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

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

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

\centerline{\bf{January 18, 1994}}

\bigskip

\flushpar Do as many as time permits.

\bigskip

\item{1.} The following theorems are easily shown to imply
one another:

\item\item{i)} The Lebesgue Dominated Convergence Theorem.

\item\item{ii)} Fatou's Lemma.

\item\item{iii)} The Lebesgue Monotone Convergence Theorem.

\item\item{ } State the three above theorems, pick one of them, and prove it. (Of course, not using the other two or some other nearly equivalent theorem.)

\bigskip

\item{2.} Give an example of a sequence of non-negative continuous functions $(f_n)$ defined on [0,1] such that

\bigskip

\item\item{(a)} $\dsize\lim_{n\to \infty} \dsize\int^1_0 f_n(x)dx = \infty$ \ .

\bigskip

\item\item{(b)} $\dsize\lim_{n\to \infty} \dsize\int^1_0 \dsize\frac{f_n(x)}{1+f_n(x)} dx = 0$ \ .

\bigskip

Verify (a) and (b).

\bigskip

\item{3.} Prove that the characteristic function of the Cantor set is Riemann integrable.

\bigskip

\item{4.} Prove that $\dsize\lim_{n\to \infty} \dsize\int^1_0 x \sin nx \ dx = 0$.

\bigskip

\item{5.} Let $m$ be Lebesgue measure on $X=[0,1]$, and let $\mu$ be a measure on the Lebesgue sets with $\mu(X)=1$, and $\mu\sim m$ (i.e. $\mu$ and $m$ have the same sets of measure zero).  Prove there exists a measurable set $A$ such that $\mu(A)=1/2$.

\newpage

\item{6.} Define a sequence of measures $(\mu_n)$ on the Lebesgue measurable subsets of [0,1] by $\mu_n(A)=\dsize\int^1_0 I_A(x) \ nx^{n-1} \ dx$, $n=1,2,3,\dots$ \ .

\bigskip

\item\item{(a)} Verify that $\dsize\lim_{n\to \infty} \mu_n([a,b])=0$ if $0<a<b<1$.

\bigskip

\item\item{(b)} Suppose $0<a_k<b_k<1$ and $[a_k,b_k]$ are disjoint, $k=1,2,\dots,r$.  Verify that $\dsize\lim_{n\to \infty} \mu_n(\dsize\bigcup^r_{k=1} [a_k,b_k])=0$.

\bigskip

\item\item{(c)} Suppose $[a_k,b_k], k=1,2,3,\dots,$ are all disjoint.  Does $\dsize\lim_{n\to \infty} \mu_n(\dsize\bigcup^\infty_{k=1} [a_k,b_k])$ exist?  Give reasoning for your answer.

\bigskip

\item{7.} Let $\lambda$ be Lebesgue measure and $\mu$ be counting measure both regarded as Borel measures on $I=[0,1]$.  Let $\Delta$ be the diagonal in $I\times I$; $\Delta=\{(x,y)|x=y\}$.

\item\item{(a)} Show that $\Delta$ is measurable (with respect to the product measure on Borel subsets of $I\times I$.

\bigskip

\item\item{(b)} Let $f$ be the characteristic function of $\Delta$.  Compute the integrals:  $\dsize\int_I(\dsize\int_I fd\lambda)d\mu$, $\dsize\int_I(\dsize\int_I fd\mu)d\lambda)$ and $\dsize\int_{I\times I} fd\mu \times d\lambda$.

\bigskip

\item\item{(c)} Reconcile with Fubini's Theorem.

\bigskip

\item{8.} Let $(X,{\cal{B}},\mu)$ be a measure space with $\mu(X)=1$.  Let $(A_n)$ be a sequence of measurable sets with $\mu(A_n)=1/2$, all $n$, and $\mu(A_n\cap A_m)=1/4$, $n\neq m$.  Let $I_i$ be the characteristic function of a set $A_i$ and let $f_n(x)=\dsize\frac{1}{n} \dsize\sum^n_{i=1} I_i(x)$.  Prove that
$$\lim_{n\to \infty} \int_X |f_n(x)-\frac{1}{2}|d\mu=0 \ .$$
Hint: Show $\|f_n-1/2\|^{^2}_{_2} \to 0$ and use H\"older's Inequality.


\bye