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\begin{document}

\centerline {\bf Math 523a - Homework 2} 

\bigskip

\begin{enumerate}

\item (problem 4, p. 14 in Folland). Let $\A$ be an algebra. 
We say $\A$ is closed under increasing countable unions if for every 
sequence $E_n \in \A$ with $E_n \subset E_{n+1}$ for $n=1,2,\cdots$, 
we have $\cup_{n=1}^\infty E_n \in \A$. Prove that $\A$ is a $\sigma$-algebra
if and only if it is closed under countable increasing unions. 

\item (problem 5, p. 14 in Folland). For any collection $\E$ of subsets 
of $X$, let $\M(\E)$ denote the $\sigma$-algebra generated by $\E$.
Prove that 
\bea
\M(\E) = \bigcup_\F \M(\F)
\eea
where the union is over all countable subsets $\F$ of $\E$. 

\item Prove that the $\sigma$-algebra of Borel sets in $\reals^2$
is generated by the collection of open balls in $\reals^2$.  

\item Let $(X,\M,\mu)$ be a measure space and $E_n \in \M$ for $n=1,2,\cdots$. 
Prove that 
\bea
\sum_{n=1}^\infty \mu(E_n) < \infty \quad  \implies \quad \mu(\limsup E_n)=0
\eea
Recall that $\limsup E_n = \cap_{n=1}^\infty \cup_{k=n}^\infty E_k$. 

\item (problem 11, p. 27 in Folland) Let $(X,\M,\mu)$ be a {\bf finite }
measure space. 

\begin{enumerate}
\item Prove that if $E,F\in \M$ and $\mu(E \Delta F)=0$ then 
$\mu(E)=\mu(F)$. 
\item Define $E \sim F$ if $\mu(E \Delta F)=0$. Prove that $\sim$ is 
an equivalence relation. 
\item Prove that $\rho(E,F)=\mu(E \Delta F)=0$ 
defines a metric on the set of equivalence classes in $\M$. 
\end{enumerate}

\newpage

\item Let $(X,\M,\mu)$ be a measure space. 
We say that it is {\it complete} if for every $N \in \M$ such that
$\mu(N)=0$, every subset of $N$ is in $\M$ (and so has measure $0$). 
If the measure space is not complete, there is a natural way to 
``complete it.'' Define 
\bea
\N &=& \{ N \in \M: \nu(N)=0\} \\
\overline{\M} &=& \{E \cup F: E \in \M \, and \, F \subset N \,  for \, 
some \, N \in \N\}
\eea
Prove that $\overline{\M}$ is a $\sigma$-algebra and there is a unique 
extension of $\mu$ to a complete measure on $\overline{\M}$.

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

