% scribe: Sivakumar Rathinam
% lastupdate: Oct. 2, 2005
% lecture: 1
% title: Ideas from measure theory
% references: Durrett, sections 1.1 and 1.2
% keywords: identification lemma, sigma-field, field, probability space, measure theory, Dynkin's pi-lambda lemma, probability measure, 
% end

\documentclass[12pt,letterpaper]{article}

\include{macros}

\begin{document}

\lecture{1}{Ideas from measure theory}{Sivakumar Rathinam}{rsiva@berkeley.edu}

\section{Probability spaces}
% keywords: identification lemma, sigma-field, field, probability space, measure theory, Dynkin's pi-lambda lemma, probability measure
% end


This lecture introduces some ideas from measure theory which are the foundation of
the modern theory of probability.  The notion of a probability space is defined,
and Dynkin's form of the monotone class theorem is presented.

\begin{definition}
Let $\Omega$ be a set of points $\omega$. In probability theory,
$\Omega$ represents all possible outcomes of an experiment or
observation.
\end{definition}

\begin{example}
Tossing a coin has a set of outcomes $\Omega = \{Head,Tail\}$.
\end{example}

\begin{example}
Position of a body in a 3-D Euclidean space belongs to the set
$\Omega = {R}^3$.
\end{example}

A subset of $\Omega$ is called an event. It is natural to ask
questions like whether an outcome of a random experiment belongs to
to a event or not. To do this, we need to define classes of
subsets of the space $\Omega$. Also, since we would be talking
about any combination of events, a systematic treatment would
require the class of sets to have the necessary set theoretic
operations: namely the sets being closed under countable unions
and intersections. The next few definitions would be in this
regard. Once we define the classes of sets we are interested in,
one can assign a probability measure to each of these sets.

\begin{definition}
A class $\mathcal{F}$ of subsets of a space $\Omega$ is called a \emph{field}
if it contains $\Omega$ itself and is closed under
complements and finite unions. That is
\begin{enumerate}
    \item $\Omega \in \mathcal{F}$
    \item $A \in \mathcal{F}$ implies $A^c \in \mathcal{F}$
    \item $A,B \in \mathcal{F}$ implies $A\cup B \in \mathcal{F}$
\end{enumerate}
\end{definition}

Note that by DeMorgan's law, given that $\mathcal{F}$ is closed under
complement, $\mathcal{F}$ is closed under unions if and only if
$\mathcal{F}$ is closed under intersections. Therefore, $A,B \in
\mathcal{F}$ implies $A\cup B \in \mathcal{F}$ in the above
definition can be replaced with $A,B \in \mathcal{F}$ implies
$A\cap B \in \mathcal{F}$.

\begin{definition}
A class $\mathcal{F}$ of subsets of $\Omega$ is a \emph{$\sigma$-field} if
it is a field and if it is closed under the formation of countable
unions. That is,
\begin{enumerate}
    \item $\mathcal{F}$ is a field.
    \item $A_1,A_2,... \in \mathcal{F}$ implies $A_1 \cup A_2
    \cup .... \in \mathcal{F}$.
\end{enumerate}
\end{definition}

A field is closed under finite set theoretic operations whereas a
$\sigma$-field is closed under countable set theoretic operations.
Usually in a problem dealing with probabilities, one is
dealing with a small class of subsets $\mathcal{A}$, for example
the class of subintervals of $(0,1]$. It is possible that when we perform
countable operations on such a class $\mathcal{A}$ of sets, we might end up
operating on sets outside the class $\mathcal{A}$. Hence, we would
like to define a class denoted by $\sigma(\mathcal{A})$. This class
is called the $\sigma$-field generated by $\mathcal{A}$. It is
defined as the intersection of all the $\sigma$-fields
containing $\mathcal{A}$ (or the smallest $\sigma$-field containing
$\mathcal{A}$). Now, we give the definition of the probability
measure.

\begin{definition}
 A set function\footnote{A set function is a real-valued function defined on some class
of subsets of $\Omega$.} $\P$ on a $\sigma$-field $\mathcal{F}$ is a
\emph{probability measure} if it satisfies the following conditions:
\begin{enumerate}
    \item $0 \leq \P(A) \leq 1$ for $A \in \mathcal{F}$.
    \item $\P(\emptyset) = 0, \P(\Omega) =1$.
    \item If $A_i \in \mathcal{F}$ is a countable union of sets, then
    $\bigcup_i A_i \in \mathcal{F}$.
\end{enumerate}
\end{definition}

If $\mathcal{F}$ is a $\sigma$-field, then the triple
$(\Omega,\mathcal{F},\P)$ is called a probability measure space or
simply a \emph{probability space}. The countable additivity of the
probability measure gives rise to the following properties that
are stated in a theorem.

\begin{theorem}\label{count}
Let $\P$ be a probability measure on a field $\mathcal{F}$.
\begin{enumerate}
    \item Continuity from below: If $A_n$ and $A$ lie in
    $\mathcal{F}$ and $A_n\uparrow A$, then $\P(A_n) \uparrow \P(A)$.
    \item Continuity from above: If $A_n$ and $A$ lie in
    $\mathcal{F}$ and $A_n\downarrow A$, then $\P(A_n) \downarrow \P(A)$.
    \item Countable subadditivity: If $A_1, A_2...$ and
    $\bigcup_{k=1}^{\infty} A_k$ lie in $\mathcal{F}$, then
    \begin{equation}
        \P\left(\bigcup_{k=1}^{\infty} A_k\right) \leq \sum_{k=1}^{\infty}\P(A_k).
    \end{equation}
\end{enumerate}
\end{theorem}


\begin{example}
If $\mathcal{A}$ is the class of subintervals of $\Omega = (0,1)$, then
 the sigma field generated by $\mathcal{A}$, denoted by $\mathcal{B}$, is
called the collection of 
\emph{Borel} sets of the unit interval. The probability space on
a unit interval is then defined as $(\Omega,\mathcal{F},\P)$, where
$\Omega = (0,1)$, $\mathcal{F} = \{A \cap (0,1): A \in
\mathcal{B}\}$ and $\P(B) = \lambda(B)$ for $B\in \mathcal{F}$. Here $\lambda$
is the Lebesgue measure $\lambda((a,b]) = b-a$ for $a<b$.
\end{example}

Now we will present a key result that would help us to extend the
results we have on a field $\mathcal{A}$ to the $\sigma$-field
generated by $\mathcal{A}$. This is stated as the
\emph{Identitification Lemma for Probabilities}:

\begin{lemma}[Identification Lemma for Probabilities]
\label{iden}
Let $P$ and $Q$ be two probability measures on
$\sigma(\mathcal{A})$ where $\mathcal{A}$ is closed under intersections.
If $P(A) = Q(A)$ for $A \in \mathcal{A}$, then $P(A)=Q(A)$ for all
$A \in \sigma(\mathcal{A})$.
\end{lemma}

Before we prove this lemma, let's state some basic tools from
measure theory that are very useful.

\begin{definition}
A collection of subsets $\mathcal{D}$ of set $\Omega$ is called a
\emph{$\lambda$-system} if 
\begin{enumerate}
    \item $\Omega \in \mathcal{D}$
    \item If $A \in B$ and $B \in \mathcal{D}$, $A \subset B \Rightarrow B-A
    \in \mathcal{D}$
    \item If $A_n \in \mathcal{D}$ and $A_n \uparrow A \Rightarrow A \in
    \mathcal{D}$
\end{enumerate}
\end{definition}

\begin{theorem}[Dynkin's $\pi$-$\lambda$ Theorem]
Suppose $\mathcal{A}$ is a collection of sets closed under
$\cap$ (a \emph{$\pi$-system}). If $\mathcal{D}$ is a $\lambda$-system
with $\mathcal{A} \subset \mathcal{D}$, then $\sigma(\mathcal{A})
\subset \mathcal{D}$.
\end{theorem}

 With this theorem let's try to prove the identification lemma on
 probabilities. \newpage

\begin{proof} (of Lemma \ref{iden})
Consider $\mathcal{D} = \{ A \in \sigma(\mathcal{A}): P(A)=Q(A)\}$.
Let's check that $\mathcal{D}$ is a $\lambda$-system.
\begin{enumerate}
    \item $\Omega \in \mathcal{D}$ because $P(\Omega)=Q(\Omega)=1$.
    \item Let $A \in D , B \in \mathcal{D}$. Then $A \subset B$ implies $B-A \in
    \mathcal{D}$. This is because $P(B)=Q(B) \Rightarrow P(B-A) + P(A) =
    Q(B-A)+ Q(A) \Rightarrow P(B-A)=Q(B-A)$.
    \item $A_n \in \mathcal{D}$ and $A_n \uparrow A$ implies $A \in \mathcal{D}$. This is
    because $P(A_n) = Q(A_n) \Rrightarrow P(A)=Q(A)$ using theorem
    \ref{count}.
\end{enumerate}

Now directly applying Dynkins $\pi-\lambda$ Theorem, we get
$\mathcal{A} \subset \mathcal{D}$ implies $\sigma(\mathcal{A})
\subset \mathcal{D}$.
\end{proof}

\end{document}
