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\def\course{MAT 324}
\def\semester{Fall 2007}

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\begin{center}
{\bf  PROBLEM SET 2a}
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\begin{enumerate}

\item Prove that the Lebesgue function $F$  satisfies
             $$ |F(x) - F(y)| \leq |x-y|^\alpha ,$$
       where $\alpha = \log 2 / \log 3$.  This is called 
       a H{\"o}lder condition of order $\alpha$.

\item If $E\subset [0,1]$ is a null set, and $f:[0,1] \to [0,1]$ is 
      continuous, does $f(E)$ have to be a null set as well?  Prove 
      this or find a counterexample.

\item Let $X=\{x+y: x,y \in C\}$ be the set of sums of numbers in the 
      Cantor middle third set.  What is $X$?

\item Prove that if $\lambda >0$ then  $m^*(\lambda E) = \lambda m^*(E)$ where 
      $\lambda E = \{ \lambda x: x \in E\}$.

\item  If $X$ is set of finite Lebesgue measure show that $m(X \cap X+t) \to 0$
       as $t \to \infty$. Here $X+t =\{ x+t: x \in X\}$.  Does there have to 
       be a value of $t$ so that $m(X \cap X+t) =0$?

\end{enumerate}

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