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\noindent
\textbf{Math 5616H} \hfill \textbf{Homework 11}

\noindent
\textbf{Posted: 4/18; Updated: 4/19; Due: Friday, 4/24/2015}\\
The problem set is due at the beginning of the class on
Friday. \smallskip

\noindent
\textbf{Reading}: Chapter 11: Review Sections 13-18, read Sections
19-25. \smallskip

\noindent
\textbf{Conventions}: For this homework, $(X, \M, \mu)$ will be a
measure space, unless stated otherwise. By definition, a
\emph{measurable set} is one from $\M$ and a \emph{measurable
  function} on $X$ is such $f: X \to [-\infty,\infty]$ that $f^{-1}(a,
\infty]$ is a measurable set.

\begin{prob}
  Let $(f_n)_{n \in \nn}$ be a sequence of measurable functions on a
  measure space $(X, \M, \mu)$. Prove that the set $\{x \in X \; | \;
  \lim_n f_n(x) \text{ exists}  \linebreak[0] \text{in } [-\infty, \infty]\}$
  is a measurable set, \emph{i.e}., belongs to the $\sigma$-algebra
  $\M$.
\end{prob}

\begin{prob}
  Prove that the characteristic function $\chi_{[0,1]}$ is not equal
  almost everywhere to a continuous function on $\nr$.
\end{prob}

\begin{prob}
  Let $f: \nr \to \nr$. Prove that if $f$ is differentiable on $\nr$,
  then $f'$ is Borel measurable, meaning the preimage of $(a,\infty]$
  is a Borel set. \emph{Hint}: Observe that $f'$ is a pointwise limit
  of functions
\[
f_n(x) = \frac{f(x+1/n) - f(x)}{1/n}.
\]
  \end{prob}

\begin{prob}
  Prove that if $f : X \to [0, \infty]$ is a nonnegative measurable
  function and $A \subset B$ are measurable sets, then $\int_A f d\mu
  \le \int_B f d\mu$.
\end{prob}

\begin{prob}
  Prove that if $\int_E f d\mu = 0$ for some $f \ge 0$ and $E \in \M$,
  then $f = 0$ almost everywhere. \emph{Hint}: Let $A_n := \{ x \in E
  \; | \; f(x) > 1/n\}$. Show that $\mu(A_n) = 0$, given the
  assumption. Deduce that $\mu(\bigcup_n A_n) = 0$.
\end{prob}

\begin{prob}
  Define $f : [0,1] \to \nr$ by $f(x) := 0$ if $x$ is rational, and
  $f(x) := d^2$ if $x$ is irrational, where $d$ is the first nonzero
  digit in the decimal expansion of $x$. Show that $\int_{[0,1]} f d m
  = 95/3$. \emph{Hint}: Is not $f$ simple?
\end{prob}

\begin{prob}
  Show that if $f : X \to [0,\infty]$ is measurable and bounded and
  $\int_{X} f d \mu < \infty$, then $\int_{X} f^2 d \mu < \infty$.
\end{prob}

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