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%  Lecture 2
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\section[Lecture 2 -- Constructing sigma-Fields and Measures]{Lecture 2 \textemdash{} Constructing \texorpdfstring{$\sigma$}{sigma}-Fields and Measures (Existence; Carath\'eodory's Extension)}
\label{sec:lec02}

\emph{Motivating question.} Suppose we declare the measure of a half-open
interval to be its length, \(\mu((a,b])=b-a\) for \(b>a\). What
\emph{else} can we then measure? The collection of all such intervals is
not a \(\sigma\)-field --- e.g.\ \((a,b]\cup(c,d]\) is not in general a
half-open interval --- so we need a procedure that grows the class of
``measurable'' sets and extends \(\mu\) to it. The extension is delivered
by \emph{Carath\'eodory's extension theorem}, the existence half of the
construction of a measure space. Uniqueness is taken up next lecture.

\subsection{Semirings, rings, and fields}

We climb a small ladder of set systems sitting below a \(\sigma\)-field.
At each rung the candidate measure has more room to manoeuvre.

\begin{definition}{Semiring}{semiring}
A collection \(\Acal\) of subsets of \(\Omega\) is a \emph{semiring} if
\begin{itemize}
  \item \(\emptyset\in\Acal\),
  \item \(A\cap B\in\Acal\) for all \(A,B\in\Acal\), and
  \item for all \(A,B\in\Acal\) the set difference splits as a finite
        disjoint union
        \[
          B\setminus A \;=\; \bigsqcup_{i=1}^{n}C_i,\qquad C_i\in\Acal.
        \]
\end{itemize}
The set difference itself need not lie in \(\Acal\), but it must be
\emph{expressible} as a finite union of members of \(\Acal\).
\end{definition}

\begin{example}[Half-open intervals form a semiring]
The collection of all half-open intervals \((a,b]\subseteq\R\) (together
with \(\emptyset\)) is a semiring: intersections of half-open intervals
are half-open intervals, and a difference \((a,b]\setminus(c,d]\) is the
union of at most two half-open intervals.
\end{example}

\begin{definition}{Ring}{ring}
A collection \(\Acal\) of subsets of \(\Omega\) is a \emph{ring} if
\begin{itemize}
  \item \(\emptyset\in\Acal\), and
  \item for all \(A,B\in\Acal\), both \(B\setminus A\in\Acal\) and
        \(A\cup B\in\Acal\).
\end{itemize}
A ring is closed under finite (set-theoretic) unions and differences.
\end{definition}

\begin{example}[Finite unions of half-open intervals]
The collection of all \emph{finite} unions of half-open intervals
\((a,b]\subseteq\R\) is a ring. It is not yet a \(\sigma\)-field --- it
fails to absorb countable unions.
\end{example}

\begin{definition}{Field}{field}
A ring \(\Acal\) is a \emph{field} (or \emph{algebra}) on \(\Omega\) if
additionally \(\Omega\in\Acal\).
\end{definition}

\begin{remark}
A field that is closed under \emph{countable} unions is a
\(\sigma\)-field. The progression
\[
\text{semiring}\;\subset\;\text{ring}\;\subset\;\text{field}\;\subset\;\sigma\text{-field}
\]
mirrors a progression in stability under set operations: pairwise
intersection only, then finite unions/differences, then \(\Omega\),
finally countable unions.
\end{remark}

\subsection{Set functions and pre-measures}

Before defining a measure we collect the regularity properties a set
function may enjoy.

\begin{definition}{Set function and its properties}{setfunction}
Let \(\Acal\) be a collection of subsets of \(\Omega\). A \emph{set
function} is any map \(\mu\colon\Acal\to[0,\infty]\) (not necessarily a
measure). For \(A,B\in\Acal\) we say:
\begin{itemize}
  \item \(\mu\) is \emph{increasing} (monotone) if
        \(A\subseteq B\;\Rightarrow\;\mu(A)\le\mu(B)\);
  \item \(\mu\) is \emph{(finitely) additive} if
        \(\mu(A\cup B)=\mu(A)+\mu(B)\) whenever \(A,B\in\Acal\) are
        disjoint and \(A\cup B\in\Acal\);
  \item \(\mu\) is \emph{countably additive} if for every pairwise
        disjoint sequence \(\{A_i\}_{i=1}^{\infty}\subseteq\Acal\) with
        \(\bigcup_i A_i\in\Acal\),
        \[
          \mu\!\left(\bigcup_{i=1}^{\infty}A_i\right)
             \;=\; \sum_{i=1}^{\infty}\mu(A_i);
        \]
  \item \(\mu\) is \emph{countably subadditive} if for every (not
        necessarily disjoint) sequence
        \(\{A_i\}_{i=1}^{\infty}\subseteq\Acal\) with
        \(\bigcup_i A_i\in\Acal\),
        \[
          \mu\!\left(\bigcup_{i=1}^{\infty}A_i\right)
             \;\le\; \sum_{i=1}^{\infty}\mu(A_i).
        \]
\end{itemize}
\end{definition}

\begin{definition}{Pre-measure}{premeasure}
A set function \(\mu\colon\Acal\to[0,\infty]\) on a ring \(\Acal\) is a
\emph{pre-measure} if \(\mu(\emptyset)=0\) and \(\mu\) is countably
additive on \(\Acal\).
\end{definition}

A pre-measure is exactly what a measure looks like \emph{before} the
underlying set system has been closed up to a \(\sigma\)-field.
Carath\'eodory's theorem will perform that closing-up.

\subsection{Outer measure and \texorpdfstring{$\mu^{*}$}{mu*}-measurability}

A pre-measure on a ring can be extended in a canonical way to
\emph{every} subset of \(\Omega\) by approximating from above.

\begin{definition}{Outer measure}{outermeasure}
Let \(\mu\) be a pre-measure on a ring \(\Acal\) on \(\Omega\). The
\emph{outer measure} induced by \(\mu\) is
\[
  \mu^{*}(E) \;=\; \inf\left\{\sum_{i}\mu(A_i)
                     \;:\; A_i\in\Acal,\;
                     E \subseteq \bigcup_{i}A_i\right\},
  \qquad E \subseteq \Omega,
\]
where the infimum runs over finite or countable covers of \(E\) by
elements of \(\Acal\).
\end{definition}

The outer measure \(\mu^{*}\) is defined on \emph{all} of
\(\Pcal(\Omega)\), but in general it is only countably
\emph{sub}additive there --- not additive. To recover countable
additivity we restrict attention to sets that ``split'' every test set
cleanly.

\begin{definition}{$\mu^{*}$-measurable set}{caratheodory-measurable}
A set \(B\subseteq\Omega\) is \emph{\(\mu^{*}\)-measurable} (in the
sense of Carath\'eodory) if for every \(E\subseteq\Omega\),
\[
  \mu^{*}(E\cap B) \;+\; \mu^{*}(E\cap B^{c})
     \;=\; \mu^{*}(E).
\]
Write \(\Mcal\) for the collection of all \(\mu^{*}\)-measurable
subsets of \(\Omega\).
\end{definition}

\begin{figure}[h]
\centering
\begin{tikzpicture}[>=Stealth, scale=1.0]
  % Omega: outer rectangle
  \draw[thick, deepnavy] (-3.6,-2.0) rectangle (3.6,2.0);
  \node[deepnavy] at (3.25,1.7) {\(\Omega\)};

  % B: a "notched" blob inside Omega (blue)
  \draw[thick, exampleblue]
    plot[smooth cycle, tension=0.9]
    coordinates {(-1.6,1.2) (0.6,1.5) (2.4,0.8)
                 (1.5,-0.2) (2.2,-1.3) (0.0,-1.4)
                 (-0.4,-0.4) (-1.8,-0.6)};
  \node[exampleblue] at (1.6,0.4) {\(B\)};
  \node[exampleblue] at (-3.0,-1.6) {\(B^{c}\)};

  % E: a red blob crossing into and out of B
  \draw[thick, highlightred, fill=highlightred, fill opacity=0.18]
    plot[smooth cycle, tension=0.9]
    coordinates {(-2.4,1.4) (-0.2,1.3) (0.9,0.4)
                 (-0.4,-0.2) (-2.2,0.0)};
  \node[highlightred] at (-1.6,0.7) {\(E\)};
\end{tikzpicture}
\caption{The Carath\'eodory criterion. A set \(B\subseteq\Omega\) is
\(\mu^{*}\)-measurable when every test set \(E\) (red) is split
additively by \(B\) and its complement: \(\mu^{*}(E)=\mu^{*}(E\cap
B)+\mu^{*}(E\cap B^{c})\).}
\label{fig:caratheodory-split}
\end{figure}

\begin{remark}
Countable subadditivity of \(\mu^{*}\) already gives ``\(\ge\)'' in the
defining identity, so the substantive condition is the reverse
inequality: \(B\) does not waste mass at its boundary. The class
\(\Mcal\) is the largest natural domain on which \(\mu^{*}\) is
genuinely additive.
\end{remark}

\subsection{Carath\'eodory's extension theorem}

We now assemble the pieces. Starting from a pre-measure on a ring, the
outer measure machinery delivers a measure on a \(\sigma\)-field
containing the original ring.

\begin{theorem}{Carath\'eodory Extension}{caratheodory}
Let \(\Acal\) be a ring on \(\Omega\) and let \(\mu\) be a pre-measure
on \(\Acal\). Let \(\mu^{*}\) be the outer measure induced by \(\mu\)
and \(\Mcal\) the collection of \(\mu^{*}\)-measurable sets. Then:
\begin{enumerate}
  \item \(\mu^{*}(\emptyset)=0\), and \(\mu^{*}\) is monotone and
        countably subadditive on \(\Pcal(\Omega)\);
  \item \(\mu^{*}\) and \(\mu\) agree on \(\Acal\), i.e.\
        \(\mu^{*}(A)=\mu(A)\) for every \(A\in\Acal\);
  \item \(\Acal\subseteq\Mcal\);
  \item \(\Mcal\) is a \(\sigma\)-field on \(\Omega\) and \(\mu^{*}\)
        restricted to \(\Mcal\) is a measure;
  \item consequently
        \[
          \Acal \;\subseteq\; \sigma(\Acal)
                \;\subseteq\; \Mcal
                \;\subseteq\; \Pcal(\Omega),
        \]
        and \(\mu^{*}\big|_{\sigma(\Acal)}\) is a measure on
        \(\sigma(\Acal)\) extending \(\mu\).
\end{enumerate}
\end{theorem}

\begin{remark}
The chain in (5) records the precise sense in which Carath\'eodory
``extends'' \(\mu\): the original ring \(\Acal\) sits inside the
generated \(\sigma\)-field \(\sigma(\Acal)\), which sits inside the
larger \(\sigma\)-field \(\Mcal\) of \(\mu^{*}\)-measurable sets,
which sits inside the full power set. Both \(\sigma(\Acal)\) and
\(\Mcal\) carry the measure \(\mu^{*}\); the ``correct'' extension is
the outer measure restricted to \(\sigma(\Acal)\). It can happen that
\(\sigma(\Acal)\) is a strict subset of \(\Mcal\) (this is the gap
filled by completion).
\end{remark}

\begin{remark}
We have shown \emph{existence}: at least one measure on
\(\sigma(\Acal)\) agreeing with \(\mu\) on \(\Acal\) exists. The
companion question --- \emph{is the extension unique?} --- is
\[
  \mu_1(A)=\mu_2(A) \;\forall A\in\Acal
  \;\;\overset{?}{\Longrightarrow}\;\;
  \mu_1(B)=\mu_2(B) \;\forall B\in\sigma(\Acal),
\]
and is the subject of the next lecture, via Dynkin's
\(\pi\)-\(\lambda\) theorem.
\end{remark}