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\noindent
\textbf{Math 5616H} \hfill \textbf{Homework 10}

\noindent
\textbf{Posted: 4/11; Updated 4/15; Due: Friday, 4/17/2015}\\
The problem set is due at the beginning of the class on
Friday. \smallskip

\noindent
\textbf{Reading}: Chapter 11: Sections 8, 10-11, 13-18. \smallskip \

\begin{prob}
  Let $f: \nr^n \to \nr^n$ be a \emph{homeomorphism}, \emph{i.e}., a
  continuous map $\nr^n \to \nr^n$ which has a continuous
  inverse. Suppose that $f$ takes sets of Lebesgue measure zero to
  sets of Lebesgue measure zero. Prove that $f$ takes Lebesgue
  measurable sets to Lebesgue measurable sets.
  \end{prob}

\begin{prob}
  If $(X, \M, \mu)$ is a measure space, $f: X \to [-\infty, \infty]$
  is measurable, and $B \subset \nr$ is a Borel set, show that the
  preimage $f^{-1}(B)$ is measurable in $X$. (I recall that a subset
  $A$ in a measure space $(X, \M, \mu)$ is called \emph{measurable},
  if $A \in \M$. Sets measurable with respect to an outer measure is a
  different notion, even though the two notions become the same, when
  the measure is constructed out of an outer measure. In this problem,
  we are talking about an arbitrary measure.)
\end{prob}

\begin{prob}
  Suppose $f: X \to [-\infty, \infty]$ is a function on a measure
  space. Show that $f$ is measurable iff the preimage $f^{-1} (a,
  \infty]$ is measurable for all $a \in \nq$.
\end{prob}

\begin{prob}
  Suppose $X$ is a measure space, $f: X \to \nr$ is measurable, and
  $g: \nr \to \nr$ is continuous. Show that $g \circ f: X \to \nr$ is
  measurable.
\end{prob}

\begin{prob}
  Describe the conditions on a subset $A \subset X$ under which the
  characteristic function $\chi_A: X \to \nr$ of $A$, defined by
\[
\chi_A(x) := 
\begin{cases}
1 & \text{if } x \in A,\\
0 & \text{otherwise},\end{cases}
\]
is measurable.
\end{prob}

\begin{prob}
  Suppose $(f_i)_{i \in \nn}$ is a sequence of measurable functions on
  a \emph{complete} measure space $X$, which means that any subset of
  a measure-zero set in $X$ is measurable. Suppose also that the
  sequence converges pointwise \emph{almost everywhere} to a function
  $f$, \emph{i.e}., for every $x \in X$, except for members of a set a
  of measure zero. Show that $f$ is measurable. You can use 11.17.
\end{prob}

\begin{prob}
  Show that Lebesgue measure is \emph{translation invariant}, that is,
\[
m(A+x) = m(A) \text{ for every Lebesgue measurable $A \subset
  \nr^n$ and $x \in \nr^n$},
\]
where $A+x := \{ a + x \; | \; a \in A\}$.
\end{prob}

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