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\begin{document}
\begin{center}
{\bf \large Math 702 Problem Set 6} \vs \\
{\bf due Friday 3/21/2014} \vs \vs
\end{center}

\noindent
Measurable sets and functions in Euclidean spaces are always meant to be Lebesgue measurable. A {\bit full measure} set is one whose complement has measure zero. \vs \vs

\noindent
{\bf Problem 1.} 
\begin{enumerate}
\item[(i)]
If $f:\RR \to \RR$ is measurable and $\ve>0$, prove that there is a continuous function $g: \RR \to \RR$ such that $m \big( \{ x: f(x) \neq g(x) \} \big) \leq \ve$. \vs

\item[(ii)]
Show by an example that there may not be such a $g$ for $\ve=0$. \vs
\end{enumerate}

\noindent
{\bf Problem 2.} Suppose $f_n:[0,1] \to \RR$ is a sequence of measurable functions such that $f_n \to 0$ a.e. in $[0,1]$. \vs

\begin{enumerate}
\item[(i)]
Show that there is a subsequence $\{ f_{n_k} \}$ for which the series $\sum_{k=1}^{\infty} |f_{n_k}|$ converges a.e. in $[0,1]$. \vs

\item[(ii)]
Can we find a subsequence $\{ f_{n_k} \}$ for which $\sum_{k=1}^{\infty} |f_{n_k}|$ converges uniformly on a full measure subset of $[0,1]$? \vs 
\end{enumerate}
(Hint: Use Egoroff's theorem.) \vs \vs

\noindent
{\bf Problem 3.} Give an example of a monotone class $\fC$ in $\RR$ such that  
\begin{enumerate}
\item[(i)]
$\fC$ is not a $\sigma$-algebra, and
\item[(ii)]
$E \in \fC$ implies $\RR \sm E \in \fC$, and 
\item[(iii)]
$\fC$ has infinitely many elements. \vs
\end{enumerate}

\noindent
{\bf Problem 4.}
Recall that $\fB_X$ is the Borel $\sigma$-algebra of the metric space $X$. \vs 
\begin{enumerate}
\item[(i)]
Show that for any pair of metric spaces $X$ and $Y$,
$$
\fB_X \otimes \fB_Y \subset \fB_{X \times Y}.  
$$
\item[(ii)]
If $X$ is separable, prove that 
$$
\fB_X \otimes \fB_Y = \fB_{X \times Y}.
$$
As a special case, for $\fB_d=\fB_{\RR^d}$ we have the relation $\fB_d \otimes \fB_k = \fB_{d+k}$ once we identify the Cartesian product $\RR^d \times \RR^k$ with $\RR^{d+k}$.  
\end{enumerate}
(Hint: For (i), show that the product of two Borel sets is Borel. For (ii), show that every open set in $X \times Y$ is a countable union of open rectangles.) \vs \vs 

\noindent
{\bf Problem 5.} Investigate integrability of the following functions on the given product space:
\begin{align*}
f(x,y) & = \frac{x^2-y^2}{(x^2+y^2)^2} \qquad \text{on} \ [0,1] \times [0,1] \vs \\
g(x,y) & = \frac{xy}{(x^2+y^2)^2} \qquad \text{on} \ [-1,1] \times [-1,1]. \vs 
\end{align*}

\noindent
{\bf Problem 6.} Suppose $f$ is integrable on $[0,1]$. Show that the function $\ds{g(x)=\int_x^1 t^{-1} f(t) \, dt}$ 
is integrable on $[0,1]$ and $\ds{\int_0^1 g = \int_0^1 f}$. \vs \vs 

\noindent
{\bf Problem 7.} (Bonus) Does there exist a full measure subset of $\RR^2$ which contains no measurable rectangle $A \times A$ of positive measure?  
 

  


\end{document}
