Area

Overview

Area is a set function mapping from a class of so-called measurable sets $\mathcal{M}$ into the real numbers.

Axioms

We assume there exists a class $\mathcal{M}$ of measurable sets in the plane and a set function $a$, whose domain is $\mathcal{M}$, with the following six properties:

Nonnegative Property

For each $S\in \mathcal{M}$, $a(S)\ge 0$.

Additive Property

If $S,T\in \mathcal{M}$, then $S\cup T$ and $S\cap T$ are in $\mathcal{M}$. Also $a(S\cup T)=a(S)+a(T)-a(S\cap T).$

Notice this last formulation is a special case of PIE.

Difference Property

If $S,T\in \mathcal{M}$ such that $S\subseteq T$, then $T-S\in \mathcal{M}$ and $a(T-S)=a(T)-a(S).$

Invariance Under Congruence

If $S\in \mathcal{M}$ and $T$ is congruent to $S$, then $T\in \mathcal{M}$ and $a(S)=a(T)$.

Choice of Scale

Every rectangle $R$ is in $\mathcal{M}$. If the edges of $R$ have lengths $h$ and $k$, then $a(R)=hk$.

Exhaustion Property

Let $Q$ be a set. If there exists exactly one $c$ such that $a(S)\le c\le a(T)$ for all step regions $S$ and $T$ satisfying $S\subseteq Q\subseteq T$, then $Q\in \mathcal{M}$ and $a(Q)=c$.

Bibliography

• Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).