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\title{Problem Set\\ MTH 70200 \\ Real Analysis}
\date{May 2, 2024}

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

\begin{document}
  \maketitle
  \begin{prob} (Folland 3.37)
    Suppose $F\colon\RR\to \mathbb{C}$. Show that there is a constant $M$ such that
    $\abs{F(x)-F(y)}\leq M\abs{x-y}$ (that is, $F$ is \emph{Lipschitz continuous})
    if and only if $F$ is absolutely continuous and $\abs{F'}\leq M$ a.e.
  \end{prob}
  
  \begin{prob}(Folland 3.39)
    Suppose that $\{F_j\}$ is a sequence of nonnegative increasing functions on $[a,b]$
    and that $F(x)=\sum_{j=1}^{\infty}F_j(x)<\infty$ for all $x\in[a,b]$.
    Prove that $F'(x)=\sum_{j=1}^{\infty}F_j'(x)$ for a.e.\ $x\in[a,b]$.
    \textit{Hint:} It suffices to assume that $F_j\in NBV$. Consider the measures $\mu_{F_j}$.
  \end{prob}
  
  \begin{prob}(Folland 3.40)
    Let $F$ denote the Cantor function on $[0,1]$, and set $F(x)=0$ for $x<0$ and $F(x)=1$ for $x>1$.
    Let $\{[a_n,b_n]\}$ be an enumeration of the closed subintervals of $[0,1]$
    with rational endpoints, and let
    \begin{align*}
      F_n(x) = F\biggl(\frac{x-a_n}{b_n-a_n}\biggr).
    \end{align*}
    Then $G=\sum_{n=1}^{\infty}2^{-n}F_n$ is continuous and strictly increasing on $[0,1]$,
    and $G'=0$ a.e. \emph{Hint:} use the previous exercise.
  \end{prob}
  
  \begin{prob}(Folland 6.7)
    Suppose that $f\in L^p\cap L^\infty$ for some $p<\infty$, so that $f\in L^q$ for $q>p$.
    Show that $\norm{f}_{\infty}=\lim_{q\to\infty}\norm{f}_q$.
  \end{prob}

  \begin{prob}(Folland 6.13)
    Recall that a metric space is called \emph{separable} if it has a countable dense subset.
    Show that $L^p(\RR^n,m)$ is separable for $1\leq p<\infty$, but $L^\infty(\RR^n,m)$ is
    not separable. \textit{Hint:} Find an uncountable collection of functions in $L^\infty$
    that all have $L^\infty$-distance at least $1$ from each other.
  \end{prob}
  
\end{document}

      
