\input vanilla.sty
\magnification=1200
\baselineskip=20pt
\nopagenumbers

\centerline{\bf University at Albany}

\centerline{\bf Department of Mathematics and Statistics}

\centerline{\bf Ph.D. Preliminary Examination}

\centerline{\bf Real Analysis}

\centerline{\bf Monday, June 3, 1996}

\bigskip

Do eight of the following ten problems.  If you do more,
please indicate which eight you wish to be graded.

\bigskip

\item{1.} Give precise statements of the following:

\itemitem{A.}  The Monotone Convergence Theorem

\itemitem{B.}  Egoroff's Theorem

\itemitem{C.}  The Radon-Nikodym Theorem (include a definition of absolute continuity for measures)

\itemitem{D.}  Fubini's theorem

\itemitem{E.}  H\"older's Inequality

\itemitem{F.}  Fatou's Lemma

\bigskip

\item{2.}  Using only properties of Lebesgue measure and the
definition of Lebesgue integral prove the Monotone
Convergence Theorem.

\bigskip

\item{3.} A.  Define what it means for a function on [0,1]
to be Absolutely Continuous.

B.  Prove that an absolutely continuous function is continuous.

C.  Prove that an absolutely continuous function is of bounded variation.

\bigskip

\item{4.}  If $f$ and $g$ are measurable functions on [0,1]
prove that $\{x:  f(x)<g(x)\}$ is measurable.

\bigskip

\item{5.} If $S$ is an infinite subset of $R$ prove that $S$
contains a countable dense subset.

\bigskip

\item{6.}  If $f$ is a differentiable function on $(-\infty,
\infty)$ with both $f$ and $f'$ in $L(-\infty, \infty)$ show
that $\dsize\int^\infty_{-\infty} f'(x) \ dx =0$.

\bigskip

\item{7.}  Prove that if $f$ and $g$ are positive,
continuous functions on $(-\infty, \infty)$ which are
periodic of period 1 that
$$\lim_{n\to \infty} \int^1_0 f(x) \ g(nx) \ dx = \int^1_0 f(x) \ dx \cdot \int^1_0 g(x) \ dx$$

\bigskip

\item{8.} Give an example of a closed bounded subset of $L^2[0,1]$ which is not compact.  Prove your answer.

\bigskip

\item{9.} Construct a non-Lebesgue-measurable subset of the real numbers.

\bigskip

\item{10.} Let $\lambda$ denote Lebesgue measure on $I=[0,1]$ and $\mu$ counting measure both regarded as Borel measures on $I$.  The diagonal:
$$\Delta = \{(x,y): x=y, (x,y) \in I\times I\} \ . $$

\itemitem{A.} Show that $\Delta$ is measurable with respect to product measure, $\lambda\times \mu$ on the Borel sets of $I\times I$.

\itemitem{B.}  Let $f(x,y) = \chi_{\Delta}(x,y)$, the characteristic function of $\Delta$.  Compute \newline $\dsize\int_I\bigg(\dsize\int_I f \ d\lambda\bigg) \ d\mu$, $\dsize\int_I\bigg(\dsize\int_I f \ d\mu\bigg) \ d\lambda$, and $\dsize\int_{I\times I} f \ d\lambda \times d\mu$.

\medskip

\itemitem{C.} Reconcile part B with Fubini's Theorem.

\bye