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\title{Problem Set II\\ MTH 70200 \\ Real Analysis}
\date{February 8, 2024}

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

\begin{document}
  \maketitle
  \begin{prob}
    Consider the following increasing, right-continuous function:
    \begin{align*}
      F(x)=\begin{cases}
      0 \ &\text{if $x<0$,}\\
      1\ &\text{if $x\geq 0$.}
      \end{cases}
    \end{align*}
    \begin{enumerate}[a)]
      \item
        Describe the associated Borel measure $\mu_F$. 
      \item Let $\mu_0$ be the premeasure defined on the algebra of disjoint unions of $h$-intervals
        induced by $F$, and let $\mu^*$ be the outer measure induced by $\mu_0$.
        (This makes $\mu_F$ the restriction
        of $\mu^*$ to the Borel $\sigma$-algebra.)
        We mentioned in class that the domain of $\mu_F$ could also be taken to be the larger
        class of $\mu^*$-measurable sets. Which sets are $\mu^*$-measurable?
    \end{enumerate}
    

  \end{prob}
  
  \begin{prob}\label{prob:atom} (Folland 1.28) Let $F$ be increasing and right-continuous, and let $\mu_F$
    be the associated measure. Show that $\mu_F(\{a\})=F(a)-F(a-)$, $\mu_F([a,b))=F(b-)-F(a-)$,
    $\mu_F([a,b])=F(b)-F(a-)$, and $\mu_F((a,b))=F(b-)-F(a)$.
    (Here $F(x-)$ denotes $\lim_{y\uparrow x}F(y)$.)
    
  \end{prob}
  Commentary on Problem~\ref{prob:atom}: 
    a point $x$ such that $\mu(\{x\})>0$ is called an \emph{atom} of $\mu$.
    Thus $\mu_F$ has no atoms if and only if $F$ is continuous.
  \begin{prob}
    For a set $E\subset\RR$, let $\diam(E)=\sup E - \inf E$. 
    Let $\mu$ be a \textbf{finite} Borel measure on $\RR$ with no atoms.
    Show that for every $\epsilon>0$, there exists $\delta>0$ such that
    \begin{align*}
      \diam(E)<\delta \Longrightarrow \mu(E)<\epsilon.
    \end{align*}
    
    
  \end{prob}
  \begin{prob}
    Let $m^*$ be the outer measure on $\RR$ induced by the premeasure on h-intervals
    associated with some increasing,
    right-continuous $F\colon\RR\to\RR$.
    Suppose that $A\subseteq \RR$ has the property that for any real $a<b$,
    \begin{align*}
      m^*\bigl(A\cap(a,b]\bigr) \leq \tfrac12 m^*\bigl((a,b]\bigr).
    \end{align*}
    Prove that $m^*(A)=0$. (\textit{Hint:} First assume that $A$ is bounded.)
  \end{prob}
  
  \begin{prob}
    (Quals Spring 2016) Let $(X,\Mm_1,\mu_1)$ and $(X,\Mm_2,\mu_2)$
       be measure spaces on the same set $X$.
       Let $\mu_1^*$ and $\mu_2^*$ be the outer measures on $X$ generated by $\mu_1$ and $\mu_2$,
       respectively. Prove that $\mu_1^*=\mu_2^*$ if and only if
       $\mu_2^*|_{\Mm_1}=\mu_1$ and $\mu_1^*|_{\Mm_2}=\mu_2$.
  \end{prob}
  
  
\end{document}
  \begin{prob}
    (Folland 1.26) Prove Proposition 1.20. (Use Theorem~1.18.)
  \end{prob}
