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\begin{document}
\begin{center}
{\bf \large Math 702 Problem Set 5} \vs \\
{\bf due Friday 3/14/2014} \vs \vs
\end{center}

\noindent
Throughout $m$ denotes Lebesgue measure on $\RR^d$. The space $L^1(m)$ of Lebesgue integrable functions on $\RR^d$ is denoted more traditionally by $L^1(\RR^d)$. \vs \vs

\noindent

\noindent
{\bf Problem 1.} Find, with justification, the limits
$$
\lim_{n \to \infty} \int_0^n \left( 1 + \frac{x}{n} \right)^n e^{-2x} \, dx \qquad \text{and} \qquad \lim_{n \to \infty} \int_{-\infty}^{\infty} \frac{n \sin(x/n)}{x(x^2+1)} \, dx. \vs \vs 
$$ 

\noindent
{\bf Problem 2.} Consider the function $g(x)=\sum_{n=1}^{\infty} 2^{-n} f(x-r_n)$, where 
$$
f(x)= \begin{cases} x^{-1/2} & \quad \text{if} \ x \in (0,1) \\
0 & \quad \text{if} \ x \notin (0,1).
\end{cases}
$$
and $\{ r_n \}$ is any enumeration of $\QQ$. Show that $g \in L^1(\RR)$ so $g<+\infty$ a.e., but $g$ is unbounded on every interval. \vs \vs 

\noindent
{\bf Problem 3.} 
\mbox{}
\begin{enumerate}
\item[(i)]
For each $0<\ve<1$ construct an open dense set $U \subset [0,1]$ such that $m(U)=\ve$. \vs
\item[(ii)]
Construct a Borel set $E \subset [0,1]$ such that 
$$
0<m(E \cap I)<m(I)
$$ 
for every interval $I \subset [0,1]$. 
\end{enumerate}
(Hint: For (i), think of fat Cantor sets. For (ii), construct $E$ as the intersection of a decreasing sequence of suitably chosen open dense sets.) \vs \vs 

\noindent
{\bf Problem 4.} 
\mbox{}
\begin{enumerate}
\item[(i)]
Let $E \subset \RR^d$. Show that $m(E)=0$ iff for every $\ve>0$ there are countably many open (Euclidean) balls $\{ B(x_n,r_n) \}$ such that $E \subset \bigcup B(x_n,r_n)$ and $\sum r_n^d < \ve$. \vs

\item[(ii)]
Suppose $f: \RR^d \to \RR^d$ is a Lipschitz map, i.e., there is an $M>0$ such that  
$$
\| f(x)-f(y) \| \leq M \| x-y \| \qquad \text{for all} \ x,y \in \RR^d. 
$$ 
Show that $m(f(E))=0$ whenever $m(E)=0$. \vs
\end{enumerate} 

\noindent
{\bf Problem 5.} Suppose $f:\RR^d \to \RR$ is Lebesgue measurable. Show that there are Borel maps $g,h: \RR^d \to \RR$ such that $g(x) \leq f(x) \leq h(x)$ for every $x \in \RR^d$ and $g(x)=f(x)=h(x)$ for a.e. $x \in \RR^d$. (Hint: It suffices to assume $f \geq 0$. Treat the case of positive simple functions first.) \vs \vs

\noindent
{\bf Problem 6.} Give an example of a translation-invariant Borel measure on $\RR^d$ that is not a multiple of Lebesgue measure $m$. Why doesn't your example violate the uniqueness theorem that we proved for $m$? \vs \vs

\noindent
{\bf Problem 7.} (Bonus) The {\bit support} of a Borel measure $\mu$ on a metric space $X$ is defined by 
$$
\supp(\mu) = \{ x \in X: \mu(B(x,r))>0 \ \text{for every} \ r>0 \}. \vs
$$ 
\begin{enumerate}
\item[(i)]
If $X$ is separable, show that $\supp(\mu)$ is the smallest closed set in $X$ whose complement has measure zero. In particular, $\supp(\mu)$ is non-empty (unless $\mu=0$). \vs

\item[(ii)]
Show that every non-empty closed set in $\RR^d$ is the support of a Radon measure. \vs 
\end{enumerate}
(Hint for (ii): Given a non-empty closed set in $\RR^d$, choose a countable dense subset of it, use this subset to define a positive linear functional on $\sC_c(\RR^d)$, and apply the Riesz representation theorem.)   


 
 

  







\end{document}
