% LaTeX file -- PM 450 Winter 2017, Assignment 4

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\begin{document}
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%%%%%%%   TITLE   %%%%%%
{\noindent\textbf{\large PM 450 \hfill Assignment 4 \hfill Due Friday March 3.}} 
\vskip 2em 

\begin{Ex}

\item Show that if $\bQ \cap [0,1]$ is covered by \textit{finitely many} intervals $I_1,\dots,I_n$, then $\sum_{i=1}^n \ell(I_i) \ge 1$.

\item Let $\{E_n\}_{n\ge0}$ be the non-measurable sets constructed in class which partition $[0,1)$.
\begin{parts}
\item Show that if $F$ is measurable and $F\subset E_n$, then $m(F)=0$.
\item Show that if $F$ is measurable and $m(F)>0$, then $F$ contains a non-measurable set.\\
\hint WLOG $F\subset [0,1)$. Consider $F = \dot\bigsqcup_{n\ge0} F\cap E_n$.
\end{parts}


\item \label{comp} Show that if $f:\bR\to \bR$ is measurable and $g:\bR\to\bR$ is continuous,
then $g \circ f$ is measurable.

\item Let $f_n: \bR \to\bC$ be a sequence of measurable functions. Define
$A = \{ x : \dlim_{n\to\infty} f_n(x)\ \text{ exists} \}$. 
Prove that $A$ is measurable.

\item A sequence $f_n:[0,1]\to\bC$ of measurable functions \textit{converges in measure} to $f$ 
if for all $\ep>0$, $\dlim_{n\to\infty} m(\{x : |f(x)-f_n(x)| \ge \ep \}) = 0$. 
Prove that if $f_n$ converges in measure to $f$, then there is a subsequence $f_{n_i}$ which converges to $f$ almost everywhere.

\item A generalized Cantor set can be constructed by removing an open interval of length $a_1$ from the middle of $[0,1]$,
then removing open intervals of length $a_2$ from the middle of the two remaining intervals, etc. The only constraint is that $a_{n+1}$ should be strictly smaller than the lengths of the intervals remaining at the $n$th stage.
\begin{parts}
\item Compute the measure of the usual Cantor set $C$.
\item Show that for any $0 \le r < 1$, there is a generalized Cantor set $K$ with $m(K)=r$.
\item Given a generalized Cantor set $K$, show that there is a homeomorphism $h$ of $[0,1]$ onto itself such that $h(C)=K$.
\item Construct a measurable set $E\subset \bR$ so that for every non-empty finite open interval $I=(a,b)$,
one has $0 < m(E\cap I) < m(I)$. \\
\hint it is enough to do this on $[0,1]$ and replicate. Repeatedly fill the gaps of a generalized Cantor set with more generalized Cantor sets.
\end{parts}
 

\item Let $f$ be the Cantor function, which is a continuous monotone function on $[0,1]$ which takes the value $1/2$ on the middle third,
values $1/4$ and $3/4$ on the middle thirds of the second level, etc. Let $g(x) = x + f(x)$.
\begin{parts}
\item Show that $g$ is a homeomorphism of $[0,1]$ onto $[0,2]$. i.e. show that $h=g^{-1}$ is a continuous function.
\item Show that $m(g(C)) = 1$.
\item Show that there is a measurable subset $A\subset [0,1]$ so that $h^{-1}(A)$ is not measurable.
\item Hence show that if $f$ is measurable and $g$ is continuous, then $f\circ g$ need not be measurable.
Compare with Q.\ref{comp}
\end{parts}




\end{Ex}
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