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\title{Problem Set\\ MTH 70200 \\ Real Analysis}
\date{March 28, 2024}

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

\begin{document}
  \maketitle
  \begin{prob}
    Let $\mu$ be counting measure on the integers.
    Let $\nu$ be a signed measure on the integers.
    What is the Radon-Nikodym derivative  $\frac{d\nu}{d\mu}$?
  \end{prob}
  \begin{prob}
    Let $f,g\colon\RR\to[0,\infty]$ be Borel-measurable.
    Let $d\mu=f\,dm$ and $d\nu=g\,dm$, where $m$ is Lebesgue measure.
    Let $S_f=\{x\in\RR\colon f(x)>0\}$ and
      $S_g=\{x\in\RR\colon g(x)>0\}$.
    \begin{enumerate}[(a)]
      \item Show that $\nu\perp\mu$ if and only if $m(S_f\cap S_g)=0$.
      \item Show that $\nu\ll\mu$ if and only if $m(S_g\cap S_f^c)=0$.
      \item Show that the Lebesgue decomposition of $\nu$ with respect to $\mu$
        is $\nu=\rho + \lambda$ where $d\rho = \frac{g}{f}\,d\mu$
        and $d\lambda = \chi_{S_f^c}g\,dm$.
      
        
    \end{enumerate}
    \begin{prob}
      (Half of Folland 3.9)
      Suppose $\{\nu_j\}$ is a sequence of positive measures and $\mu$ is a positive measure,
      all on the same space.
      If $\nu_j\perp\mu$ for all $j$, then $\sum_{j=1}^{\infty}\nu_j\perp\mu$.
    \end{prob}
    
    \begin{prob}
      (Folland 3.11)
      Let $\mu$ be a positive measure. A collection of functions
      $\{f_\alpha\}_{\alpha\in A}\subseteq L^1(\mu)$ is called \emph{uniformly integrable}
      if for every $\epsilon>0$, there exists $\delta>0$ such that for all $\alpha\in A$,
      \begin{align*}
        \mu(E)<\delta \Longrightarrow \abs[\bigg]{\int_E f_\alpha\,d\mu}<\epsilon.
      \end{align*}
      
      \begin{enumerate}[(a)]
        \item Prove that any finite subset of $L^1(\mu)$ is uniformly integrable.
        \item Prove that if $\{f_n\}$ is a sequence in $L^1(\mu)$ that converges in the $L^1$-metric
          to some $f\in L^1(\mu)$, then $\{f_n\}$ is uniformly integrable.
      \end{enumerate}
      
    \end{prob}
    
  \end{prob}
  
\end{document}

% compute Radon-Nikodym derivatives in some basic situations
% 3.11
% #8, #9?
