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\begin{document}

\centerline {\bf Math 563 - Fall 15 - Homework 2} 

\smallskip

\no 1. This problem is an introduction to the Kolmogorov extension
theorem which we will prove later. 
Let $\Omega$ be the set of real valued sequences. So an element of $\Omega$ 
looks like $\omega=(\omega_1,\omega_2, \omega_3, \cdots)$ where the 
$\omega_i$ are real numbers. A subset $E$ is called a finite 
dimensional rectangle if there is a positive integer $n$ and 
$-\infty \le a_i \le b_i \le \infty$ for $i=1,2,\cdots,n$ such that \\  
$E=\{ \omega: \omega_i \in (a_i,b_i] \, for \, i=1,2,\cdots,n\}$. 

\no (a) Show that the collection of finite dimensional
rectangles is a semi-algebra. (not hard)

\no (b) For $n=1,2,3,\cdots$, let $\mu_n$ be a 
probability measure on the Borel sets in $\reals^n$. 
We say they are {\it consistent} if for all $n$ and 
$-\infty \le a_i<b_i \le \infty$ for $i=1,2,\cdots,n$, we have
\bea
\mu_{n+1}((a_1,b_1] \times (a_2,b_2] \times \cdots \times (a_n,b_n] 
\times \reals) = 
\mu_n((a_1,b_1] \times (a_2,b_2] \times \cdots \times (a_n,b_n])
\eea
Let $\field$ be the $\sigma$-field in $\Omega$ 
generated by the finite dimensional rectangles. 
The Kolmogorov extension theorem says there is a unique probability
measure $P$ on $(\Omega,\field)$ such that 
\bea
P(\{ \omega: \omega_i \in (a_i,b_i] \, for \, i=1,2,\cdots,n\})
=\mu_n((a_1,b_1] \times (a_2,b_2] \times \cdots \times (a_n,b_n])
\eea
Prove the uniqueness part of the theorem, i.e., if two probability 
measures on $(\Omega,\field)$ both satisfy the above then they 
are equal on $\field$. 

\smallskip

\no 2. (from Resnick) 
A {\it countable partition} of $\Omega$ is a countable disjoint 
collection of sets whose union is all of $\Omega$. 
Let $\{E_n\}_{n=1}^\infty$ be such a partition and let $\field$ be the 
$\sigma$-field generated by the collection sets 
$\{E_n\}_{n=1}^\infty$. 

\no (a) Given an explicit description of the sets in $\field$. 

\no (b) Let $X : \Omega \ra \reals$. Prove that $X$ is measurable 
with respect to $\field$ if and only if there are constants $c_n$ 
such that 
\bea
X = \sum_{n=1}^\infty c_n 1_{E_n}
\eea


\smallskip

\no {\bf Notation comment:} If $X$ is a random variable, then 
the distribution of $X$ is a probability measure on the 
Borel sets in $\reals$ which I will denote $\mu_X$. It is 
defined by $\mu_X(B)= P(X \in B)$ where $B$ is a Borel set. 
Resnick denotes it by $P \circ X^{-1}$. 

\smallskip

\no 3. Let $X$ be a simple random variable on a probability space
$(\Omega,\field,P)$. (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, and your answer can be quite short.)

\smallskip

\no 4. (from Durrett) 
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 Borel measurable function $f : \reals \ra \reals$ such that 
$Y=f(X)$. 

Hint: First reduce to the case that $Y \ge 0$. 
Show that \\
$\{\omega: m 2^{-n} \le Y(\omega) < (m+1)2^{-n} \}=X^{-1}(B_{n,m})$,
where $B_{n,m}$ is a Borel set in $\reals$. 
Now let $f_n(x)=m2^{-n}$ for $x \in B_{n,m}$. Show that 
$f_n(x)$ converges pointwise and let $f(x)$ be the limit. 


\smallskip

\no 5. 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 $\field$ 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 $\field$ 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 6. {\bf (Optional: I will not grade this one.)}
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}
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