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\centerline{\bf Ph.D. Qualifying Exam: Real Analysis}
\bigskip
\centerline{April 12, 2008}
\bigskip
\noindent {\bf Instructions:} Do six of the 9 questions.
No materials are allowed.
\noindent {\bf Examiners:} Rao Nagisetty; Denis White.
\begin{enumerate}
\item %1
Suppose that $(\Omega,\mathcal{F},\mu)$ is a finite measure space and
that $f_n$ is a sequence in $L^1(\Omega,\mathcal{F},\mu)$ which converges
to 0 in
$L^1(\Omega,\mathcal{F},\mu)$.
\begin{enumerate}
\item
Give an example to show that $f_n$ need not converge to 0 almost everywhere.
\item
Show that $f_n$ converges in measure to 0.
\item
Suppose that some subsequence of the $f_n$ converges pointwise almost everywhere to some function $f$.
Must $f=0$ almost everywhere? Explain.
\end{enumerate}
\item % 2
Suppose $f$ and $g$ are nonnegative integrable functions defined on a measure space
$(\Omega,\mathcal{F},\mu)$.
\begin{enumerate}
\item
Show that $ \displaystyle{\min\{ \int_{\Omega} f \, d\mu, \int_{\Omega} g \, d\mu \} \ge
\int_{\Omega} \min\{f,g\} \, d\mu}$.
\item
If equality holds then what can be said about the relationship between $f$ and $g$?
\end{enumerate}
\item % 3
\begin{enumerate}
\item % 3a
Give an example of a sequence of bounded functions which are Riemann integrable
on a compact interval $[a,b]$ and the sequence converges pointwise
to a function which is not Riemann integrable.
\item %3b
Give an example of a function $f$ which is not Lebesgue measurable on $[a,b]$ but $f^2$ is.
\item
Give an example of a function $f$ which is Lebesgue integrable on $[a,b]$ but $f^2$ is not.
\end{enumerate}
\item %4
\begin{enumerate}
\item
State the Baire Category theorem. If you use the terminology ``first category'' or
``second category'' then you should define those terms.
\item
Suppose that $E$ is a complete metric space with metric $d$
(CORRECTION: Suppose $E$ is a Banach space). Suppose that
$X \subseteq E$ has the property that it complement $X^c$ is countable.
Show that $X$ is set of the second category.
\end{enumerate}
\item %5
Prove or disprove.
\begin{enumerate}
\item
Every absolutely continuous function defined on [0,1] is of bounded variation.
\item
Every continuous function defined on [0,1] is of bounded variation.
\item
If $f$ is continuous and increasing on [0,1] then $f(1)-f(0) = \int_0^1 f'(x) \, dx.$
\end{enumerate}
\item %6
Suppose that $(\Omega,\mathcal{F},\mu)$ is a measure space and $f_n$
is a sequence of real valued Borel measurable functions
$f_n:\Omega \to \mathbb{R}$ such that
$$
\sum_{n \in \mathbb{N}} \int_{\Omega} |f_n|\, d\mu < \infty
$$
Show that $\sum_n f_n(x)$ converges $\mu$-almost everywhere to
a function $f(x)$ say and $f \in L^1(\mu)$ and
\[
\int_{\Omega} f \, d\mu = \sum_{n \in \mathbb{N}} \int_{\Omega} f_n \, d\mu
\]
\item % 7
Consider the sequence $f_n(x) = e^{-n\sqrt{x}}$. Show that, for any $a>0$ $f_n$ converges to 0
uniformly on $[a,\infty)$ but $f_n$ does not converge uniformly on $(0,\infty)$. Compute
$$
\lim_{n \to \infty} \int_0^{\infty} f_n(x) \, dx
$$
and explain your answer.
\item %8
Let $I=[0,1]$ and $K=I^n$. Fix $\alpha$ such that $0<\alpha<1$. Let $S$ be the family
of all real valued functions on $K$ for which
$$\|f\|_\alpha=\left(\sup_K|f(x)|+\sup_{K\times K}\frac{|f(x)-f(y)|}{|x-y|^\alpha}\right)\leq 1.$$
Show that the closure of $S$ in $C(K)$, the space of continuous functions on $K$ with the
supremum norm, is compact.
\item %9
Let $f$ be a continuous function on $[0,\infty)$ and $\int_0^\infty|f(x)|dx<\infty.$. Assume
that $\int_0^\infty f(x)e^{-nx}dx=0$ for all integers $n$ sufficiently large. Show that $f\equiv 0$.
\end{enumerate}
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