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\title
\bf \bf Ph.D. Comprehensive Examination\\
Real Analysis Section %Real Analysis Qualifying Exam
\endtitle
\author
\bf Spring 1999
\endauthor
\endtopmatter
\document

\bf Justify carefully all reasoning.\rm

\def\diam{\text{diam}}
\def\Union{\bigcup}

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

\s{I.1} {
Define the Lebesgue measure $|A|$ of a set $A\subset\bold R$.
Show that 
$$|A|=\inf\left\{\sum_{n=1}^\infty \diam(A_n): 
A\subset\Union_{n=1}^\infty A_n,A_n\text{ arbitrary}\right\}.$$
Here $\diam(A)=\sup\{|x-y|:x,y\in A\}$ is the diameter of $A$.
} 
\s{I.2} 
{Show that
$$F(x)=\int_0^\infty\frac{\sin\left(xt^2\right)}{1+t^2}\,dt,\qquad x\in\bold R$$
is continuous, where ``$dt$'' denotes Lebesgue measure on $\bold R$. }

\s{I.3}{Let $f_n$ be a sequence of absolutely convergent continuous functions
in $[a,b]$ such that $f_n(a) = 0$. Suppose that $f'_n$ is a Cauchy sequence in
$L^1[a,b]$. Show that there exists $f$, absolutely continuous in $[a,b]$, such
that $f_n \to f$ uniformly in $[a,b]$.}

\s{I.4} {Let $f$ be a non-negative function on $\bold R$, let
$g(x,y)=f(4x)f(x-3y)$, and let
$\mu_n$ denote Lebesgue measure on $\bold R^n$.  Suppose that 
$\int _{\bold R^2} g \,d\mu_2 = 2$.
Calculate $\int_{\bold R}fd\mu_1$.}

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

\s{II.1}
{Let $f:[0,1]\to\bold R$ satisfy $1\le f(x)\le 2$ and let 
$$N(p)=\left(\int_0^1f(x)^p\,dx\right)^{1/p},\qquad p\not=0.$$
\roster
\item Compute $\lim_{p\to\infty} N(p)$.
\item Compute $\lim_{p\to0} N(p)$.
\item Compute $\lim_{p\to-\infty} N(p)$.
\endroster}

\s{II.2}
{Use the DCT on $(0,\infty)$ to compute 
$$\lim_{n\to\infty}\int_0^n \(1-\frac{t}n\)^ne^{it}\,dt.\qquad\qquad
(i=\sqrt{-1})$$ }

\s{II.3}
{Let $E$ be the set of $x\in[0,2\pi]$ such that $\lim_{n\to\infty}e^{inx}$
exists. Show that the Lebesgue measure of $E$ is zero.}

\enddocument

euclid%