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\coverheader{\large R.~de Santiago}{\large Math 561}{\large }
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\begin{document}
%%%%% COVER PAGE %%%%%
\begin{center} California State University, Long Beach \\
Department of Mathematics and Statistics \vspace{1cm} \\

{\Large \bf Real Analysis Comprehensive Examination } \\
Feb.~2026 \vspace{1cm} \\

{\bf Choose six of the nine problems, and mark the problems that you want graded on the table below. }\vspace{1cm}\\
%\vsword{}
%\gradetable
\begin{tabular}{ |c|c| } 
 \hline
 Question & To Be Graded (\checkmark) \\ \hline
 1 &   \\ \hline
 2 &   \\ \hline
  3 &   \\ \hline
   4 &   \\ \hline
 5 &   \\ \hline
  6 &   \\ \hline
 7 &   \\ \hline
  8 &   \\ \hline
 9 &   \\ \hline
 

\end{tabular}
\end{center}
Notation:
\begin{itemize}
    \item If $A\subseteq\R$,  $|A|$ denotes the Lebesgue Outer Measure of $A$.
    \item $\mathcal{B}$ denotes the Borel measurable subsets of $\R$
    \item $\lambda$ denotes the Lebesgue Measure on the Real line
    \item $(X,\mathcal{S},\mu)$ will denote a measure space
    \item $\mathcal{L}^1(X,\mathcal{S},\mu):=\set{ f \text{ measurable}  :  \int_X|f|\, d\mu<\infty}$
    \item $\|f\|_1:=\int_X |f|\, d\mu$
    
\end{itemize}
\newpage 
\begin{questions}

\question Let $E\subseteq \R$ be measurable with $\lambda(E)<\infty$. 
    \begin{parts}
        \part Let $\set{t_n}_{n=1}^\infty$ be a sequence of real number with $\lim_{n\to \infty}t_n=0$.  Show that
        \[\lim_{n\to\infty} \lambda(E\cap (E+t_n))=0\]
        \part Conclude that that \[
\lim_{t\to0} \lambda(E\Delta (E+t))=0.
\]
    \end{parts}

\question Let $t\in \R$. Show that for any Lebesgue measurable set, we have that $E+t$ is also Lebesgue measurable.

\item Prove or give a counterexample: if $f\in \mathcal{L}^1(\R,\mathcal{B},\lambda)$, then $\lim_{x\to\infty} f(x)=0$.

\question Let $f:X\to \R$ be measurable and bounded. Show that for each $\varepsilon>0$. there exists a bounded measurable step function $g:X\to \R$ so that for all $x\in X$,
\[ |f(x)-g(x)|<\varepsilon\]


\question Let $f:\R\to \R$ be such that $f$ is absolutely continuous, $f'\in \mathcal{L}^1(\R,\mathcal{B},\lambda)$. Show that $\lim_{x\to\infty} f(x)$ exists.


\question Let $f:\R\to\R$ be such that for all $c\in \R$ and all $r>0$, we have $
\int_{c-r}^{c+r}f(x)\, d\lambda(x)\geq 2r$. Show $f(x)\geq 1$ almost everywhere.

\question Suppose that $f_n,f\in \mathcal{L}^1([0,1],\mathcal{B},\lambda)$, $f_n\to f$ pointwise on $[0,1]$. Does it follow that $\int_{[0,1]} f_n\, d\lambda\to \int_{[0,1]} f\, d\lambda$.  

\question Let $f_n:[0,1]\to \R$ be the following function:
\[
f_n(x)= nx\chi_{[0,1/n]}(x)
\]
Does $f_n$ converge to the zero function pointwise? what about in $\|\cdot \|_1$?

\question Let $f_n:[0,1]\to \R$ be given as follows:
\begin{itemize}
    \item $f_1(x)=1$;
    \item $f_2(x)=1$ if $x\in [0,1/2]$, zero otherwise;
    \item $f_3(x)=1$ if $x\in [1/2,1]$, zero otherwise;
    \item $f_4(x)=1$ if $x\in [0,1/3]$, zero otherwise;
    \item $f_5(x)=1$ if $x\in [1/3,2/3]$, zero otherwise;
    \item $f_6(x)=1$ if $x\in [2/3,1]$, zero otherwise;
    \item $f_7(x)=1$ if $x\in [0,1/4]$, zero otherwise;
\end{itemize}
    \begin{parts}
        \part Does $f_n$ converge to the zero function  pointwise?
        \part What is $\lim_{n\to\infty}\|f_n\|_1$?
    \end{parts}

\question Suppose $(X,\mathcal{S},\mu)$ is a measure space $f:X\to \R$ is measurable, and assume that $\int_X|f|^pd\mu<\infty$. Show that for any $k>0$ we have
\[\mu(\set{x: |f(x)|\geq k})\leq \frac{1}{k^p}\int_X |f|^p\, dx \]
%\newpage
\question Let $E\subseteq \R$ be Lebesgue measurable with $0<\lambda(E)<\infty$. Set $f:\R\to \R$ to $f(x)=\lambda((-\infty,x)\cap E)$. 
    \begin{parts}
        \part Show that $f$ is continuous on $\R$.
        \part Show that for any $b\in (0,\lambda(E))$, there exists some $x\in \R$ so that $f(x)=b$.
    \end{parts}
%\vspace{\stretch{1}}

\question Let $(X,\mathcal{S},\mu)$  be a measure space and  $f_n,f:X\to \R$ measurable.
    \begin{parts}
        \part State Fatou's Lemma.
        \part State the Dominated Convergence Theorem.
        \part Derive the Dominated Convergence Theorem as a consequence of Fatou's lemma.
    \end{parts}
%\vspace{\stretch{1}}\newpage
\question Compute
\[\lim_{n\to\infty } \int_0^1\left[1-(-1)^ne^{-x^2/n}\right]x^{-1/2}\, d\lambda(x)\]


%\vspace{\stretch{1}}
%\textcolor{red}{My Solution\\
%Let $f(x)=e^{-x}x$.  Notice that $f(x)\geq0$ for all $x\in [0,\infty)$ and that (using integration by parts) $\int_{[0,\infty) }xe^{-x}\, d\lambda(x) =1$.  Thus, $f\in\mathcal{L}^1([0,\infty)]) $. Let $f_n(x)= (1-(x/n))^nx\chi_{[0,x]}(x)$. Then by the hint, $|f_n(x)|\leq f(x)$ for all $n$ and $x\in [0,\infty)$.  Since the pointwise limit of $f_n(x)= e^{-x}x$,  by the Lebesgue dominated convergence theorem, we see  that
%\[\lim_{n\to\infty}\int_0^n\left(1-\frac{x}{n}\right)^nx\, d\lambda(x)=\lim_{n\to\infty }\int_{[0,\infty)} f_n\, d\lambda=\int_{[0,\infty)} f\, d\lambda=1\]}




\question Let $(X,S,\mu)$ be a measure space.
\begin{parts}
    \part State Egoroff's theorem.

    \part 
  {Alternate question: Let $f_n,f:X\to \R$ be measurable, $f_n\to f$ pointwise,  and $X=\bigcup_{k=1}^\infty X_k $ where $\mu(X_k)<\infty$ for all $k\in\N$. Show that for all $\varepsilon>0$, there exists a a set $E\in S$ so that 
    \begin{itemize}        \item $\mu(X\setminus E)<\varepsilon $, and 
        \item $f_n\to f$ uniformly on $E$.
    \end{itemize}}
\end{parts}
%\vspace{\stretch{1}}
\end{questions}


\end{document}