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\begin{document}
\begin{center}
{\bf \large Math 701 Problem Set 10} \vs \\
{\bf due Friday 12/6/2013} \vs \vs
\end{center}

\noindent
$\fM$ and $m$ will denote the $\sigma$-algebra of Lebesgue measurable sets in $\RR$ and Lebesgue measure, respectively. \vs \vs   

\noindent
{\bf Problem 1.} For $E \subset \RR$ and $\lambda >0$, set $\lambda E = \{ \lambda x: x \in E \}$. Show that if $E \in \fM$, then $\lambda E \in \fM$ and $m(\lambda E) = \lambda m(E)$. (Hint: Verify that the map $t \mapsto \lambda t$ preserves $G_\delta$'s, $F_\sigma$'s, and sets of measure zero.) \vs 

\noindent
{\bf Problem 2.} Imitate the construction of the middle-thirds Cantor set to show that for every $0<\alpha<1$ there is a Cantor set $K \subset [0,1]$ with $m(K)=\alpha$. \vs

\noindent
{\bf Problem 3.} Construct a measurable set $E \in [0,1]$, with $m(E)=1/2$, such that both $E$ and $[0,1] \sm E$ are dense in $[0,1]$. \vs    

\noindent
{\bf Problem 4.} Let $E \subset [0,1]$ be the set of numbers whose unique non-terminating decimal expansion does not contain the digit $7$. Show that $E \in \fM$  and find $m(E)$. \vs 

\noindent
{\bf Problem 5.} Let $\{ E_n \}_{n \geq 1}$ be a sequence in $\fM$ such that $\sum_{n \geq 1} m(E_n) < +\infty$. Show that almost every point of $\RR$ belongs to at most finitely many of the $E_n$. (Hint: Let $A$ be the set of points in $\RR$ that belong to infinitely many of the $E_n$. Verify that 
$$
A = \bigcap_{k \geq 1} \bigcup_{n \geq k} E_n
$$  
and conclude that $m(A)=0$.) \vs 

\noindent
{\bf Problem 6.} Let $E \subset [0,1]$ be a measurable set with $m(E)>0$. Show that the arithmetical difference set 
$$
E - E = \{ x-y : x,y \in E \}
$$
contains a neighborhood of the origin. (Hint: Given a small $\ve>0$, find an open set $U \supset E$ with $m(E)>(1-\ve) m(U)$. Write $U$ as the disjoint union of open intervals. One of these intervals, say $I$, must satisfy $m(E \cap I)> (1-\ve) m(I)$. Show that if $|t|<\delta=(1-2\ve)|I|$, then $E \cap I$ meets its translate by $t$. This will prove $E-E$ contains $(-\delta,\delta)$.)\vs   




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