\begin{assignment}{2022-09-14}{2022-09-22}
  \item
    Let $\mu, \nu$ be two measures on $(X, \Sigma)$. Suppose $\mathcal C \subseteq \Sigma$ is a $\pi$-system such that $\mu = \nu$ on $\mathcal C$.
  \begin{parts}
    \item Suppose $\exists C_i \in \mathcal C$ such that $\bigcup_1^\infty C_i = X$ and $\mu(C_i) = \nu(C_i) < \infty$. Show that $\mu = \nu$ on $\sigma(\mathcal C)$.
    \item If we drop the finiteness condition $\mu(C_i) < \infty$ is the previous subpart still true? Prove or find a counter example.
  \end{parts}

  \item
    Let $X$ be a metric space and $\mu$ a Borel measure on $X$. Suppose there exists a sequence of sets $B_n \subseteq X$ such that $\bar B_n \subseteq \mathring B_{n+1}$, $\bar B_n$ is compact, $X = \cup_1^\infty B_n$ and $\mu(B_n) < \infty$.
    Show that $\mu$ is regular.
    Further, for any $A \in \mathcal B(X)$ and $\epsilon > 0$ show that there exists $U$ open and $C$ closed such that $C \subseteq A \subseteq U$ and $\mu(U - C) < \epsilon$.
    If $\mu(A) < \infty$, then also show that $C$ above can be chosen to be compact.
    \par
    \notes{[Note that the Lebesgue measure on $\R^d$ satisfies the assumptions in this problem, and thus for any $A \in \mathcal L(\R^d)$ and $\epsilon > 0$ there exists $U$ open and $C$ closed such that $C \subseteq A \subseteq U$ and $\lambda(U - C) < \epsilon$.]\par}

  \item
    \begin{parts}
      \item
	Find $E \in \mathcal B(\R)$ so that for all $a < b$, we have $0 < \lambda(E \cap (a, b)) < b - a$.
      \item
	Let $\kappa \in (0, 1/2)$.
	Does there exist $E \in \mathcal B(\R)$ such that for all $a < b \in \R$, we have $\kappa(b - a) \leq \lambda(E \cap (a, b)) \leq (1-\kappa) (b - a)$?
	Prove it.
    \end{parts}

  \item If $A \in \mathcal L(\R^d)$ must $\dlim_{x \to 0} \lambda( (A + x) \cap A) = \lambda(A)$?
    Prove it, or find a counter example.

  \item
    \begin{parts}
      \item Prove $\mathcal B(\R^{m+n}) =  \sigma( \{ A \times B \st A \in \mathcal B(\R^m) \mathop\& B \in \mathcal B(\R^n) \})$.
      \item Prove $\mathcal L(\R^{m+n}) \supsetneq  \sigma( \{ A \times B \st A \in \mathcal L(\R^m) \mathop\& B \in \mathcal L(\R^n) \})$.
      \item Show $\mathcal L(\R^2) \supsetneq \mathcal B(\R^2)$.
    \end{parts}
\end{assignment}
