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\begin{document}
\begin{center}
  \bfseries \Large Second Homework Assignment \\ Math 73/103 \\ Due
  TBA.
\end{center}

\begin{ques}
  Page 32 of the text, problem \#6.\footnote{``The text'' means
    Rudin's \emph{Real \& Complex Analysis}, third edition.}  (Note
  that we have already shown that $\M$ is a $\sigma$-algebra so there
  is no need to show it again.  In the third edition, you are also
  asked to decribe the $\M$-measurable functions and their integrals.
  The key is to notice that they must be constant $\mu$-almost
  everywhere.)
\end{ques}


\begin{ques}
  Page 32 of the text, problem \#7.
\end{ques}

\begin{ques}
  Page 32 of the text, problem \#10.
\end{ques}


\begin{ques}
  Page 32 of the text, problem  \#12.  (This is easy if $f$ is
  bounded.  If $E_{n}=\set{x\in X:|f(x)|\le n}$ and
  $f_{n}=\charfcn_{E_{n}}\cdot f$, show that
  $\int_{X}|f-f_{n}|\,d\mu\to 0$.)
\end{ques}




\begin{ques}
  Suppose that $Y$ is a topological space and that $\M$ is a
  $\sigma$-algebra in $Y$ containing all the Borel sets.  Suppose in
  addition, $\mu$ is a measure on $(Y,\M)$ such that for all $E\in\M$
  we have
  \begin{equation}
    \label{eq:1}
    \mu(E)=\inf\set{\mu(V):\text{$V$ is open and $E\subset V$}}.
  \end{equation}
Suppose also that
\begin{equation}
  \label{eq:2}
  Y=\bigcup_{n=1}^{\infty}Y_{n}\quad\text{with $\mu(Y_{n})<\infty$ for
  all $n\ge1$.}
\end{equation}
In \$25 words, $\mu$ is a \emph{$\sigma$-finite outer regular measure} on
$(Y,\M)$.
\begin{enumerate}
\item Show that Lebesgue measure $m$ is a
  $\sigma$-finite outer regular measure on $(\R,\M)$.
  
\item Suppose $E$ is a $\mu$-measurable subset of $Y$.
  \begin{enumerate}
  \item Given $\epsilon>0$, show that there is an open set $V\subset
    Y$ and a closed set 
    $F\subset Y$ such that $F\subset E \subset V$ and $\mu(V\setminus
    F)<\epsilon$.
    

  \item Show that there is a $G_{\delta}$-set $G\subset Y$ and a
    $F_{\sigma}$-set $A\subset Y$ such that $A\subset E\subset G$ and
    $\mu(G\setminus A)=0$.
    
  \end{enumerate}
\item Argue that $(\R,\M,m)$ is the completion of the restriction of
  Lebesgue measure to the Borel sets in $\R$.
  

\end{enumerate}
\end{ques}

\begin{ques}
  Let $m$ be Lebesgue measure on $\R$ and suppose that $E$ is a set of
  finite measure.  Given $\epsilon>0$, show that there is a finite
  \emph{disjoint} union $F$ of open intervals such that $m(E\triangle
  F)<\epsilon$ where $E\triangle F:=(E\setminus F )\cup( F\setminus E)$ is
  the symmetric difference.  (This illustrates the first of
  Littlewood's three principles: ``Every Lebesgue measurable set is
  nearly a disjoint union of open intervals''.)
\end{ques}



\goodbreak

\begin{ques}
  Let $(X,\M,\mu)$ be a measure space, and let $(X,\M_{0},\mu_{0})$ be
  its completion.
  \begin{enumerate}
  \item If $f:X\to \C$ is $\mu_{0}$-measurable, show that there is a
    $\mu$-measurable function $g:X\to \C$ such that $f=g$
    a.e.~$[\mu_{0}]$.
    
  \item In part~(a), is there necessarily a $\mu$-null set $N$ such
    that $f(x)=g(x)$ for all $x\notin N$?
    

  \item What does this result say about Lebesgue measurable functions
    and Borel functions on $\R$?  (Compare with problem \#14  on page
    59 of the text.)
    

  \end{enumerate}

\end{ques}

\begin{ques}
  Suppose that $(X,\M,\mu)$ is a measure space.  Recall that $E\in\M$
  is called $\sigma$-finite if $E$ is the countable union of sets of
  finite measure.  Let $f\in\mathcal{L}^{1}(\mu)$.
  \begin{enumerate}
  \item Show that $\set{x\in X:f(x)\not=0}$ is $\sigma$-finite.
   

  \item Suppose that $f\ge0$.  Show that there are (measurable) simple
    functions $\phi_{n}$ such that $\phi_{n}\nearrow f$ everywhere and
    there is a single $\sigma$-finite set outside of which the
    $\phi_{n}$ vanish.
    

  \item Given $\epsilon>0$ show that there is simple function such
    that\label{item:1}
    \begin{equation*}
      \int_{X}|f-\phi|\,d\mu<\epsilon.
    \end{equation*}
   

\item If $(X,\M,\mu)=(\R,\M,m)$ is Lebesgue measure, show that we can
  take the simple function $\phi$ in part~(\ref{item:1}) to be a step
  function --- that is, a finite linear combination of characteristic
  functions of \emph{intervals}.
  

  \end{enumerate}
\end{ques}
\end{document}

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