%% ================================================================
\section{Introduction}

Objectives for the paper:

\begin{itemize}
\item Make some brief mention of the history.
\item Focus mainly on a survey of modern treatments, including terminology
	and order of presentation.
\end{itemize}

Objectives for the construction:

\begin{itemize}
\item Compatibility with Riemann integral --- e.g. FTC, area, linearity
\item Monotone convergence/countable additivity
\item Ability to integrate functions the Riemann integral cannot;
extending the concept of measure beyond integration.
% sums? probability?  what else?
\end{itemize}

Constructions:

\begin{itemize}
\item Riemann and Riemann-Stieltjes as background
\item Original Lebesgue construction?  He is quoted as saying
	he ``partitioned the range rather than the domain'',
	although the Daniell construction as presented by Faris
	does not work this way.
\item Carath\'eodory
\item Daniell
\item Other?
\end{itemize}

Preconditions:
\begin{itemize}
\item Topological notions such as compactness of domain
\item Under what conditions do we require the domain $X$ to be merely a set?
	A topological space?  A metric space?
\end{itemize}

My notes:
\begin{itemize}
\item Explain the reason for the terms \emph{ring} and \emph{algebra}.
	These are suggestive of the terms from abstract algebra.  Why were these
	words chosen?
\item Ring and algebra for sets as well as functions.
\end{itemize}

Faris notes:

\begin{itemize}

\item Given a pre-measure on a ring, or a pre-integral on a vector lattice, how
you get a measure or an integral?

\item How to show that an additive set function on a ring, or a linear function
on a vector lattice, actually satisfies monotone convergence or countable
additivity.  Can it be done without compactness or some other topological
notion?

\item Catalog the terminologies used throughout the literature.  What sources?
	\begin{itemize}
	\item Rudin
	\item Royden
	\item Folland
	\item Dudley
	\item Who else?
	\end{itemize}
	Have one pair of sections for the basics of the two constructions.  Then,
	\emph{separately}, catalog the terminologies.

\item Always describe \emph{reasons} for each definition and precondition:
	\emph{why} do we define it this way?
\end{itemize}

%% ================================================================
\section{Measures-first approach}

\begin{itemize}
\item $\mu(0) = \emptyset$
\item $E_i \cap E_j = \emptyset \implies \mu(\cup E_i) = \sum \mu(E_i)$.
\item xlat'n invar't
\item $\mu(Q)=1$.
\end{itemize}

The last two are geometric; the first two are properties for general measures.

Monotonicity is a consequence: $E \subseteq F \implies \mu(E) \le \mu(F)$.

Subadditivity is a consequence: $\mu(\cup E_i) \le \sum \mu(E_i)$.

Set MCT, set DCT

To \emph{construct}:

Def outer measure.

Then, prove properties are satisfied.

algebra to $\sigma$-algebra:  def pre-measure:  same first two props,
but only on an algebra.

$\mu(g-f)=\mu(g)-\mu(f)$: by linearity.
Order preservation:
$g\ge f \implies \mu(g) \ge \mu(f)$, i.e.
$g-f \ge 0 \implies \mu(g-f) \ge 0$: orientation?

%% ================================================================
\section{Functions-first approach}

We start with a pre-integral on a Stone vector lattice, then extend this to an
integral on a $\sigma$-algebra.  Definitions of these terms are as follows.
(Please see appendix \ref{appx:notation} for notation such as $f \wedge g$.)

%% - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
\begin{defn}
Let $X$ be a (set?  topological space?  metric space?).
Let $L$ be a set of functions from $X$ to $\R$.
Then $L$ is a \bfidx{Stone vector lattice} if the following conditions
are satisfied:

\begin{itemize}
\item Vector space:  If $f,g\in L$ and $a \in \R$, then
	$f+g, af \in L$.
\item Lattice:  If $f,g\in L$ then $f \wedge g, f \vee g \in L$.
\item Stone property:  If $f \in L$ then $f \wedge 1 \in L$.
	(Thus $L$ has functions which are \bfidx{locally constant},
	without needing to have constant functions.)
\end{itemize}
\end{defn}

\begin{ex}
Take $L$ to be the set of compactly supported continuous functions from $\R$ to
$\R$.
\end{ex}

%% - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
\begin{defn}
Let $\mu: L \to \R$.  Then $\mu$ is a \bfidx{pre-integral} if the following
conditions are satisfied:

\begin{itemize}
\item Linear:  If $f,g\in L$ and $a \in \R$ then $\mu(f+g) = \mu(f)+\mu(g)$
	and $\mu(af) = a\mu(f)$.
\item Order-preserving:  If $f < g$ then $\mu(f) < \mu(g)$.
\item Monotone convergence:  If $f_n \ge 0$ and $f_n \uparrow f$ then $\mu(f_n)
	\upto \mu(f)$.
\end{itemize}

\end{defn}

%\begin{ex}
%The Riemann integral?
%\end{ex}

%% - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
\begin{defn}
A \bfidx{$\sigma$-algebra} of functions $\mcF$ satisfies the following
conditions:

\begin{itemize}
\item $\mcF$ is a Stone vector lattice which contains constant functions.
\item $\mcF$ is closed under pointwise convergence.
\item (Note:  a $\sigma$-ring is the same except it needn't have
	constant functions.)
\end{itemize}

\end{defn}

%% - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
\begin{defn}
Let $\mu: \mcF^+ \to [0, +\infty]$.
Then $\mu$ is an \bfidx{integral} if the following conditions are
satisfied:
\begin{itemize}
\item asdfadsr
\item asdfadsr
\item asdfadsr
\end{itemize}
\end{defn}

%% ================================================================
\section{To be filed}

finite meas/int; $\sigma$-finite meas/int.

measure space

$\sigma$-ring and $\sigma$-algebra on \emph{sets} and \emph{functions}.

$$
\sum f(x_i)\Delta x_i \textrm{ vs. }
\sum y_i \mu(f^{-1}y_i)?
$$

%% ================================================================
\section{Terminologies}

%% ================================================================
\appendix
\newpage
%% ================================================================
\section{Notation}
\label{appx:notation}

\begin{notation}
We write $\N$ for the non-negative integers, $\Z$ for the integers,
$\Q$ for the rationals, and $\R$ for the reals.
\end{notation}

Throughout this section, let $X$ be a set.  Let $a \in \R$ and $f,g:X \to \R$.

\begin{defn}
The \bfidx{characteristic function} on $X$ is given by
$$
	1_X(x) = \begin{cases}
		1 & x \in X \\
		0 & x \not\in X \\
	\end{cases}
$$
\end{defn}

Note that some authors use a $\chi$ or $K$ in place of 1.

\begin{defn}
We define $f+g$ and $af$ to be
$$
	(f+g)(x) = f(x)+g(x), \quad (af)(x) = af(x) \textrm{ for } x \in X.
$$
\end{defn}

\begin{notation}
If
$$
	f(x) < g(x) \quad \textrm{ for all } x \in X
$$
then we write
$$
	f < g.
$$
Likewise for $f \le g$, $f>g$, and $f \ge g$.
\end{notation}

\begin{defn}
We define $f \wedge g$ to be the function
$$
	(f \wedge g)(x) = \inf\{f(x), g(x)\} \textrm{ for } x \in X.
$$
Likewise,
we write $f \vee g$ for the function
$$
	(f \vee g)(x) = \sup\{f(x), g(x)\} \textrm{ for } x \in X.
$$
\end{defn}

\begin{mnem}
The second letter of \emph{infimum} is \emph{n}, which looks something like
$\wedge$.
The second letter of \emph{supremum} is \emph{u}, which looks something like
$\vee$.
\end{mnem}

\begin{notation}
We write $f_n$ for a sequence of functions from $X$ to $\R$, for
$n = 0, 1, 2, \ldots$.
\end{notation}

\begin{notation}
Let $f_n$ be a sequence of functions from $X$ to $\R$.
We write $f_n \upto f$ or $f_n \downto f$ if:
\begin{itemize}
\item For all $n$, $f_{n} \le f_{n+1}$ or $f_{n} \ge f_{n+1}$,
	respectively.
\item For all $n$, $f_{n} \le f$ or $f_{n} \ge f$, respectively.
\item $f_n$ converges to $f$ \plainidx{pointwise}.
\end{itemize}

\end{notation}
