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\title{Real Analysis}
\author{Homework 2}
\date{due September 15, 2008}
\maketitle

\begin{enumerate}
\item (From Lieb and Loss.) Let $(\Omega,\Sigma,\mu)$ be a measure
  space.  It is natural to define $\mu(A)=0$ if $A$ is the subset of a
  nullset, even though $A$ may not be contained in the
  $\sigma$-algebra.

  Show that if we extend $\Sigma$ by adding or subtracting such sets
  of measure zero to the sets in $\Sigma$, the resulting collection of
  sets is again a $\sigma$-algebra, and the extended measure is again
  a measure on the larger $\sigma$-algebra.

\item Complete the proof of Theorem 1.4 in Lieb and Loss by writing
  out the proof for the $\sigma$-finite case, filling in the two
  missing arguments marked ``Why?'' in the text.

\item (From Folland.) Let $\mu$ denote the Lebesgue measure on the
  real line.  If $E \in \mathcal B_{\mathbb R}$ and $\mu(E)<\infty$,
  then for every $\varepsilon>0$ there exists a set $A$ which is a
  finite union of open intervals such that
\[
  \mu ((E \setminus A) \cup (A \setminus E)) < \varepsilon \,.
\] 

\end{enumerate}

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