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\leftheadtext{Ph.D.~Real Analysis Exam, Spring 1995}
\topmatter
\title
\bf \bf Ph.D. Comprehensive Examination\\
Real Analysis  Section %Real Analysis Qualifying Exam
\endtitle
\author
\bf January 1995
\endauthor
\endtopmatter
\document

%\s{NOTE}
%{Please state on the outside of your examination book which
%problems  you want graded.
%All reasoning should be justified.}
\vglue\baselineskip

\t{Part I}{Do three (3) of these problems.}

\s{I.1}
{Give an example of a closed set which contains no interval and has Lebesgue
measure equal to $2$.}

\s{I.2}
 {State each of the following inequalities. In each instance comment on the case
where equality is achieved:
 \roster
 \item H\"older inequality for $L^p$, $1 \leq p < \infty$
 \item Minkowski's inequality for $L^p$, $1 \leq p < \infty$
 \item Bessel's inequality for $L^2$
 \endroster}
 
 As an alternative to answering {\it one} of the items above, you may
demonstrate that Minkowski's inequality holds for $0< p < 1$.
 

\s{I.3}
{Give an example of a sequence of functions $\{f_n\}_{n=1}^\infty$ defined on
$[0,1]$ such that $f_n$ converges to some function $f$ in measure but $f_n$ does
not converge to $f$ a.~e.}


\s{I.4}
{A function $f:\R \to \R$ is measurable if $\{x: f(x) > \alpha\}$ is Lebesgue
measurable for each $\alpha$. Give an example of a measurable function $f$ and
a Lebesgue measurable set $E$ such that $f^{-1}(E)$ is nonmeasurable.
Hint: Consider the Cantor-Lebesgue function.}

\vglue\baselineskip
\vfill
\eject
\t{Part II}{Do two (2) of these problems.}

\s{II.1}
{For each $\alpha$, $\theta$, $0\leq \alpha \leq 1$, $0\leq \theta \leq \pi/2$,
let $\ell_{\alpha,\theta}:[0,1] \to \Bbb R$ be the function
illustrated:}\vskip1.5in

Let $S$ be the set of all finite linear combinations $\sum_{k=1}^n a_k
\ell_{\alpha_k,\theta_k}$ of the $\ell_{\alpha,\theta}$ for all possible
choices of $a_k$, $\alpha_k$, $\theta_k$. Find the uniform closure of $S$ ({\it
i.e.}, the closure of $S$ with respect to the sup norm). Prove your result.

\s{II.2}
{Prove: Every nonempty open, bounded, convex set in $\Bbb R^2$ that is symmetric
about the origin is the open unit ball for some norm on $\Bbb R^2$. Hint: use
euclidean distance to define $\|x\|$ properly and demonstrate that it is indeed
a norm.}

\s{II.3}
{Let $f$ be a nonnegative measurable function defined on $\Bbb R$, with
$\int_{\Bbb R} f< 1$. Let $f_n = f*f*\dots*f$, convolution $n$ times. 
\roster
\item Show that $f_n \to 0$ in $L^1(\Bbb R)$, as $n \to \infty$
\item Prove that $f_n \to 0$ a.~e., $n \to \infty$.
\endroster}
\enddocument