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\begin{document}

\centerline {\bf Math 563 - Fall 14 - Homework 1} 

\smallskip

\no 1. Let $E_n$ be a sequence of events. We define a new event :
\bea
\{ \omega : \exists \, \, infinite \, \, I \subset \naturals \, \, such 
\, that \, \, i \in I \Rightarrow \omega \in E_i \}
\eea
This event is sometimes written $E_n \, i. o.$, where $i. o.$ stands for 
``infinitely often.''

\no (a) Show that 
$E_n \, i.o. = \cap_{n=1}^\infty \cup_{k=n}^\infty E_k$

\no (b) Prove that if $\sum_{n=1}^\infty P(E_n) < \infty$, then
$P(E_n \, i.o.) = 0$. This is sometimes called the ``easy half'' of the 
Borel Cantelli lemma. 

\smallskip

\no 2. Let $X$ be a simple random variable on a probability space
$(\Omega,\f,P)$. (This means that the range of $X$ is finite.)
Let $c_1,c_2,\cdots,c_n$ be the values that $X$ 
takes on. Let $p_j = P(X=c_j)$. Let $\mu_X$ be the distribution of $X$.
Give an explicit description of $\mu_X$ in terms of the $c_j$ and $p_j$. 
(This is not a hard problem.)

\smallskip

\no 3. Let $X$ be a real valued function on $\Omega$ 
and let $\sigma(X)$ be the $\sigma$-field generated by the sets
$X^{-1}(B)$ where $B$ is a Borel set in $\reals$. 
(This is the smallest $\sigma$-field with respect to which $X$ is measurable.)
Let $Y$ be a real valued function on $\Omega$. 
Prove that $Y$ is measurable with respect to $\sigma(X)$ if and only if
there is a measurable function $f : \reals \ra \reals$ such that 
$Y=f(X)$. This is problem 2.8 in Durrett. For a hint look at problem
2.7 or 2.9. 

\smallskip

\no 4. We flip a fair coin infinitely many times. Let $X_n$ be $1$ if the 
$n$th flip is heads, and $0$ if the $n$th flip is tails. 
The sample space $\Omega$ consists of all sequences of heads and tails. 
$X_n$ is a real valued function on $\Omega$. In this problem we assume
that there is a $\sigma$-field $\f$ and a probability measure $P$ such 
that $X_n$ is a random variable and the probability measure agrees
with your intiution. (We will eventually prove such an $\f$ and $P$ 
exist.) Define
\bea
X = \sum_{n=1}^\infty {X_n \over 2^n}
\nonumber
\eea
Note that $0 \le X \le 1$. Find the distribution $\mu_X$ of $X$. 
Hint: find $P(X \in E)$ when $E$ is an interval of the form
$((k-1)/2^n,k/2^n)$ for integers $k$ and $n$. 

\smallskip

\no 5. Let $X_n$ be as in the last problem. Now define
\bea
Y = \sum_{n=1}^\infty {2 X_n \over 3^n}
\nonumber
\eea
NB: It is $3^n$ in the denominator, not $2^n$.

(a) Prove that the distribution function $F_Y$ is continuous.

(b) Prove that $F_Y$ is differentiable a.e. with the derivative equal 
to $0$ a.e. Hint: prove that $F_Y$ is constant on the 
complement of the Cantor set.

(c) Let $\mu_Y$ be the distribution of $Y$. Let $m$ be Lebesgue 
measure on the real line. Prove that $\mu_Y$ and $m$ are mutually 
singular. This means that there is a Borel set $A$ 
with $m(A)=0$ and $\mu_Y(A^c)=0$. 

\end{document}
\end



\no 1. Prove the following proposition. (It was stated in class.)

\no {\bf Proposition:} Let $X: \Omega  \ra \Omega^\prime$ where 
$(\Omega,\f)$ and $(\Omega^\prime,\f^\prime)$ are
measureable spaces. 
Suppose that $\e \subset \f^\prime$ and $\e$ generates $\f^\prime$. 
Suppose also that $X^{-1}(B) \in \f$ for all $B \in \e$. Prove that $X$ is 
measurable. 

\smallskip

