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%  Lecture 1
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\section[Lecture 1 -- Measures and sigma-Fields]{Lecture 1 \textemdash{} Measures and \texorpdfstring{$\sigma$}{sigma}-Fields}
\label{sec:lec01}

The opening question of the course is disarmingly simple: \emph{what is a
measure?} Intuitively, a measure assigns a size \textemdash{} length,
area, volume, probability \textemdash{} to a set. To pin this down, we
have to settle two things up front: which subsets of an ambient space
\(\Omega\) we are even allowed to talk about, and what rules the
size-assignment must obey. The first question is answered by a
\(\sigma\)-field; the second by the definition of a measure. The
relationship of \(\R^n\) to its norm sits in the background as the
running prototype.

\subsection{Notation and set-theoretic preliminaries}

Throughout the course \(\Omega\) denotes a fixed ambient set, the
\emph{sample space}. For \(A\subseteq\Omega\) we write
\[
A^{c} \;=\; \{\,x\in\Omega : x\notin A\,\} \;=\; \Omega\setminus A
\]
for the complement, and use \(\Omega\setminus A\) and \(A^c\)
interchangeably. A countable collection \(\{A_i\}_{i=1}^{\infty}\) of
subsets of \(\Omega\) is \emph{pairwise disjoint} if
\[
A_i\cap A_j \;=\; \emptyset \qquad\text{whenever } i\neq j.
\]

The \emph{power set} of \(\Omega\), denoted \(\Pcal(\Omega)\), is the
collection of all subsets of \(\Omega\). When \(\Omega\) has \(n\)
elements this collection has \(2^n\) elements, which is the source of
the alternative notation \(2^{\Omega}\).

\begin{remark}
Symmetric difference \(A\triangle B=(A\setminus B)\cup(B\setminus A)\)
is the set-theoretic counterpart of the logical \texttt{XOR}. It will
not feature heavily in this lecture, but is worth keeping in the
toolkit.
\end{remark}

\subsection{\texorpdfstring{$\sigma$}{sigma}-fields}

The first object we need is a class of subsets that is closed under the
operations we plan to perform on it: complement, countable union, and
(as a consequence) countable intersection.

\begin{definition}{$\sigma$-field}{sigma-field}
For a set \(\Omega\), a \emph{\(\sigma\)-field} (or
\emph{\(\sigma\)-algebra}) is a collection \(\Fcal\) of subsets
\(A\subseteq\Omega\) such that
\begin{enumerate}
  \item \(\emptyset,\ \Omega\in\Fcal\);
  \item if \(A\in\Fcal\) then \(A^{c}\in\Fcal\);
  \item for any countable collection \(\{A_i\}_{i=1}^{\infty}\) with
        \(A_i\in\Fcal\) for all \(i\in\N\),
        \[
          \bigcup_{i=1}^{\infty} A_i \;\in\; \Fcal.
        \]
\end{enumerate}
\end{definition}

The pair \((\Omega,\Fcal)\) is then called a \emph{measurable space}.

\begin{remark}
Combining (2) and (3) via De Morgan's laws delivers closure under
countable intersection: for \(\{A_i\}_{i=1}^{\infty}\subseteq\Fcal\),
\[
\Bigl(\bigcap_{i=1}^{\infty} A_i\Bigr)^{c}
   \;=\; \bigcup_{i=1}^{\infty} A_i^{c} \;\in\; \Fcal,
\]
and applying (2) once more yields
\(\bigcap_{i=1}^{\infty} A_i\in\Fcal\).
\end{remark}

\begin{remark}
Any \(\sigma\)-field on \(\Omega\) is a subcollection of the power set,
\(\Fcal\subseteq\Pcal(\Omega)\). The power set itself is a
\(\sigma\)-field, but it is typically too large to be useful: on
\(\R\), for example, it is too generous a collection to admit a
translation-invariant length-like measure (a fact made precise in
Lecture~4).
\end{remark}

\subsection{Measures}

With a class of admissible sets in hand, a measure is a rule for
assigning a non-negative size to each.

\begin{definition}{Measure}{measure}
Let \((\Omega,\Fcal)\) be a measurable space. A \emph{measure} is a
function \(\mu:\Fcal\to\R^{+}\) satisfying
\begin{enumerate}
  \item \(\mu(\emptyset)=0\);
  \item \(\mu\) is \emph{countably additive}: for any pairwise disjoint
        countable collection \(\{A_i\}_{i=1}^{\infty}\subseteq\Fcal\),
        \[
          \mu\!\left(\bigcup_{i=1}^{\infty} A_i\right)
            \;=\; \sum_{i=1}^{\infty}\mu(A_i).
        \]
\end{enumerate}
The triple \((\Omega,\Fcal,\mu)\) is then called a \emph{measure space}.
\end{definition}

\begin{remark}
Take care to distinguish a \emph{measurable space}
\((\Omega,\Fcal)\) from a \emph{measure space}
\((\Omega,\Fcal,\mu)\): the former specifies only \emph{which} sets
can be sized, the latter also specifies \emph{how}.
\end{remark}

\begin{remark}
There also exist \emph{signed measures}, which are allowed to take
values in \(\R\) rather than \(\R^{+}\). These will not concern us in
this lecture.
\end{remark}

\subsection{Special classes of measures}

Three size-on-the-whole-space conditions get repeated names.

\begin{definition}{Probability, finite, and $\sigma$-finite measures}{special-measures}
Let \((\Omega,\Fcal,\mu)\) be a measure space.
\begin{itemize}
  \item \(\mu\) is a \emph{probability measure} if \(\mu(\Omega)=1\); we
        then call \((\Omega,\Fcal,\mu)\) a \emph{probability space} and
        usually write \(\mu=\P\).
  \item \(\mu\) is a \emph{finite measure} if \(\mu(\Omega)<\infty\).
  \item \(\mu\) is a \emph{\(\sigma\)-finite measure} if there exists a
        countable cover \(\Omega=\bigcup_{i=1}^{\infty}A_i\) with
        \(A_i\in\Fcal\) and \(\mu(A_i)<\infty\) for every \(i\).
\end{itemize}
\end{definition}

\begin{example}[Length on $\R$]
Take \(\Omega=\R\) and (anticipating Lecture~2) define \(\mu([a,b])=b-a\).
Then \(\mu\) is not finite, but it is \(\sigma\)-finite: writing
\[
\R \;=\; \bigcup_{i=1}^{\infty}\bigl([\,i-1,i\,]\,\cup\,[-i,-i+1]\bigr),
\]
each piece has finite length. This is the prototype of \emph{Lebesgue
measure}.
\end{example}

\subsection{Examples on a finite sample space}

\begin{example}[Counting measure]
Let \(\Omega=\{1,2,\ldots,n\}\) and take \(\Fcal=\Pcal(\Omega)\), which
has \(2^{n}\) elements (hence the notation \(2^{\Omega}\)). The
\emph{counting measure} is
\[
\mu(A) \;=\; \#A, \qquad A\in\Fcal.
\]
Thus \(\mu(\{1,3,7\})=3\) and \(\mu(\Omega)=n\). The normalised version
\[
\nu(A) \;=\; \tfrac{1}{n}\,\mu(A)
\]
is the uniform probability measure on \(\{1,\ldots,n\}\).
\end{example}

\begin{example}[Binomial probability measure]
By contrast, on \(\Omega=\{0,1,\ldots,n\}\) the binomial \((n,p)\)
distribution assigns to each point \(i\) the weight
\[
\mu(\{i\}) \;=\; \binom{n}{i}\,p^{i}\,(1-p)^{\,n-i},
\qquad p\in(0,1),
\]
extended additively to \(\Pcal(\Omega)\). This gives a probability
measure on \(\Omega\) which is \emph{not} uniform.
\end{example}
