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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?
\item Suppose $E $ is measurable set of real numbers and let $f(t) = m(E \cap (t-1,t+1))$.
Show that $f$ is continuous.
\item Suppose $E$ is a closed set in the upper half-plane whose vertical projection onto the
real line is surjective (onto). For each real number $x$ let
$y(x) $ be the closest point of $E \cap L_x$ to $x$ (here $L_x$ is the vertical line
through $x$). Show that $y$ is a measureable function, but need not be continuous.
\item Given a real number $x \in [0,1]$ let $x_n(x) \in \{ 0,1\}$ be its
$n$-binary digit (if this is not unique, choose the epansion ending in all
$0$'s). Let $f(x) = \limsup_{n \to \infty} \frac 1n \sum_{k=1}^n x_n(x)$.
Show that $f$ is measureable. Where is $f$ continuous?
Can you guess what $\int_0^1 f (x) dx $ is?
\end{enumerate}
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