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\begin{document}

\title{Real Analysis}
\author{Homework 1}
\date{due September 8, 2008}
\maketitle

\begin{enumerate}
\item Let $\Omega$ be a set and $\Sigma$ a $\sigma$-algebra on
  $\Omega$.  Show that
  \begin{enumerate}
  \item $\Sigma$ is closed under countable intersection,
  \item $\Omega \in \Sigma$ and $\emptyset \in \Sigma$,
  \item $A_1, A_2 \in \Sigma$ implies $A_1 \setminus A_2 \in \Sigma$.
  \end{enumerate}

\item The \emph{Borel $\sigma$-algebra} on a metric space $\Omega$ is
  defined as the $\sigma$-algebra generated by the open subsets of
  $\Omega$.

  Show that the Borel $\sigma$-algebra on $\mathbb R$ is also
  generated by the set of open intervals $\{(a,b) \colon a<b \}$.

  (The proof is essentially given in Folland, the task here is to
  write it out in a clear, self-contained manner.)

\item (From Folland.) If $(\Omega, \Sigma, \mu)$ is a measure space
  and $A_j \in \Sigma$ for $j \in \N$, show that
\[
  \mu (\liminf_{j \to \infty} A_j) \leq 
  \liminf_{j \to \infty} \mu(A_j) \,.
\]

\item (From Folland.) If $(\Omega, \Sigma, \mu)$ is a measure space
  and $E,F \in \Sigma$, show that
\[
  \mu(E) + \mu(F) = \mu (E \cup F) + \mu (E \cap F) \,.
\]

\end{enumerate}

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