\input vanilla.sty
\magnification=1200
\baselineskip=10pt
\nopagenumbers

\centerline{\bf{Department of Mathematics and Statistics}}

\bigskip

\centerline{\bf{Ph.D. Program}}

\bigskip

\centerline{\bf{Preliminary Examination in Real Analysis}}

\bigskip

\centerline{\bf{Friday, January 20, 1995}}

\bigskip

\centerline{\bf{Do all eight problems.}}

\bigskip

\item{1.} State the following theorems:

\medskip

Fatou's lemma,

\medskip

Lebesgue dominated convergence theorem,

\medskip

Fubini's theorem,

\medskip

Egoroff's theorem,

\medskip

H\"older's inequality,

\medskip

The Radon-Nikodym theorem.

\bigskip

\item{2.} State and prove the Monotone convergence theorem
(without relying on any of the other convergence theorems).

\bigskip

\item{3.} Prove that the sum of two measurable functions is
a measurable function.

\bigskip

\item{4.} Show that if $f$ is a monotone function on $[a,b]$
then $f$ has at most a countable number of discontinuities.

\bigskip

\item{5.} Let $f_n$ be a sequence of measurable functions
such that $f_n(x)\to f(x)$ almost everywhere, and suppose
that \ \ \ $\sup \dsize\int^1_0 |f_n(x)|dx < \infty$.

\medskip

(a) Show that $f$ is measurable and that $\dsize\int^1_0
|f(x)|dx < \infty$.

\medskip

(b) Does $\dsize\lim_{n\to \infty} \dsize\int^1_0 f_n(x) dx
= \dsize\int^1_0 f(x)dx$?

\bigskip

\item{6.} Show how to construct a non-constant continuous
function on [0,1] which is differentiable at each rational
point and such that for every rational number $x$,
$$f'(x)=0 \ .$$

\bigskip

\item{7.} Let $f(x)$ be a measurable function on
$[0,\infty)$ such that
$$\int^\infty_0 [f(x)]^ndx = c \ \ \text{for} \ n=2,3,4 \
.$$
Show that $f(x)=\chi_A(x)$ almost everywhere for some
measurable set $A\subseteq [0,\infty)$.

\bigskip

\item{8.} Show that of $f\in L^1(X,\mu)$ then
$$\int^\infty_0 \mu\{x: |f(x)|>t\} \ dt = \int_X|f(x)| \ d\mu(x) \ .$$


\bye