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\begin{document}

\centerline {\bf Math 563 - Fall 15 - Homework 1} 

\smallskip

\no 1. Let $E_n$ be a sequence of events. Let 
\bea
A=\limsup_{n \ra \infty} E_n =  \cap_{n=1}^\infty \cup_{k=n}^\infty E_k \\
B=\liminf_{n \ra \infty} E_n =  \cup_{n=1}^\infty \cap_{k=n}^\infty E_k 
\eea
Prove that 
\bea
1_A = \limsup_{n \ra \infty} 1_{E_n} \\
1_B = \liminf_{n \ra \infty} 1_{E_n} 
\eea

\medskip

\no 2. Let $E_n$ be a sequence of events. 
Recall that  $\limsup_{n \ra \infty} E_n$ is defined
to be $\cap_{n=1}^\infty \cup_{k=n}^\infty E_k$. 
This event can be thought of as the event that $E_n$ occurs 
infinitely often. 
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. 

\medskip

\no 3. Let $E_n$ be a sequence of events with $P(E_n)=1$ for all $n$. 
Prove that 
\bea
P(\cap_{n=1}^\infty E_n)=1
\eea

\medskip

\no 4. (from Resnick) Let $\field$ be a $\sigma$-field in $\Omega$. 
Suppose $Q$ is a function from $\field$ into $[0,1]$ such that 
$Q(\Omega)=1$, $Q$ is finitely additive on $\field$, and 
$Q$ has the following property. 
If $E_n \in \field$ are pairwise disjoint and their union is $\Omega$, 
then $\sum_{n=1}^\infty Q(E_n)=1$.  
Prove that $Q$ is a probability measure. 

\medskip

\no 5. (from Resnick)
Let $P_1,P_2$ be two probability measures on $(\Omega,\field)$.
You might think that if they agree on a collection $\C$ of events, and 
the $\sigma$-field generated by $\C$ is $\field$, then the two probability
measures agree on all of $\field$. This is not true. Show that the 
following gives a counterexample. 

$\Omega=\{a,b,c,d\}$ and $\C=\{ \{a,b\}, \{c,d\},  \{a,c\}, \{b,d\} \}$
\bea
P_1(\{a\}) = P_1(\{d\}) = P_2(\{b\}) = P_2(\{c\}) = \frac{1}{6}, \\
P_1(\{b\}) = P_1(\{c\}) = P_2(\{a\}) = P_2(\{d\}) = \frac{1}{3}
\eea

\medskip

\no 6. (from Resnick)
Let $P$ be a probability measure on $\borel(\reals)$, the Borel sets
in $\reals$. Prove that for any $E \in \borel(\reals)$ and any $\epsilon>0$
there exits a finite union of disjoint intervals $A$ 
such that $P(E \Delta A)<\epsilon$. 
 
Hint: Define $\field$ to be the collection of Borel sets such that for
all $\epsilon>0$ there exits a finite union of disjoint intervals $A$ 
such that $P(E \Delta A)<\epsilon$.  What can you prove about $\field$?








\end{document}
\end


\documentclass[12pt]{article}
\usepackage{graphicx}% Include figure files
\usepackage{amssymb}
\usepackage{amsfonts}
%\usepackage{amsmath}
\usepackage{amsthm}
\usepackage{amsfonts}

\newcommand{\ba}[1]{\begin{array}{#1}}
\newcommand{\ea}{\end{array}}
\newcommand{\arr}[2]{\ba{#1} #2 \ea}
\newcommand{\be}{\begin{equation}}
\newcommand{\ee}{\end{equation}}
\newcommand{\bea}{\begin{eqnarray*}}
\newcommand{\eea}{\end{eqnarray*}}
\newcommand {\no}  {\noindent}
\newcommand{\borel}{\mathcal{B}}
\newcommand{\f}{\mathcal{F}}
\newcommand{\e}{\mathcal{E}}
\newcommand{\ra}{\rightarrow}
\newcommand{\reals}{\mathbb{R}}
\newcommand{\naturals}{\mathbb{N}}

\begin{document}

\centerline {\bf Math 563 - Fall 15 - Homework 1} 

\smallskip

\no 1. Let $E_n$ be a sequence of events. Let 
\bea
A=\limsup_{n \ra \infty} E_n =  \cap_{n=1}^\infty \cup_{k=n}^\infty E_k \\
B=\liminf_{n \ra \infty} E_n =  \cup_{n=1}^\infty \cap_{k=n}^\infty E_k 
\eea
Prove that 
\bea
1_A = \limsup_{n \ra \infty} 1_{E_n} \\
1_B = \liminf_{n \ra \infty} 1_{E_n} 
\eea

\smallskip

\no 2. Let $E_n$ be a sequence of events. 
Recall that  $\limsup_{n \ra \infty} E_n$ is defined
to be $\cap_{n=1}^\infty \cup_{k=n}^\infty E_k$. 
This event can be thought of as the event that $E_n$ occurs 
infinitely often. 
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 3. Let $E_n$ be a sequence of events with $P(E_n)=1$ for all $n$. 
Prove that 
\bea
P(\cap_{n=1}^\infty E_n)=1
\eea

\smallskip

\no 4. Let $P$ be a probability measure on $\borel(\real)$, the Borel sets
in $\reals$. Prove that for any $E \in \borel(\real)$ and any $\epsilon>0$
there exits a finite union of disjoint intervals $A$ 
such that $P(E \Delta A)<\epsilon$. 
 
Hint: Define $\field$ to be the collection of Borel sets such that for
all $\epsilon>0$ there exits a finite union of disjoint intervals $A$ 
such that $P(E \Delta A)<\epsilon$.  What can you prove about $\field$?

\smallskip

\no 5. (from Resnick) Let $\field$ be a $\sigma$-field in $\Omega$. 
Suppose $Q$ is a function from $\field$ into $[0,1]$ such that 
$Q(\Omega)=1$, $Q$ is finitely additive on $\field$, and 
$Q$ has the following property. 
If $E_n \in \field$ are pairwise disjoint and their union is $\Omega$, 
then $\sum_{n=1}^\infty Q(E_n)=1$.  
Prove that $Q$ is a probability measure. 

\smallskip

\no 6. Let $P_1,P_2$ be two probability measures on $(\Omega,\field)$.
You might think that if they agree on a collection $\C$ of events and 
the $\sigma$-field generated by $\C$ is $\field$, then the two probability
measures agree on all of $\field$. This is not true. Show that the 
following gives a counterexample. 




\end{document}
\end


