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

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

This property lets us prove $\varnothing$ is measurable with the expected area:

This property also lets us state the monotone property:

Invariance Under Congruence

If $S\in \mathcal{M}$ and $T$ is shape 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 $a(Q)=c$.

Regions Between Graphs

If two functions $f$ and $g$ are related by the inequality $f(x)\le g(x)$ for all $x$ in an interval $[a,b]$, we write $f\le g$ on $[a,b]$. If $f\le g$ on $[a,b]$, the set $S$ consisting of all points $⟨x,y⟩$ satisfying the inequalities

is called the region between the graphs of $f$ and $g$.

If $f$ and $g$ are integrable functions on $[a,b]$, then the area of the region between the graphs of $f$ and $g$ satisfies

Similarity Transformations

Let $S$ be a set of points on the plane. Let $kS$ denote the set obtained by multiplying the coordinates of each point of $S$ by $k>0$. Then $kS$ is similar to $S$ and the process by which we produced $kS$ is called a similarity transformation.

Let $f$ be nonnegative and integrable on $[a,b]$. Let $S$ be its ordinate set. Then for $k>0$,