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\def\course{MAT 324}
\def\semester{Fall 2006}

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{\bf  PROBLEM SET 3}
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\begin{enumerate}

\item If $f$ is measurable, show that the set of local maxima of $f$ is 
      a measureable set ($x$ is a local maximum of $f$ if there is an 
      interval  $I$ centered at $x$ so that $f(x) = \max_{y\in I} f(y)$).

\item Suppose $f$ is continuous on the reals and let $f^(2) = f \circ f$ 
and $f^{(n)} = f\circ f^{(n-1)} = f \circ f \circ \dots \circ f$ $n$ times.
Let $F(x) =0$ if $ \{ x_n\}=\{f^{n}(x)\}_{n=1}^\infty$ is bounded and
$F(x)=1$ if the sequence is unbounded. Show $F$ is measurable.

\item A function is called simple if it only takes on finite number of 
      different values. If $g$ is bounded and measurable, and $\epsilon >0$ 
      is given, show there is a measurable simple function $f$ so that 
      $\sup_x |g(x) - f(x)| \leq \epsilon$.  Is this true if $g$ is not 
      bounded?

\end{enumerate}

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