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\begin{document}
\title{Stochastic Processes}
\author{Homework 1}
\date{Due in class Thursday, February 25}
\maketitle

Each problem is worth 5 points.  

\begin{enumerate}
\item Show that any open subset of $\R$ is a countable union of open
intervals.

\item Show that 
\begin{enumerate}
\item all one-point sets $\{ x \}$ with $x \in \R$ 
\item $\mathbb Q$
\end{enumerate}
belong to $\mathcal B(\R)$, the Borel $\sigma$-field on $\R$.

\item Let $\Omega = [0,1]$ and $A \subsetneq \Omega$ nonempty.  What
is the $\sigma$-field $\F$ generated by $\{A\}$?  What are the
$\F$-measurable functions?

\item Let $(\Omega, \F, P)$ be a probability space and $\xi \colon
\Omega \to \R$ a random variable which is nonnegative a.e., in other
words, $P\{ x \in \Omega \colon \xi(x)<0 \} = 0$.  Show that
\[
  \int_\Omega \xi \, \d P \geq 0 \,.
\]

\item Let $\Omega = [0,1]$, $\F=\mathcal B(\R)$, and $P$ the Lebesgue
measure.  Give an example of a sequence of nonnegative Borel functions
$\{f_n \}_{n \in \N}$ defined on $[0,1]$ such that
\[
  \int_\Omega f_n \, \d P \to 0 \,,
\]
but $f_n \nrightarrow 0$ on a set of positive measure.

\end{enumerate}

\end{document}
