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\centerline{\bf Ph.D. Qualifying Exam: Real Analysis}
\bigskip
\centerline{April 10, 2010}
\bigskip
\textbf{Examiners: Z. \v{C}u\v{c}kovi\'{c}, D. A. White}

\noindent {\bf Instructions:}  
Do 6 problems  of 8.
If you attempt more than 6, indicate which are to be graded.  
  \begin{enumerate} 
   \item % 1 
Let $\phi \in L^{\infty}(\mathbb{R})$.  (The measure on $\mathbb{R}$ is
Lebesgue measure.)  Show that
$$
\lim_{n \to \infty}\left( \int_{\mathbb{R}} \frac{|\phi(x)|^{n}}{1+x^2} \, dx \right)^{1/n} = \|\phi\|_{\infty}
$$  
% \item %2
%Suppose that $f \in L^1(0,1)$ has the property that,
%for every rational number $r$, $0<r<1$, $\int_0^rf(x) \, dx = r$.
%Show that $f(x) = 1 $ almost everywhere.
  \item %2
Suppose that $f:\mathbb{R} \to \mathbb{R}$ is differentiable and 
$\lim_{x \to \infty}f'(x) =A$ exists.  Show that
$$
\lim_{x \to \infty} \frac{f(x)}{x} = A
$$
%  \item %3
%  \begin{enumerate}
%   \item
%Show that $L^{2}(0,1)$ is a proper subset of $L^{1}(0,1)$.
%   \item
%Suppose that $\| \cdot \|_1$ is the $L^1$ norm and $g_0\in L^1(0,1)$.
%Compute $\mbox{dist}(g_0,L^2(0,1))= inf\{\|f-g_0\|_1:f \in L^2(0,1)\}$.
%   \item
%Suppose that $F$ is a finite dimensional subspace of $L^{1}(0,1)$ 
%and $g_0 \in L^1(0,1)$.
%Show there is $f_0\in F$ so that $\|f_0-g_0\|_1 = \mbox{dist}(g_0,F)$.
%(You may use the fact $F$ is closed in  $L^1(0,1)$ without proof.)   
%  \end{enumerate}
 \item%(Texas; Jan 2009, no 9)
     Let $K : [0, 1]\times [0, 1] \to \mathbb{R}$ be continuous. If 
$f \in L^1 (0, 1)$, set
$$                                        
      (T f )(x) =     \int_0^1K(x, y)f (y)\, dy,  
$$ 
for all $x \in [0,1]$.                                    
(a) Show $T f \in C([0, 1])$.
(b) Let $B$ be the unit ball of $L^1 (0, 1)$ and show that $T (B)$ is 
relatively compact in $C([0, 1])$.
  \item %(Texas; Jan 2010, no 5)
In $C[0, 1]$, let
$$
        \mathcal{A} = \mbox{span} \{x^n (1 - x) : n \ge 1\}.
$$
Prove that $\mathcal{A}$ is an algebra whose uniform closure is
$\{f \in C[0, 1] : f (0) = f (1) = 0\}$.
\item
Suppose that $f\in L^1(\mathbb{R})$ and $A$ is a Borel subset of $\mathbb{R}$.
Show that the mapping 
$$
t \mapsto \int \chi_{A+t}f(x) \, dx
$$
is continuous from $\mathbb{R}$ to itself.  Here $A+t = \{x+t:x\in A\}$.
(Suggestion: Begin with the case that $A$ is an interval.)
   \item
Suppose that $f_n$, $n \in \mathbb{N}$ is a sequence  of functions 
defined on $A \subseteq \mathbb{R}$ and uniformly convergent to a 
function $f$. Assume $x_0$ is a limit point of $A$ and 
$\lim_{x\to x_0}f_n(x)$ exists for all $n\in \mathbb{N}$. Prove
that 
$$
\lim_{n \to \infty}\lim_{x \to x_0}f_n(x) = \lim_{x \to x_0}f(x)
$$ 
(in the sense that one limit exists if and only if the other does and they
are equal).
%\item
%  Suppose that $a_n>0$, show that $\ds \sum_{n=1}^\infty a_n$ converges if and only if $\ds
%\sum_{n=1}^\infty\arctan(a_n)$ converges.
\item
Let $f$ be a non-negative element of $L^1(0, \infty)$ and let  $A$ be a Borel 
subset of $(0,\infty)$.
  \begin{enumerate}
   \item
Suppose the Lebesgue measure $m(A)$ of $A$ is finite.
 Prove that
$$                                                                    
  \lim_{n \to \infty}      \int_A (f (x))^{1/n}dx = m(\{x\in A : f (x) > 0\})  
    $$
 \item
Consider the case when $m(A) = \infty$.  What can be said  about   
$$                                                                    
  \lim_{n \to \infty} \int_A (f (x))^{1/n}dx 
    $$
in this case?  
    \end{enumerate} 
    \item
Suppose that $f\in L^1(\mathbb{R})$.  Show that, for almost all $x$. 
$$
\lim_{h \to 0}\frac{1}{h} \int_{|y-x|<h}|f(y)-f(x)| dy=0
$$
Suggestion: Begin with the case that $f$ is continuous and of compact 
support.     
  \end{enumerate} 
\end{document}
\item  %1
Assume that $f:\mathbb{R} \to \mathbb{R}$ is uniformly continuous.  
 \begin{enumerate}
  \item  Prove that there exists constants $A$ and $B$ such that $|f(x)| \le A + B|x|$. 
  \item  Is the converse true? In other words if there exists constants $A$ and $B$ such that $|f(x)| \le A + B|x|$
is $f(x)$ uniformly continuous?
 \end{enumerate}
\item %6
   Prove or disprove the following statements.  (The understood measure here is Lebesgue measure.)
  \begin{enumerate}
   \item If $f_n$ is a decreasing sequence of functions which converges pointwise  almost everywhere to   $f$ and if
$f_n$ and $f$ all belong to $L^1(\mathbb{R}$ then 
$$
\lim_{n \to \infty}\int_{\mathbb{R}}f_n dx = \int_{\mathbb{R}}f dx 
$$ 
   \item $L^2(0,1) \subseteq L^1(0,1)$ 
    \item $\cap_{p>1}L^p(\mathbb{R})\subseteq L^1(\mathbb{R})$
  \end{enumerate}

%\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 (Texas Aug 2009)
    Let (X, \mathcal{F}, μ) be a measure space with μ(X) < \infty. Given 
sets A_i\in \mathcal{F}, i ≥ 1, prove that
                                                  n
                                   \infty
                                      
                                                  = lim_n μ(\cap_{i=1}^n A_i)
                               μ(\cap A_i)                     
                                           n→∞
                                  i=1            i=1
Give an example to show that this need not hold when μ(X) = ∞.
\item (Texas Exam)
     Let $f \in L^1 (0, \infty)$ and define
$$                                              
               h(x) = \int_0^{\infty}(x+y)^{-1}f(y) \, dy
$$                                         
for x > 0. Show that h is differentiable at all x > 0 and show h′ ∈ L1 (r, ∞) for every r > 0.
What about for r = 0? (Justify your answer.)