dx.$$

\begin{exer}{6.19}  Let $f_n (x) : = {{x^n} \over {n!}} e^{-x}$ for $n \in
N_0 : = \{0, 1, 2, 3\ldots \}$.
\itemitem{a}  Show that $\lim_{x \rightarrow \infty} f_n (x) = 0$ for
all $x > 0$.
\itemitem{b}  Show that $f_n \in L^1 (0, \infty)$ with $\|
f_n \|_1 = 1$ for
all $n \in N_0$.
\itemitem{c}  Show that $\lim_k \rightarrow \infty \int_{0}^{k} {{x^n}
\over {n!}} (1 - {{x} \over {k}} )^k dx = 1$ for all $n \in N_0$.

\begin{exer}{6.20}  Prove that
$$\lim_{b \rightarrow \infty} \int_{0}^{b} {{\sin x} \over {x}} dx$$
\item{}  exists but that the function ${\sin x} \over x$ is not integrated
in $(0, \infty)$.

\begin{exer}{6.21}  Compute the following limit and justify your calculations:
$$\lim_{n \rightarrow \infty} \int_{0}^{infty} \left (1 + {{x} \over {n}}
\right )^{-n} \sin \left ( {x \over n} \right ) dx$$

\begin{exer}{6.22}  Let $f$ be a continuous nonnegative function on $[a,b]$ where
$a < b$.  Let $M = max \{f(x) : a \leq x \leq b \}$.  Show
$$\lim_{n \rightarrow \infty} \left (\int_{a}^{b}  f(x)^n dx \right )^{{1} \over {n}}
 = M.$$

\begin{exer}{6.23}  Find (and justify) the limits (a) $\lim_{n \rightarrow \infty}
\int_{0}^{n} {{\sin x} \over {1 + nx^2}} dx$ and (b) $\lim_{n \rightarrow \infty}
\int_{0}^{e^n} {x \over {1 + nx^2}} dx$.

\begin{exer}{6.24}  Let $f_n(x) = \sum_{i = 0}^{n - 1} {1 \over n} f (x + {i \over n})$,
where $f$ is a continuous function on $\RR$.  Show that the sequence
of functions $\left\langle  f_n \right\rangle$ converges uniformly on every finite
segment $[a,b]$ to the function $F(x) = \int_{x}^{x + 1} f(s) ds$.

\begin{exer}{6.25}  Let $f \varepsilon L^1 (\RR)$  with respect to the
Lebesgue measure such that $\int_R | x | | f(x) | dx < \infty$.
Show that the following function
$$g(y) = \int_R e^{fxy} f(x) dx$$
\item{}differentiable at every $y \varepsilon \RR$.

\begin{exer}{6.26}  Prove that if $f$ is a real-valued Lebesgue integrable function on
{\bf R}, then
$$\lim_{x \rightarrow 0} \int | f(x + t) - f(t) | dt = 0.$$

\begin{exer}{6.27}  Let $f \in L^1 ({\bf R}).$
\itemitem{a}  Prove:  $\lim_{n \rightarrow \infty} \int_{0}^{1 \slash n} f(x)
dx = 0.$
\itemitem{b}  Must $\lim_{n \rightarrow \infty} \int_{n}^{\infty} f(x) dx
= 0.$

\begin{exer}{6.28}  Give an example of a sequence of uniformly bounded measurable
functions $f_n (x)$ on $[0.1]$ such that $m \{ x | f_n (x) \neq 0 \}
\rightarrow 0$ as $n \rightarrow \infty$, but the sequence $\{f_n (x) \}$ does
not converge for any $x \in [0,1].$

\begin{exer}{6.29}  Let
$$f_n(x)=\cases \endcases $$
$$f_n(x)=\cases \endcases $$
$$f_n(x)=\cases \endcases $$
$$f_n(x)=\cases \endcases $$
$$f_n(x)=\cases \endcases $$
$$f_n(x)=\cases \endcases $$
$$f_n(x)=\cases \endcases $$
$$f_n (x) = \cases{
$4n^2x$, & for $0 \leq x \leq 1 \slash (2n)$; \cr
$2n - n^2$ $(x - 1 \slash n)$, & for $1 \slash n \leq x \leq 2 \slash n$; \cr
$0$, & for $2 \slash n \leq x \leq 1$. \cr
}
$$
\item{} Sketch the graphs $







f_1$ and $f_2$.  Prove that if $g$ is a continuous
real-valued function on $[0,1]$, then.
$$\lim_{n \rightarrow \infty} \int_{0}^{1} f_n (x) g (x) dx = g (0).$$
\item{}  {\bf Hint:}  First show that $\int_{0}^{1} f_n dx = 1.$

\begin{exer}{6.30}  Assume that the real-valued measurable function $f(t,x)$ and its
partial derivative ${{\partial} \over {\partial t}} f (t,x)$ are bounded on $[0,1]^2$.
Show that for $t \in (0,1)$
$${{d} \over {d t}} \left [{\int_{0}^{1} f(t,x) dx} \right ] = \int_{0}^{1} {{\partial}
\over {\partial t}} f (t,x) dx$$
\item{}({\bf HINT:}  Consider the difference quotient for the derivative on the left.)

\begin{exer}{6.31}  Prove that if $f_n$ is Lebesgue integrable on $[0,1]$ for each $n$,
and $\sum_{n = 1}^{\infty} \int_{0}^{1}  | f_n (x) | dx < \infty$, then
$\sum_{n = 1}^{\infty} f_n (x)$ is convergent {\it a.e.}, and
$$\int_{0}^{1} \sum_{n = 1}^{\infty} f_n (x) dx = \sum_{n = 1}^{\infty} \int_{0}^{1} f_n (x) dx.$$

\begin{exer}{6.32}  Let $f \in L^1 (\RR)$.  Prove that $\lim_{n
\rightarrow \infty} \int_{0}^{n} e^{-n x} f(x) dx = 0.$

\begin{exer}{6.33}  Assume that $A \geq 0$, $B > 0$, and $f$ continuous and
nonnegative on $[a,b]$.  Assume that $f(t) \leq A + B \int_{a}^{t}
f(x) ds$ for $a \leq t \leq b$.  Prove that $f(t) \leq A e^{B (t - a)}$ for
$a \leq t \leq b$.
\vfill
\eject
\centerline{\bf {7. Absolutely continuous functions and the fundamental theorem of calculus}}
\pni
\begin{exer}{7.1}  Show that  the product of two absolutely continuous functions on
a closed finite interval $[a,b]$ is absolutely continuous.

\begin{exer}{7.2}     \hfill
\itemitem{a}  Show that a Lipschitz function is absolutely continuous.
\itemitem{b}  Show that an absolutely continuous function $f$ is Lipschitz if
and only if $f^{\prime}$ is bounded.

\begin{exer}{7.3}  Let $f$ be absolutely continuous in the interval $[\in, 1]$
for each $\in > 9$.  Does the continuity of $f$ at $0$ imply that $f$ is
absolutely continuous on $[0,1]$?  what if $f$ is also of bounded variation
on $[0,1]$?

\begin{exer}{7.4}  A function $f : [0,1] \rightarrow L^1  [0,1]$ is called Lipschitz
continuous if there exists $M > 0$ such that $\| f(t) - f (s) \|_1
\leq M | t - s |$ for all $t, s \in [0,1]$.  It called
differentiable at a point $s \in (0,1)$ if the differential
quotients ${{f (t) - f (s)} \over {t~-~s}}$ converge in $L^1 [0,1]$
as $t \rightarrow s$.  Let $f : [0,1] \rightarrow L^1 [0,1]$ be
given by $f (t) = \chi_{[0,t]}$, where $\chi_{[0,t]}$ denotes the
characteristic function of the interval $[0,t]$.  Show that $f$ is
Lipschitz continuous and nowhere differentiable. \vfill \eject
\centerline{\textbf{8. Basic properties of $L^p$-spaces, Riesz
representation for $L^p$-spaces}} \pni
\begin{exer}{8.1}  Show that $(L_p [0,1],\| \cdot \|_p)$ is separable for $1 \leq p < \infty$, but not separable for $p = \infty$.

\begin{exer}{8.2}  Show that $L^p (0,1) \subset L^q (0,1)$ for any $p > 1 \geq 1$.
Here the integrability is with respect to the Lebesgue measure.  Is the inclusion
map for $L^p (0,1)$ into $L^q (0,1)$ continuous?

\begin{exer}{8.3}  Prove or disprove the equality $L^{\infty} [0,1] = \cap_{1 \leq
p < \infty} L^p [0,1]$.

\begin{exer}{8.4}  Let $f \in L_p (\RR), 1 \leq p < \infty$.  Show
that
$$\int_{| x | > n} | f (x) |^p dx \rightarrow 0~{\rm f o r}~n \rightarrow \infty.$$

\begin{exer}{8.5}  Let $g_n = n \chi_{[0,n^{-3}]}$.  Show
$$\int f g + n \rightarrow 0~~~~{\rm f o r~a l l} f \in L^2 [0,1],$$
\item{}  but not all $f \in L^1 [0,1]$.


\vfill
\bye

\itemitem{c}  Show that the set $\{f \in L_ {\alpha} | \| f
\|_{\alapha} \geq 1, f(a) = o \}$ is compact in $C([a, b])$ (with the
supremum norm).   {\bf 3}



\begin{exer}{1.2}  Let $(E, \|\cdot\|)$ be a Banach space and $F$ a finite dimensional subspace.
Show that $(F, \|\cdot\|)$ is a Banach space.

\itemitem{(c)} Let $C[0,1]$ denote the Banach space of all continuous real functions on $
[0,1]$ equipped with the sup-norm and $C_0[0,1]$ the closed subspace of $
C[0,1]$ consisting of those $f$ with $f(0)=0$. Let $R[0,1]$ denote the completion of $
X$. Consider the antiderivative operator $I:C[0,1]\to C_0[0,1]$ defined by $
If=F$, where $F(t)=\int_0^tf(s)\,ds$. Show that $I$ extends to an isometric isomorphism between $
R[0,1]$ and $C_0[0,1]$.
\end

