bookshelf/Common/Real/Geometry/Area.tex

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\documentclass{article}
\input{../../../preamble}
\newcommand{\lean}[2]{\leanref{./Area.html\##1}{#2}}
\begin{document}
\header{Axiomatic Framework of Area}{Tom M. Apostol}
We assume there exists a class $\mathscr{M}$ of measurable sets in the plane and
a set function $a$, whose domain is $\mathscr{M}$, with the following
properties:
\section*{\defined{Nonnegative Property}}%
\label{sec:nonnegative-property}
For each set $S$ in $\mathscr{M}$, we have $a(S) \geq 0$.
\begin{axiom}
\lean{Nonnegative-Property}{Nonnegative Property}
\end{axiom}
\section*{\defined{Additive Property}}%
\label{sec:additive-property}
If $S$ and $T$ are in $\mathscr{M}$, then $S \cup T$ and $S \cap T$ are in
$\mathscr{M}$, and we have $a(S \cup T) = a(S) + a(T) - a(S \cap T)$.
\begin{axiom}
\lean{Additive-Property}{Additive Property}
\end{axiom}
\section*{\defined{Difference Property}}%
\label{sec:difference-property}
If $S$ and $T$ are in $\mathscr{M}$ with $S \subseteq T$, then $T - S$ is in
$\mathscr{M}$, and we have $a(T - S) = a(T) - a(S)$.
\begin{axiom}
\lean{Difference-Property}{Difference Property}
\end{axiom}
\section*{\defined{Invariance Under Congruence}}%
\label{sec:invariance-under-congruence}
If a set $S$ is in $\mathscr{M}$ and if $T$ is congruent to $S$, then $T$ is
also in $\mathscr{M}$ and we have $a(S) = a(T)$.
\begin{axiom}
\lean{Invariant-Under-Congruence}{Invariance Under Congruence}
\end{axiom}
\section*{\defined{Choice of Scale}}%
\label{sec:choice-scale}
Every rectangle $R$ is in $\mathscr{M}$.
If the edges of $R$ have lengths $h$ and $k$, then $a(R) = hk$.
\begin{axiom}
\lean{Choice-of-Scale}{Choice of Scale}
\end{axiom}
\section*{\proceeding{Exhaustion Property}}%
\label{sec:exhaustion-property}
Let $Q$ be a set that can be enclosed between two step regions $S$ and $T$, so
that
\begin{equation}
\label{sec:exhaustion-property-eq1}
S \subseteq Q \subseteq T.
\end{equation}
If there is one and only one number $c$ which satisfies the inequalities
$$a(S) \leq c \leq a(T)$$ for all step regions $S$ and $T$ satisfying (1.1),
then $Q$ is measurable and $a(Q) = c$.
\begin{axiom}
\lean{Exhaustion-Property}{Exhaustion Property}
\end{axiom}
\end{document}