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\begin{document}

\begin{center}
\begin{large}
Math 70100: Functions of a Real Variable I \\
Homework 9, due Wednesday, November 12th.
\end{large}
\end{center}

{\bf Name:} Insert your name here.

\begin{questions}
\question {\em (Folland \S 1.5 \# 30)} Let $\lambda$ be Lebesgue measure on $\R$, and let $E \subset \R$ be a Lebesgue measurable set with $\lambda(E)>0$. Show that for any $\alpha<1$, there is an open interval $I$ so
that $\lambda(E \cap I) > \alpha \lambda(I)$.  ({\em Hint:} An open set in $\R$ is a countable union of disjoint open intervals.)
\begin{sol}
Insert answer here.
\end{sol}

\question Imitate the construction of the middle-thirds Cantor set to show that for every
$c$ with $0<c<1$, there is a Cantor set $K \subset [0,1]$ whose Lebesgue measure is $c$. 
\begin{sol}
Insert answer here.
\end{sol}

\question {\em (Borel-Cantelli Lemma)} Let $(X,\Sigma, \mu)$ be a measure space with $\mu(X)<\infty$. 
Suppose the sequence of sets $\{E_n \in \Sigma~:~n \in \N\}$ satisfies $\sum_{n=1}^\infty \mu(E_n)<\infty$. Show that the set
$$A=\{x \in X~:~\text{there are infinitely many $n \in \N$ so that $x \in E_n$}\}$$
is measurable (lies in $\Sigma$) and has measure zero.
\begin{sol}
Insert answer here.
\end{sol}

\question Construct a Lebesgue measurable set $E \subset [0,1]$ of Lebesgue measure $\frac{1}{2}$
such that both $E$ and $[0,1] \smallsetminus E$ are dense in $[0,1]$.
\begin{sol}
Insert answer here.
\end{sol}

\question {\em (Folland \S 1.2 \# 10)} Show that if $(X,\Sigma,\mu)$ is a measure space
and $E \in \Sigma$, then the function $\mu_E:\Sigma \to [0,\infty]$ defined by $\mu_E(A)=\mu(A \cap E)$
is a measure.
\begin{sol}
Insert answer here.
\end{sol}

\question {\em (Based on Folland \S 1.2 \# 7)} Let $\Sigma$ be a $\sigma$-algebra on the set $X$. Show that the collection of
all measures on $(X,\Sigma)$ is a closed convex cone in the sense that
if $\mu_1$ and $\mu_2$ are measures on $(X,\Sigma)$  and $c_1,c_2 \geq 0$, then so is $c_1 \mu_1+c_2 \mu_2$. 
Are these measures closed under countable sums?
\begin{sol}
Insert answer here.
\end{sol}

\question {\em (Folland \S 1.4 \# 18)}
Let ${\mathcal A} \subset {\mathcal P}(X)$ be an algebra on $X$. 
Let ${\mathcal A}_\sigma \subset {\mathcal P}(X)$ denote the collection of all countable unions of 
sets in ${\mathcal A}$, and let ${\mathcal A}_{\sigma \delta}$ denote the collection of all countable
intersections of sets in ${\mathcal A}_\sigma$. Let $\mu_0$ be a premeasure on ${\mathcal A}$
and let $\mu^\ast$ be the induced outer measure.
\begin{parts}
\part Show that for any $E \subset X$ and any $\epsilon>0$, there is an $A \in {\mathcal A}_\sigma$
with $\mu^\ast(A) \leq \mu^\ast(E)+\epsilon$. 
\begin{sol}
Insert answer here.
\end{sol}
\part Suppose $\mu^\ast(E)<\infty$. Show that $E$ is measurable if and only if there is a $B \in {\mathcal A}_{\sigma \delta}$ with $E \subset B$
and $\mu^\ast(B \smallsetminus E)=0.$
\begin{sol}
Insert answer here.
\end{sol}
\part Recall that $\mu_0$ is {\em $\sigma$-finite} if there is a countable collection $\{C_i\} \subset {\mathcal A}$ with $\bigcup_i C_i=X$ and $\mu_0(C_i)< \infty$ for all $i$. Show that if $\mu_0$ is $\sigma$-finite, then 
even if $\mu^\ast(E)=\infty$ the statement from part (b) still holds.
\begin{sol}
Insert answer here.
\end{sol}
\end{parts}
\end{questions}
\end{document}
