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\newcommand{\Bb}{\mathcal{B}}
\newcommand{\RR}{\mathbb{R}}
\newcommand{\QQ}{\mathbb{Q}}
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\title{Problem Set\\ MTH 70200 \\ Real Analysis}
\date{April 12, 2024}

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

\begin{document}
  \maketitle
  \begin{prob}
    Let $\mu$ be a finite signed Borel measure on $\RR$ and let $F(x)=\mu\bigl((-\infty,x]\bigr)$.
    Show that $\abs{\mu}(\RR)$ is equal to $T_F(\infty)$, the total variation of $F$ on $\RR$.
    
    \textit{Hint:} Show that $\abs{\mu}(\RR)\geq T_F(\infty)$
    and $\abs{\mu}(\RR)\leq T_F(\infty)$. The hard direction is $\abs{\mu}(\RR)\leq T_F(\infty)$.
    Let $\RR=P\cup N$ be a Hahn decomposition for $\mu$. Approximate $P$ by a finite disjoint
    union $A$ of h-intervals so that $\abs{\mu}(A\bigtriangleup P) = \abs{\mu}(A^c\bigtriangleup N)<\epsilon$
    (you can use Proposition~1.20 together with continuity from below).
    
  \end{prob}
  \begin{prob}
    Let $\mu$ be a finite signed Borel measure on $\RR$ and let $F(x)=\mu\bigl((-\infty,x]\bigr)$.
    Let $F=F^+-F^-$ be the Jordan decomposition
    of $F$. Show that $\mu^+=\mu_{F^+}$ and $\mu^- = \mu_{F^-}$, i.e., the Jordan
    decomposition of $\mu$ corresponds to the Jordan decomposition of $F$.
    
    \textit{Hint:} Apply the previous exercise to the restriction of $\mu$ to $(-\infty,x]$
    and use Folland Exercise~3.4. Your solution should be very short if you do it
    right.
  \end{prob}
  \begin{prob}
    (Folland 3.30) Construct an increasing function on $\RR$
    whose set of discontinuities is $\QQ$.
  \end{prob}
  \begin{prob}
    (Folland 3.33) Let $F\colon\RR\to\RR$ be increasing. Prove that
    $F(b)-F(a)\geq \int_a^b F'(t)\,ft$.
  \end{prob}
  
\end{document}
% Jordan decomposition, parts 1 and 2 (proving that mu^+ and mu^- correspond to F+, F-
% 30, 32, 33
