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\topmatter
\title
\bf \bf Ph.D. Comprehensive Examination\\
Real Analysis Section %Real Analysis Qualifying Exam
\endtitle
\author
\bf Fall 1995
\endauthor
\endtopmatter
\document




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

\s{I.1} {Let $\mu$ be the Lebesgue measure on $\Bbb R$. Let $\phi(x) = x^2$.
Define a measure $\nu$ by 
 $$
 \nu(A) = \mu(\phi^{-1}(A)), \text{ for all Lebesgue measurable sets }A.
 $$
 Find the Radon-Nikodym derivatives $\frac {d\mu}{d\nu}$ and $\frac
{d\nu}{d\mu}$ if they exist.}

\s{I.2} {Let $\Gamma(x) = \int_0^\infty e^{-t}t^{x-1}dt$. Show:
 \roster
 \item $\Gamma(x) < \infty$ for all $ x > 0$;
 \item $\Gamma'(x) = \int_0^\infty e^{-t} t^{x-1} \ln t\, dt$ if $x > 0$.
 \endroster}

\s{I.3}{Let $A \subset \Bbb R$ be a measurable set with positive measure. Show
there is an interval $I$ such that the measure of $I\cap A$ is larger than
$99$\% percent of the measure of $I$.}

\s{I.4} {\roster
 \runinitem Give an example of a sequence of functions $f_n$ defined on $\Bbb R$
such that, as $n \to \infty$, $f_n \to 0$ in measure, but $f_n$ does not
converge to $0$ almost everywhere.
 \item Give an example of a sequence of functions $f_n \in L^2(\Bbb R)$ such
that 
 $$
  \lim_{n\to \infty}\int_{\Bbb R} f_n g\,dx = 0 \text{ for all } g \in
L^2(\Bbb R),
 $$
 but $f_n$ does not converge to $0$ as $n \to \infty$, in $L^2(\Bbb R)$.
 \endroster}

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

\s{II.1}
{Let $f \in L^p([0,1],dx)$, $1 < p < \infty$. Let $F(x) = \int_0^x f(t)\,dt$.
Show that 
 $$
 \lim_{h\to 0} \frac{F(x+h) - F(x)}{h^{1 - \frac 1 p}} = 0.
 $$
 }

\s{II.2}
{Let $H$ be a Hilbert space with inner product $(\ ,\ )$. Let
$\{e_\alpha\}_{\alpha \in I}$ be an orthonormal basis for $H$. Consider a
sequence of elements $\{x_n\}$ in $H$. Show that
 $$
 \lim_{n \to \infty}x_n = x \text{ in the weak topology }
 $$
 if and only if
 \roster
 \item"{\it i})" $\lim_{n\to \infty} (x_n,e_\alpha) = (x,e_\alpha)$ for all
$\alpha \in I$;
 \item"{\it ii})" $\sup_n \|x_n\| < \infty$
 \endroster}

\s{II.3}
{Given a Lebesgue-integrable function $f$ on $\Bbb R$, set
$F(a)=\int_{-\infty}^\infty f(x)\cos(ax)\,dx$. Show that $F$
is uniformly continuous at every point.}

\enddocument