Add TeX for axiomatic area definition.
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\newcommand{\link}[1]{\lean{../..}
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{Bookshelf/Apostol/Chapter\_1\_11} % Location
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{Apostol.Chapter\_1\_11.#1} % Namespace
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{Apostol.Chapter\_1\_11.#1} % Fragment
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{Chapter\_1\_11.#1} % Presentation
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}
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@ -4,7 +4,7 @@
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\newcommand{\link}[1]{\lean{../..}
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{Bookshelf/Apostol/Chapter\_I\_03} % Location
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{Apostol.Chapter\_I\_03.#1} % Namespace
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{Apostol.Chapter\_I\_03.#1} % Fragment
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{Chapter\_I\_03.#1} % Presentation
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}
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@ -3,9 +3,9 @@
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\input{../../preamble}
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\newcommand{\link}[1]{\lean{../..}
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{Bookshelf/Enderton/Chapter_0} % Location
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{Enderton.Chapter_0.#1} % Namespace
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{Chapter_0.#1} % Presentation
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{Bookshelf/Enderton/Chapter\_0} % Location
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{Enderton.Chapter\_0.#1} % Fragment
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{Chapter\_0.#1} % Presentation
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}
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\begin{document}
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@ -0,0 +1,97 @@
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\documentclass{article}
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\input{../../preamble}
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\newcommand{\link}[2]{\lean{../..}
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{Common/Real/Geometry/Area} % Location
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{#1} % Fragment
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{#2} % Presentation
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}
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\begin{document}
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\header{Axiomatic Framework of Area}{Tom M. Apostol}
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We assume there exists a class $\mathscr{M}$ of measurable sets in the plane and
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a set function $a$, whose domain is $\mathscr{M}$, with the following
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properties:
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\section*{\verified{Nonnegative Property}}%
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\hyperlabel{sec:nonnegative-property}%
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For each set $S$ in $\mathscr{M}$, we have $a(S) \geq 0$.
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\begin{axiom}
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\link{Nonnegative-Property}{Nonnegative Property}
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\end{axiom}
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\section*{\verified{Additive Property}}%
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\hyperlabel{sec:additive-property}%
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If $S$ and $T$ are in $\mathscr{M}$, then $S \cup T$ and $S \cap T$ are in
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$\mathscr{M}$, and we have $a(S \cup T) = a(S) + a(T) - a(S \cap T)$.
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\begin{axiom}
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\link{Additive-Property}{Additive Property}
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\end{axiom}
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\section*{\verified{Difference Property}}%
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\hyperlabel{sec:difference-property}%
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If $S$ and $T$ are in $\mathscr{M}$ with $S \subseteq T$, then $T - S$ is in
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$\mathscr{M}$, and we have $a(T - S) = a(T) - a(S)$.
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\begin{axiom}
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\link{Difference-Property}{Difference Property}
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\end{axiom}
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\section*{\verified{Invariance Under Congruence}}%
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\hyperlabel{sec:invariance-under-congruence}%
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If a set $S$ is in $\mathscr{M}$ and if $T$ is congruent to $S$, then $T$ is
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also in $\mathscr{M}$ and we have $a(S) = a(T)$.
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\begin{axiom}
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\link{Invariant-Under-Congruence}{Invariance Under Congruence}
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\end{axiom}
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\section*{\verified{Choice of Scale}}%
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\label{sec:choice-scale}
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Every rectangle $R$ is in $\mathscr{M}$.
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If the edges of $R$ have lengths $h$ and $k$, then $a(R) = hk$.
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\begin{axiom}
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\link{Choice-of-Scale}{Choice of Scale}
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\end{axiom}
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\section*{\proceeding{Exhaustion Property}}%
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\hyperlabel{sec:exhaustion-property}%
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Let $Q$ be a set that can be enclosed between two step regions $S$ and $T$, so
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that
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\begin{equation}
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\label{sec:exhaustion-property-eq1}
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S \subseteq Q \subseteq T.
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\end{equation}
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If there is one and only one number $c$ which satisfies the inequalities
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$$a(S) \leq c \leq a(T)$$ for all step regions $S$ and $T$ satisfying (1.1),
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then $Q$ is measurable and $a(Q) = c$.
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\begin{axiom}
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\link{Exhaustion-Property}{Exhaustion Property}
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\end{axiom}
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\end{document}
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@ -4,7 +4,7 @@
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\newcommand{\link}[1]{\lean{../../..}
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{Common/Real/Sequence/Arithmetic} % Location
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{Real.Arithmetic.#1} % Namespace
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{Real.Arithmetic.#1} % Fragment
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{Real.Arithmetic.#1} % Presentation
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}
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@ -4,7 +4,7 @@
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\newcommand{\link}[1]{\lean{../../..}
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{Common/Real/Sequence/Geometric} % Location
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{Real.Geometric.#1} % Namespace
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{Real.Geometric.#1} % Fragment
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{Real.Geometric.#1} % Presentation
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}
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@ -17,7 +17,7 @@
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% the root of the workspace (i.e. where this `preamble.tex` file is located).
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% #1 - Path to root
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% #2 - Location
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% #3 - Namespace
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% #3 - Fragment
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% #4 - Presentation
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\newcommand{\lean}[4]{\href{#1/#2.html\##3}{#4}}
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\newcommand{\hyperlabel}[1]{%
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@ -28,6 +28,9 @@
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% Environments
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% ========================================
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\newenvironment{axiom}{%
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\paragraph{\normalfont\normalsize\textit{Axiom.}}}
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{\hfill$\square$}
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\newcommand{\divider}{%
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\vspace{10pt}
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\hrule
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