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\centerline{\bf Ph.D. Qualifying Exam in Real Analysis}
\bigskip
\centerline{January 23, 2009}
\centerline{A. Arsie, Z. \v{C}u\v{c}kovi\'{c}, D. A. White}
\bigskip
\noindent {\bf Instructions:} Do 6 problems of 9. No materials are allowed. Complete
explanations are expected.
\begin{enumerate}
\item %1
\begin{enumerate}
\item Define equicontinuity.
\item State the Arzel\'{a} Ascoli Theorem.
\item Let $\{a_n\}$, $n \in \mathbb{N}$ be a sequence of nonzero real numbers. Prove that the sequence of functions
$$
f_n(x) = \frac{1}{a_n}\sin(a_n x) + \cos(x+a_n)
$$
has a subsequence convergent to a continuous function.
\end{enumerate}
\item %2
Let $f\in L^{1}(\mathbb{R})$ and suppose that there is a countable set $S \subseteq \mathbb{R}$
so that
$$
\int_{p}^{q} f(x) \, dx = 0
$$
whenever $p$ and $q$ are \textit{not} in $S$. Prove that $f=0$
almost everywhere.
\item %3
Let $f: \mathbb{R} \to \mathbb{R}$ be a continuous function and suppose that $x_n$ and $y_n$, $n = 1,2, \ldots$ are
two sequences such that $\lim_{n \to \infty}|x_n-y_n|=0$. Does it follow that $\lim_{n \to \infty} |f(x_n)-f(y_n)|=0$?
Prove or give a counterexample.
\item % 4
Let $C(X)$ denote the continuous real valued functions defined on a compact set $X \subseteq \mathbb{R}$ and
endowed with the sup norm topology.
\begin{enumerate}
\item
Suppose that $T_0 \subseteq C(X)$ consists of all polynomials of
the form $p(x)= a_0 + a_1x^2 + a_2x^4 + \ldots a_nx^{2n}$ for some
real coefficients $a_j$, $0 \le j \le n$.
Describe the closure of $T_0$ in $C(X)$ if $X=[0,2]$
\item
Describe the closure of $T_0$ in $C(X)$ if $X=[-2,2]$
\item
Suppose that $T_1 \subseteq C([0,2])$ consists of all polynomials
of the form $q(x)= a_0x + a_1x^3 + a_2x^5 + \ldots a_nx^{2n+1}$.
Describe the closure of $T_1$ in $C([0,2])$.
\end{enumerate}
\item %5
Let $f_n(x)=\dfrac{n \sin x}{x(1+n^2x^2)}$. Evaluate $\lim_{n \to \infty}\int_0^1 f_n \, dx$ or show that the limit does not exist.
\item %6
Let $f:[1,\infty) \to [0,\infty)$ be a non-increasing function. Prove that
$$
\int_1^{\infty}f(x) \, dx < \infty \hspace{3mm} \mbox{ if and only if }\hspace{3mm} \sum_{k=0}^{\infty}2^kf(2^k)<\infty.
$$
\item %7
Let $\mathcal{F}$ be a $\sigma$-algebra on a set $\Omega$. For each $x \in \Omega$ define
$$
A_x= \cap\{B: B\in \mathcal{F} \mbox{ and } x \in B\}.
$$
(such a set is called an atom.) Prove that for all $x,y \in \Omega$, $A_x$ and $A_y$ are either identical or disjoint.
\item %8
Let $\mu$ be a finite measure on a set $\Omega$. Suppose $f$ is a nonnegative measurable function defined on $\Omega$ such that
$f^n$ is integrable for all $n =1,2,\ldots$ and that
$$
\int_{\Omega} f^n\,d\mu = \int_{\Omega} f\, d\mu
$$
for all $n$. Show that $f= \chi_{E}$ a.e. for some measurable set $E \subseteq \Omega$.
Is the result true if we do not assume that $f$ is nonnegative?
\item %9
Let $f:(0,\infty) \to \mathbb{R}$ be a convex function such that $\lim_{x \to 0}f(x) = 0$. Show that the function
$x \mapsto \dfrac{f(x)}{x}$ is increasing on $(0,\infty)$.
\end{enumerate}
\end{document}
%\item %3
%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
5$L^1(\Omega,\mathcal{F},\mu)$.
% \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 on $A$.
% \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 % 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 $d$. 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.
\end{enumerate}
\end{document}