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\title{Problem Set\\ MTH 70200 \\ Real Analysis}
\date{April 4, 2024}

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

\begin{document}
  \maketitle
  \begin{prob}
        Define $f\colon\RR^n\to\RR$ by 
        \begin{align*}
          f(x) = \frac{1}{m(B_1(0))}\chi_{B_1(0)}(x).
        \end{align*}
        Show that the Hardy-Littlewood maximal function $Hf$ is not integrable.
        \textit{Hint:} Show that $Hf(x)\geq c\abs{x}^{-n}$ for some $c$ and all large
        enough $\abs{x}$. You can use Corollary~2.52 even though we didn't officially cover it.\\
        \textit{Commentary: this problem shows that we can have $f\in L^1$ but $Hf\notin L^1$.}
    
  \end{prob}
  \begin{prob} (Folland 3.23)
    A useful variant of the Hardy-Littlewood maximal function is
    \begin{align*}
      H^*f(x) = \sup\biggl\{ \frac{1}{m(B)}\int_B\abs{f(y)}\,dy\colon\text{$B$ is an open ball containing $x$}\biggr\}.
    \end{align*}
    Show that $Hf\leq H^*f\leq 2^n Hf$ for any $f\in L^1_{\text{loc}}(\RR^n)$.
  \end{prob}
  \begin{prob} (Folland 3.24)
    Show for any $f\in L^1_{\text{loc}}(\RR^n)$ that if $f$ is continuous at $x$, then
    $x$ is in the Lebesgue set of $f$.
  \end{prob}
  \begin{prob} (Folland 3.25)
    If $E$ is a Borel set in $\RR^n$, the \emph{density} $D_E(x)$ of $E$ at $x$ is defined
    as
    \begin{align*}
      D_E(x) = \lim_{r\to0} \frac{m\bigl(E\cap B_r(x)\bigr)}{m\bigl(B_r(x)\bigr)}
    \end{align*}
    whenever the limit exists.
    \begin{enumerate}[(a)]
      \item Show that $D_E(x)=1$ for a.e.~$x\in E$ and $D_E(x)=0$ for a.e.~$x\in E^c$.
      \item Find examples of $E$ and $x$ such that $D_E(x)$ is a given $\alpha\in(0,1)$
        and such that $D_E(x)$ doesn't exist.
    \end{enumerate}
    
    
    \textit{Note:} This may or may not be helpful---there is more than one way to construct
    examples in part~(b)---but feel free to compute integrals on $\RR^2$ using polar coordinates (Theorem~2.49 in Folland, though
    that's in greater generality).
  \end{prob}
  
  
\end{document}
