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

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

\centerline{\bf{Ph.D. Preliminary Examination in Real
Analysis}}

\centerline{\bf{Saturday, January 23, 1993}}

\bigskip

\centerline{\bf{Do all 8 problems.}}

\bigskip

\item{1.} State the following theorems.

\bigskip

Fatou's lemma

\bigskip

Lebesgue Dominated Convergence Theorem

\bigskip

Lebesgue Monotone Convergence Theorem

\bigskip

Egoroff's Theorem

\bigskip

Minkowski's Inequality

\bigskip

The Radon-Nikodym Theorem

\bigskip

\item{2.} Prove Fatou's Lemma from basic principles.

\bigskip

\item{3.} Let $E$ be the subset of $[0,1]$ such that $x\in
E$ if and only if there is only one 9 in the decimal
expansion of $E$.  Prove that $E$ has Lebesgue measure 0.

\bigskip

\item{4.} Calculate
$$\lim_{h\to \infty} \int^1_0 \frac{h^{3/2}
x^{3/2}}{1+h^2x^2}dx \ .$$
Justify your calculation.

\bigskip

\item{5.} Let $\mu$ be a finite measure on the Borel sets of
$(-\infty, \infty)$.  Let
$$f(x) = \int^\infty_{-\infty} e^{itx} d\mu(t) \ .$$
Prove or give a counterexample:  $f(x)$ is uniformly
continuous on $(-\infty, \infty)$.

\bigskip

\item{6.} Let $f(x)\geq 0$ be a function $[0,1]$ and let
$E=\{(x,y): 0\leq x\leq 1, \ 0\leq y\leq f(x)\}$.  Prove
that if $E$ is a 2-dimensional Lebesgue measurable set than
$f$ is a Lebesgue measurable function.

\bigskip

\item{7.} Let $f(x)$ be a Lebesgue integrable function such
that $\dsize\int^1_0 f(x) x^n dx=0$ for all $n\geq 2$.
Prove or give a counterexample:  $f(x)=0$ almost everywhere.

\bigskip

\item{8.} Let $A$ and $B$ be Lebesgue measurable sets of
finite non-zero measure.  Let \newline
$\varphi(x)=|A\cap (B+x)|$ where absolute value denotes Lebesgue measure and \newline
$B+x=\{y:y=b+x$ for some $b\in B\}$.  Prove or give a counterexample:  \newline
$\varphi(x)$ is continuous.

\bye