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\topmatter
\title
\bf \bf Ph.D. Comprehensive Examination\\
Real Analysis Section
\endtitle
\author
\bf Spring 1994
\endauthor
\endtopmatter
\document




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

\s{I.1} {Let 
$$
f_n(x) = \left\{ \matrix
 n & \text{ if } 0 < x < {\tfrac 1 n}\\
 0 & \text{ if } {\tfrac 1 n} < x < 1.
 \endmatrix\right.
 $$
 \roster
 \item "(a)" Find $\lim_{n\to \infty} f_n(x)$.
 \item "(b)" Find $\lim_{n\to \infty} \int_0^1 f_n(x)\,dx$.
 \item "(c)" Is there a function $g(x) \in L^1(0,1)$ such that $g(x) \geq
f_n(x)$ for all $n$? Explain briefly.
 \endroster
} 
\s{I.2} {Let $f_n(x)$ be a sequence of functions on $\Bbb R$ with the propery
that for every fixed $x \in \Bbb R$ and every subsequence $f_{n_k}$ of $f_n$,
$f_{n_k}(x)$ has a convergent subsequence. Let $S$ be a countable subset of
$\Bbb R$. Show that there exists a subsequence of $f_n$ that converges at all
points of $S$.}

\s{I.3}{Show that given $\delta$, with $0< \delta < 1$, there exists a set
$E_\delta \subset [0,1]$, which is perfect [i.e., closed and such that every
point of the set is a limit point of the set], nowhere dense and $|E_\delta| =
1 - \delta$.}

\noindent Hint: The construction is similar to the construction of the Cantor
set, except that at the $k$-th stahe each interval removed has lenght $\delta$
times the length of the intervals used in Cantors construction.

\s{I.4} {If the {\it iterated} integrals
 $$
\int_0^1\left(\int_0^1 f(x,y)\, dy\right)dx \quad \text{and} \quad
\int_0^1\left(\int_0^1 f(x,y)\, dx\right)dy
 $$
exists as finite integrals and are equal to each other, does it
necessarily follow that the {\it multiple} integral
 $$
 \iint f(x,y)\,dx\,dy
 $$
 exists as a finite integral on the square?
 Cite a theorem, or provide a counterexample to verify a negative response.}
 
\newpage
\vglue\baselineskip
\t{Part II}{Do two (2) of these problems.}

\s{II.1}{Let $f\geq 0$ in $\Bbb R$, and set 
 $$
 g(x) = \sum_{n=-\infty}^\infty f(x+n)
 $$
 Show that if $g \in L(\Bbb R)$ then $f = 0$ a.e.}

\s{II.2}{For a given function $f$ on $[a,b]$, suppose that there is some $M
\geq 0$ such that for all distinct $x,y$
 $$
 \left|\frac{f(x) - f(y)}{x-y}\right| \leq M.
 $$
 Explain why
 \roster
 \item"(a)" $f$ must be measurable.
 \item"(b)" $f$ must be differentiable a.e. in $[a,b]$.
 \endroster}

\s{II.3}
{Let $f$ be a continuous function in $[-1,2]$. Given $x$, with $0\leq x\leq 1$,
and $n \geq 1$ define the sequence of functions
 $$
 f_n(x) = \frac n 2 \int_{x-\frac 1 n}^{x+\frac 1 n} f(t)\, dt.
 $$
 Show that $f_n$ is continuous in $[0,1]$ and $f_n$ converges uniformly to $f$
in $[0,1]$}

\enddocument

euclid%