\begin{assignment}{2022-09-21}{2022-09-29}
  \item Let $C \subseteq \R^d$ be convex.
    Must $C$ be Lebesgue measurable?
    Must $C$ be Borel measurable?
    Prove or find counter examples.
    \noteb{The cases $d = 1$ and $d > 1$ are different.}


  \item
    Let $f:[0, 1] \to [0, 1]$ be the Cantor function, and $g (y) = \inf\{f = y\}$.
    \begin{parts}
      \item Show that $f$ is H\"older continuous with exponent $\ln 2 / \ln 3$.
      \item Show that $g\colon [0, 1] \to C$ is injective. (Here $C$ is the Cantor set.)
    \end{parts}


  \item Let $(X, \Sigma)$ be a measure space, and $f, g:X \to [-\infty, \infty]$ be measurable.
    Suppose whenever $g=0$, $f\neq 0$, and whenever $f = \pm \infty$, $g \in (-\infty, \infty)$.
    Show that $\frac{f}{g}:X \to [-\infty, \infty]$ is measurable.
    \par
    \notes{%
      [Note that by the given data you will never get a `meaningless' quotient of the form $\frac{0}{0}$ or $\frac{\pm \infty}{\pm \infty}$.
      The remainder of the quotients (e.g. $\frac{1}{\infty}$) can be defined in the natural manner.]\par}

  \item Let $f_n:X \to \R$ be a sequence of measurable functions such that $f_n \to f$ almost everywhere (a.e.).
    Let $g:\R \to \R$ be a Borel function.
    If for a.e.\ $x \in X$, $g$ is continuous at $f(x)$, then show $g \circ f_n \to g\circ f$ a.e.

  \item
    Let $(X, \Sigma, \mu)$ be a measure space, and $(X, \Sigma_\mu, \bar\mu)$ it's completion.
    Show that $g:X \to [-\infty, \infty]$ is $\Sigma_\mu$-measurable if and only if there exists two $\Sigma$-measurable functions $f,h:X \to [-\infty, \infty]$ such that $f = h$ $\mu$-almost everywhere, and $f \leq g \leq h$ everywhere.
\end{assignment}
