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\title{Problem Set\\ MTH 70200 \\ Real Analysis}
\date{March 7, 2024}

\theoremstyle{definition}
\newtheorem{prob}{Problem}

\begin{document}
  \maketitle
  \begin{prob}(Folland 2.23)
    Given a bounded $f\colon[a,b]\to\RR$, let
    \begin{align*}
      H(x) = \lim_{\delta\to 0}\sup_{\abs{y-x}\leq\delta}f(y),\qquad h(x) = \lim_{\delta\to 0}\inf_{\abs{y-x}\leq\delta}f(y).
    \end{align*}
    Prove that $f$ is Riemann integrable if and only if its set of discontinuities has Lebesgue measure $0$
    by establishing the following lemmas.
    \begin{enumerate}[a.]
      \item $H(x)=h(x)$ iff $f$ is continuous at $x$.
      \item In the notation of the proof of Theorem~2.28a, $H=G$ a.e.\ and
        $h=g$ a.e. Hence $H$ and $h$ are Lebesgue measurable,
        and $\int_{[a,b]} H\,dm = \bar{I}^b_a(f)$
        and $\int_{[a,b]}h\,dm = \underbar{I}^b_a(f)$.
    \end{enumerate}
    \textit{You should work out for yourself why this proves the statement, but you don't
    need to write it up.}
  \end{prob}
  \begin{prob}(Folland 2.26)
    Prove that if $f\in L^1(m)$ and $F(x)=\int_{-\infty}^xf(t)\,dt$, then $F$ is continuous on $\RR$.
  \end{prob}
  \begin{prob}(Folland 2.33)
    Show that if $f_n\in L^+$ and $f_n\to f$ in measure, then
    \begin{align*}
      \int f \leq\liminf_{n\to\infty}\int f_n.
    \end{align*}
    
  \end{prob}
  \begin{prob}(Folland 2.34)
    Suppose $\abs{f_n}\leq g\in L^1$ and $f_n\to f$ in measure.
    \begin{enumerate}[a.]
      \item Show that $\int f = \lim_{n\to\infty}\int f_n$.
      \item Show that $f_n\to f$ in $L^1$.
    \end{enumerate}
    
  \end{prob}
  \begin{prob} (Quals 2022, \#1)
    Let $f$ be a measurable function on some measure space $(X,\mu)$.
    Prove that the limit (possibly infinite)
    \begin{align*}
      \lim_{n\to\infty}\int\abs{f}^{1/n}\,d\mu
    \end{align*}
    exists and find it.
  \end{prob}

  
\end{document}


