\begin{assignment}{2022-08-31}{2022-09-07}
  \item
    Let $\mu$ be a positive measure on $(X, \Sigma)$.
    \begin{parts}
      \item
	If $A_i \in \Sigma$ are such that $A_i \subseteq A_{i+1}$, show that $\mu(\bigcup_{i=1}^\infty A_i) = \lim_{i \to \infty} \mu(A_i)$.
      \item
	If $A_i \in \Sigma$ are such that $A_i \supseteq A_{i+1}$, show that $\mu(\bigcap_{i=1}^\infty A_i) = \lim_{i \to \infty} \mu(A_i)$, provided $\mu(A_1) < \infty$. Is it still true if $\mu(A_1) = \infty$?
    \end{parts}

  \item
    Prove any open subset of $\R^d$ is a countable union of cells.

  \item
    For each of the following sets, compute the Lebesgue outer measure.
    \begin{mcparts}[3]
      \item Any countable set.
      \item The Cantor set.
      \item $\{x \in [0, 1] \st x \not \in \Q \}$.
    \end{mcparts}

  \item
    \begin{parts}
      \item If $V \subset \R^d$ is a subspace with $\dim(V) < d$, then show that $\lambda^*(V) = 0$.
      \item If $P \subset \R^2$ is a polygon show that $\operatorname{area}(P) = \lambda^*(P)$.
    \end{parts}

  \item
    Does there exist a $\sigma$-algebra whose cardinality is countably infinite?
    Disprove, or find an example.


  \fullwidth{\noindent\emph{Optional problems and details I omitted in class.}}

  \itemopt
    Define $\mu(A)$ to be the number of elements in $A$.
    Show that $\mu$ is a measure on $(X, \mathcal P(X))$.
    (This is called the counting measure.)

  \itemopt
    Let $x_0 \in X$ be fixed.
    Define $\delta_{x_0}(A) = 1$ if $x_0 \in A$ and $0$ otherwise.
    Show that $\delta_{x_0}$ is a measure on $(X, \mathcal P(X))$.
    (This is called the delta measure at $x_0$.)
    
  \itemopt Show that $\lambda^*(a + E)  = \lambda^*(E)$ for all $a \in \R^d$, $E \subset \R^d$.

  \itemopt
    Show that $\lambda^*(I) = \ell(I)$ for all cells. (I only proved it for closed cells in class.)

  \itemopt
    Show that $\mathcal B(\R)$ has the same cardinality as $\R$.


  \itemopt
    \emph{(Challenge)}
    Suppose $f_n:[0,1] \to [0, 1]$ are all Riemann integrable, $0 \leq f_n \leq 1$ and $(f_n) \to 0$ pointwise.
    Show that $\dlim_{n \to \infty} \int_0^1 f_n = 0$, using only standard tools from Riemann integration.
\end{assignment}
