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\def\course{MAT 324}
\def\semester{Fall 2006}

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{\bf  PROBLEM SET 2b}
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\begin{enumerate}

\item Suppose $\{f_n\}$ is a sequence of continuous functions and let
       $E$ be the set of $x$'s where the seqeunce converges. Show
      that $E$ is Borel, and hence Lebesgue measurable.

\item Is the set of rational numbers equal to the intersection of some 
      countable collection of open sets?

\item Is a subset of a Borel set Borel?

\item What is the conditional probability that a $x \in [0,1]$ has 
      first decimal digit even, given that its second binary digit 
      is $1$?

\item Let $X_n \subset [0,1]$ be the set of $x$'s, whose $n$th binary 
      digit is $1$.  Show these events are independent.

\end{enumerate}

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