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%  Lecture 1
%  Source: handwritten notes (Mathpix-converted) + Kashlak STAT 571
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\section[Lecture 1 -- Measures and \texorpdfstring{$\sigma$}{sigma}-Fields]{Lecture 1 \textemdash{} Measures and \texorpdfstring{$\sigma$}{sigma}-Fields}
\label{sec:lec01}

We innately understand the concept of a \emph{measure} through lengths, areas
and volumes: a measure assigns a non-negative value to a set. To make this
precise we must say (i) which sets we are allowed to measure, and (ii) what
rules the measuring map must satisfy. The first question is answered by a
\(\sigma\)-field; the second by the definition of a measure.

\medskip

\textbf{Notation.} Throughout, \(\Omega\) is a fixed ambient set (the
\emph{sample space}). For \(A\subseteq\Omega\), the complement is
\(A^c=\{x\in\Omega : x\notin A\}=\Omega\setminus A\). A collection
\(\{A_i\}_{i=1}^{\infty}\) of subsets is \emph{pairwise disjoint} if
\(A_i\cap A_j=\emptyset\) whenever \(i\neq j\). The \emph{power set}
\(\Pcal(\Omega)\) is the collection of all subsets of \(\Omega\).

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

A \(\sigma\)-field is the answer to the question: \emph{which subsets of
\(\Omega\) am I allowed to measure?}

\begin{definition}{$\sigma$-Field}{sigma-field}
For a set \(\Omega\), a \emph{\(\sigma\)-field} (or \emph{\(\sigma\)-algebra})
\(\Fcal\) is a collection 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 every \(i\), one has
        \(\displaystyle\bigcup_{i=1}^{\infty}A_i\in\Fcal\).
\end{enumerate}
\end{definition}

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

\begin{remark}
Combining (2) and (3) with De Morgan's laws gives closure under countable
intersections: if \(\{A_i\}_{i=1}^\infty\subseteq\Fcal\) then
\[
\Bigl(\bigcap_{i=1}^{\infty} A_i\Bigr)^c
   = \bigcup_{i=1}^{\infty} A_i^c
\quad\Longrightarrow\quad
\bigcap_{i=1}^{\infty} A_i\in\Fcal.
\]
\end{remark}

\begin{remark}
Always \(\Fcal\subseteq\Pcal(\Omega)\). The power set is itself a
\(\sigma\)-field, but it is typically too large to be useful (e.g.\ on
\(\R\) it contains non-measurable sets).
\end{remark}

\subsection{Measures}

Once we have a \(\sigma\)-field of admissible sets, a measure is the
assignment of size to each.

\begin{definition}{Measure}{measure}
For a measurable space \((\Omega,\Fcal)\), a \emph{measure} is a function
\(\mu:\Fcal\to\R^{+}\) such that
\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 a \emph{measure space}.
\end{definition}

\begin{remark}
Be careful not to confuse a \emph{measurable space} \((\Omega,\Fcal)\) with
a \emph{measure space} \((\Omega,\Fcal,\mu)\); the former specifies only
which sets can be measured, while the latter also specifies how.
\end{remark}

\begin{definition}{Special classes of measures}{special-measures}
Let \((\Omega,\Fcal,\mu)\) be a measure space. We say that
\begin{itemize}
  \item \(\mu\) is a \emph{probability measure} if \(\mu(\Omega)=1\); in
        this case \((\Omega,\Fcal,\mu)\) is a \emph{probability space} and
        \(\mu\) is usually written \(\P\);
  \item \(\mu\) is a \emph{finite measure} if \(\mu(\Omega)<\infty\);
  \item \(\mu\) is a \emph{\(\sigma\)-finite measure} if there exist
        \(\{A_i\}_{i=1}^{\infty}\subseteq\Fcal\) with
        \(\Omega=\bigcup_{i=1}^{\infty} A_i\) and \(\mu(A_i)<\infty\) for
        every \(i\).
\end{itemize}
\end{definition}

\subsection{Examples}

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

\begin{example}[Counting measure]
Let \(\Omega=\{1,2,\dots,n\}\) and take \(\Fcal=\Pcal(\Omega)\), sometimes
written \(2^{\Omega}\) since it has \(2^n\) elements. The
\emph{counting measure} is
\[
\mu(A) \;=\; \#A, \qquad A\in\Fcal.
\]
Then \(\mu(\{1,3,7\})=3\) and \(\mu(\Omega)=n\). Normalising,
\(\nu(A)=\tfrac{1}{n}\mu(A)\) is the uniform probability measure on
\(\{1,\dots,n\}\).
\end{example}

\begin{example}[Discrete probability measures]
Instead of weighting each integer by \(1/n\), one can assign each
\(i\in\{0,1,\dots,n\}\) the binomial weight
\(\binom{n}{i}p^{i}(1-p)^{n-i}\) for \(p\in(0,1)\); this defines the
binomial probability measure. Letting \(n\to\infty\) with the appropriate
scaling yields the Poisson probabilities \(e^{-\lambda}\lambda^{i}/i!\)
for \(\lambda>0\).
\end{example}

\begin{remark}
There also exist \emph{signed measures}, taking values in \(\R\) rather
than \(\R^{+}\); these will not concern us until later.
\end{remark}