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\def\FF{\mathcal F}
\def\R{\mathbb R}
\def\Pr{\mathbf P}

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Math 831 -- Spring 2012\\[10pt]{\textbf{Homework 1\\[15pt]}}
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\noindent{\textbf {Due:  Thursday, September 13th, 2012}. }\\



\begin{enumerate}

\item Suppose $f$ is a measurable mapping from one measurable space $S$ to another measurable space $U$.  If $A$ is a measurable subset of $S$, does it follow that the image $f(A)$ is a measurable subset of $U$?
\item Exercise 1.1.1 from text.  
\begin{enumerate}[(i)]
\item If $\FF_i$, $i \in I$ are $\sigma$-fields, then $\cap_{i \in I}\FF_i$ is also. ($I$ is an arbitrary index set-- possibly uncountable.)
\item Use $(i)$ to show that if we are given a set $\Omega$ and a collection $\mathcal A$ of subsets of $\Omega$, then there is a smallest $\sigma-$field containing $\mathcal A$.  This is called the $\sigma$-field generated by $\mathcal A$ and we denote it by $\sigma({\mathcal A})$.
\end{enumerate}


\item Exercise 1.1.2 from text.  Let $\Omega = \R$, and let $\FF$ be all subsets of $\R$ such that either $A$ or $A^c$ is countable.  Define $\Pr(A) = 0$ if $A$ is countable and $\Pr(A) = 1$ if uncountable.  Show that $(\Omega,\FF,\Pr)$ is a probability space.

\item Exercise 1.2.1 from text.  Suppose $X$ and $Y$ are random variables on $(\Omega,\FF,\Pr)$ and let $A \in \FF$.  Show that if we let $Z(\omega) = X(\omega)$ for $\omega \in A$ and $Z(\omega) = Y(\omega)$ for $\omega \in A^c$, then $Z$ is a random variable.


\item Exercise 1.2.5 from text.  Suppose $X$ has a continuous density $f$, $P(\alpha \le X\le \beta) = 1$ and $g$ is a function that is strictly increasing and differentiable on $(\alpha,\beta)$.  Then $g(X)$ has density 
\[
	\frac{f(g^{-1}(y))}{g'(g^{-1}(y))}
\]
for $y \in (g(\alpha),g(\beta))$ and 0 otherwise.  When $g(x) = ax + b$ with $a>0$, $g^{-1}(y) = (y-b)/a$, so the answer is $(1/a)f((y-b)/a)$.
\item Exercise 1.3.1.  Show that if $\mathcal A$ generates $\mathcal S$, then $X^{-1}(\mathcal A) \equiv \{\{X \in A\}:A \in \mathcal A\}$ generates $\sigma(X) = \{\{X\in B\}:B\in \mathcal S\}$.


\end{enumerate}



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