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\noindent
\textbf{Math 5616H} \hfill \textbf{Homework 8}

\noindent
\textbf{Posted: 3/29; Updated: 4/1; Due: Friday, 4/3/2015}\\
The problem set is due at the beginning of the class on
Friday. \smallskip

\noindent
\textbf{Reading}: Chapter 11: Sections 1-8, 12. \smallskip \
\begin{prob}
  Show that if we add one more property to the definition of a
  $\sigma$-ring $\mathfrak{M}$ on a set $X$ in 11.1 of the text:
  $\mathfrak{M}$ contains the ambient set $X$, we will get a notion
  equivalent to the notion of a $\sigma$-algebra from class (a
  collection of subsets containing the empty set $\varnothing$ and
  closed under countable unions and complements).
  \end{prob}

\begin{prob}
Is the Cantor set a Borel set on $\nr$?
\end{prob}

\begin{prob}
  Let $X$ be a separable metric space and let $\mathfrak{M}$ be the
  $\sigma$-algebra generated by open balls in $X$. Show that
  $\mathfrak{M}$ contains all the open sets in $X$ and all the closed
  sets. Describe some sets in an example $\mathfrak{M}$ that are
  neither open nor closed. The $\sigma$-algebra $\mathfrak{M}$ is
  called the $\sigma$-algebra of \emph{Borel sets} in
  $X$. \emph{Hint}: Show that every open set in $X$ is a countable
  union of open balls.

\end{prob}

\begin{prob}
  Can a $\sigma$-algebra in a set $X$ have cardinality $\aleph_0$?
  \emph{Hint}: Show that if a $\sigma$-algebra is infinite, then it
  contains a countable collection of pairwise disjoint subsets. Deduce
  that the $\sigma$-algebra will also contain arbitrary unions of
  these dijoint subsets.
\end{prob}

\begin{prob}
  Show that if $\mu$ is a measure on a set $X$ with a $\sigma$-algebra
  $\mathfrak{M}$ and there is a set $A \in \mathfrak{M}$ such that $0
  < \mu(A) < \infty$, then $\mu(\varnothing) = 0$.
\end{prob}

\begin{prob}
  Show that the $\sigma$-algebra generated by the collection of open
  rectangles (called \emph{intervals} in the text) is the same as the
  $\sigma$-algebra generated by the collection of half-open
  rectangles.
\end{prob}

\begin{prob}
  Show that the Lebesgue outer measure of a face $I_1 \times \dots
  \times I_{i-1} \times \{a\} \times I_{i+1} \times \dots \times I_n$
  of a rectangle $I_1 \times \dots \times I_n \subset \nr^n$ is
  zero. Here $I_1, \dots, I_n$ are open intervals in $\nr$ and $a$ is
  one of the endpoints of $I_i$.
\end{prob}

\begin{prob}
  Show that the restriction $\mu^*|_E$ of an outer measure $\mu^*$ on
  a set $X$ to a subset $E \subset X$ defined by $\mu^*|_E (A) :=
  \mu^* (E \cap A)$ for any $A \subset X$ is an outer measure on
  $X$. Show that any set that is measurable with respect to $\mu^*$ is
  measurable with respect to $\mu^*|_E$ (regardless of whether $E$ is
  measurable).
\end{prob}

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