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\def\coursename{Statistics 330b/600b, Math 330b spring 2013}
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\def\Lpsi{\ll^\Psi(\xx,\aa,\mu)}
\def\psinorm#1{\norm{#1}_\Psi}
\begin{document}
\HW{4}{Thursday 14 February}
{\it Please attempt at least the starred problems.}
\starproblem[RR2]
Let $D= \{(x,y)\in \RR^2: x=y\}$.
\ppart
For each pair $A,B$ in~$\bb(\RR)$, show that $D\ne A\times B$.
\ppart
Show that $D\in \bb(\RR)\otimes \bb(\RR)$.
\problem[field]
Suppose $\ee_0$ is a (nonempty) collection of subsets of some set~$\xx$.
For each $n\in\NN$ define $\ee_n$ to be the collection of all subsets of the form~$E_1^c$ or $E_1\cup E_2$ or~$E_1\cap E_2$ with $E_i\in\ee_{n-1}$. Define $\ff:= \BIGCUP_{n\in\NN}\ee_n$.
\ppart
Show that $\ff$ is a field on~$\xx$. (That is, $\emptyset\in\ff$ and $\ff$ is stable under finite unions, finite intersections, and complements.)
\ppart
If $\gg$ is a field on~$\xx$ for which $\gg\supseteq \ee_0$, show that $\gg\supseteq \ff$.
\ppart
If $\ee_0$ is countable show that $\ff$ is countable.
\ppart
A sigma field $\aa$ on~$\xx$ is said to be \newdef{countably generated} if $\aa=\sig(\ee)$ for some countable~$\ee$. Explain why~$\ee$ can always be assumed to be a field.
\starproblem[graph]
Suppose $\aa$ is a sigma-field on a set~$\xx$ and~$\bb$ is a countably generated sigma-field on a set~$\yy$. Suppose also that~$\bb$ separates the points of~$\yy$, in the sense of HW1.2: if $y_1\ne y_2 $ then there exists a set~$B\in\bb$ for which~$y_1\in B$ and $y_2\in B^c$. For each $\aa\bsl\bb$-measurable map~$T$ from~$\xx$ to~$\yy$ show that the set $D=\{(x,y)\in\xx\times\yy: y=Tx\}$ belongs to~$\aa\otimes \bb$. Hint: If~$\bb$ is generated by a countable field~$\ff$, consider the set $\BIGCUP_{F\in \ff}(T^{-1}F^c)\times F$.
\problem[count.gen]
Suppose $\aa$ and $\bb$ are countably generated sigma-fields, on the sets~$\xx$ and $\yy$ respectively. Show that $\aa\otimes \bb$ is countably generated.
\problem[Borel.product]
Let $\xx$ and $\yy$ be topological spaces equipped with their Borel
sigma-fields~$\bb(\xx)$ and~$\bb(\yy)$. Equip
$\xx\times\yy$ with the product topology and its Borel
sigma-field~$\bb(\xx\times\yy)$. (The open sets in the product space are, by definition, all possible unions of
sets $G\times H$, with $G$ open in~$\xx$ and $H$ open in~$\yy$.)
\ppart
Show that
$\bb(\xx)\otimes\bb(\yy)\subseteq\bb(\xx\times\yy)$.
\ppart
A topology~$\gg_1$ is said to be countably generated if there exists a countable~$\gg_2\subseteq \gg_1$ such that $G =\cup\{H\in\gg_2: H\subseteq G\}$ for each $G\in \gg_1$.
If both $\xx$ and $\yy$ have countably generated topologies, prove equality of
the two sigma-fields (from part~(ii)) on the product space.
\ppart
Show that $\bb(\RR^n)=\bb(\RR^k)\otimes\bb(\RR^{n-k})$.
\starproblem[indicator]
Suppose $X$ is a real valued random variable, defined on a set~$\Om$ equipped with a sigma-field~$\ff$. Show that the set
$\{(\om,t)\in \Om\times \RR: X(\om)>t\}$ belongs to~$\ff\otimes \bb(\RR)$.
\problem[diag]
Suppose $\mu$ and $\nu$ are finite measures on~$\bb(\RR)$. Let $f$ be the indicator function of the set~$\{(x,y)\in\RR^2: x=y\}$.
\ppart
Show that the set $A_\nu :=\{x\in\RR: \nu\{x\}>0\}$ contains at most countably many points.
\ppart
Show that $\mu\otimes \nu f(x,y) = \SUM_{x\in A_\nu}\mu\{x\}\nu\{x\}$. Hint: First write $\nu^y f(x,y)$ as a sum of at most countably many terms.
\end{document}