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\begin{document}
\begin{center}
{\bf \large Math 702 Problem Set 1} \vs \\
{\bf due Friday 2/14/2014} \vs \vs
\end{center}

\noindent
{\bf Problem 1.} When the relation $\fM_1 \subset \fM_2$ holds between two $\sigma$-algebras on the same set, we say that $\fM_1$ is {\bit smaller} (or {\bit coarser}) than $\fM_2$, and $\fM_2$ is {\bit larger} (or {\bit finer}) than $\fM_1$. \vs

\begin{enumerate}
\item[(i)]
Given a map $f:X \to Y$ and a $\sigma$-algebra $\fM$ on $X$, verify that 
$$
f_{\ast} \fM = \{ E \subset Y: f^{-1}(E) \in \fM \}
$$ 
is the largest $\sigma$-algebra on $Y$ which makes $f$ measurable. \vs

\item[(ii)] 
Given a map $f:X \to Y$ and a $\sigma$-algebra $\fN$ on $Y$, verify that 
$$
f^{\ast} \fN = \{ f^{-1}(E): E \in \fN \}
$$
is the smallest $\sigma$-algebra on $X$ which makes $f$ measurable. \vs

\item[(iii)]
If $f: \RR^2 \to \RR$ is the projection $f(x,y)=x$, describe elements of $f^{\ast}\fB_\RR$ and $f_{\ast} \fB_{\RR^2}$. Here $\fB_\RR$ and $\fB_{\RR^2}$ are the Borel $\sigma$-algebras of $\RR$ and $\RR^2$. \vs 
\end{enumerate}

\noindent
{\bf Problem 2.} 
\begin{enumerate}
\item[(i)]
If $f,g : X \to [-\infty,+\infty]$ are measurable, show that the sets
$$
\{ x \in X: f(x)<g(x) \} \qquad \text{and} \qquad \{ x \in X: f(x)=g(x) \}
$$
are measurable (Caution: Forming $f-g$ can be problematic.) \vs

\item[(ii)]
Suppose $f_n :X \to [-\infty,+\infty]$ is a sequence of measurable functions. Show that the set of points at which $\lim_{n \to \infty} f_n$ exists is measurable. \vs
\end{enumerate} 

\noindent
{\bf Problem 3.} Consider the measurable space $(X,\fM)$ in which $X$ is an uncountable set and $\fM$ is the $\sigma$-algebra of all $E \subset X$ such that either $E$ or $E^c$ is countable. Describe measurable functions $X \to \RR$. \vs

\noindent
{\bf Problem 4.} (Borel-Cantelli Lemma) Let $(X,\fM,\mu)$ be a measure space and $\{ E_n \}$ be a sequence in $\fM$ such that $\sum_n \mu(E_n) < +\infty$. Show that almost every point of $X$ belongs to at most finitely many of the $E_n$. (Hint: Let $A$ be the set of points in $X$ that belong to infinitely many of the $E_n$. Use
$$
A = \bigcap_{k \geq 1} \bigcup_{n \geq k} E_n
$$  
to prove that $\mu(A)=0$.) \vs 

\noindent
{\bf Problem 5.} Recall that $\delta_p$ is the unit mass (=Dirac measure) at $p$. \vs
\begin{enumerate}
\item[(i)]
If $f: X \to Y$ is a map and $p \in X$, what is the push-forward $f_{\ast} \delta_p$? \vs

\item[(ii)]
Let $\TT= \{ z \in \CC: |z|=1 \}$ and $f: \TT \to \TT$ be the $120^\circ$ rotation defined by $f(z)=e^{2\pi i/3} z$. What is the limit of the probability measures 
$$
\frac{1}{n} \sum_{i=0}^{n-1} (f^{\circ i})_\ast \delta_1
$$
as $n \to \infty$? (As usual, $f^{\circ i}$ is the $i$-th iterate of $f$.)\vs 
\end{enumerate} 

\noindent
{\bf Problem 6.} Show that every infinite $\sigma$-algebra has uncountably many elements. (Hint: Suppose $\fM$ is a countably infinite $\sigma$-algebra on $X$, and for $x \in X$ let $A_x \in \fM$ be the intersection of all elements of $\fM$ that contain $x$. Show that $A_x \cap A_y \neq \es$ implies $A_x=A_y$, so every element of $\fM$ is the disjoint union of a collection of $A_x$'s. Use this to construct a bijection between $\fM$ and the set of all subsets of $\NN$, which would give the desired contradiction.) \vs  






\end{document}
