20.10: Quadrable sets

A set $\mathcal{S}$ in the plane is called quadrable if for any $\epsilon>0$ there are two polygonal sets $\mathcal{P}$ and $\mathcal{Q}$ such that

$\mathcal{P} \subset \mathcal{S}\subset\mathcal{Q} \quad \text{and} \quad \text{area }\mathcal{Q}-\text{area } \mathcal{P} < \epsilon.$

If $\mathcal{S}$ is quadrable, its area can be defined as the necessarily unique real number $s=\text{area }\mathcal{S}$ such that the inequality

$\text{area }\mathcal{Q}\le s\le \text{area }\mathcal{P}$

holds for any polygonal sets $\mathcal{P}$ and $\mathcal{Q}$ such that $\mathcal{P} \subset \mathcal{S} \subset \mathcal{Q}$.

Exercise $\PageIndex{1}$

Let $\mathcal{D}$ be the unit disc; that is, $\mathcal{D}$ is a set that contains the unit circle $\Gamma$ and all the points inside $\Gamma$.

Show that $\mathcal{D}$ is a quadrable set.

Hint

Let $\mathcal{P}_n$ and $\mathcal{Q}_n$ be the solid regular $n$-gons so that $\Gamma$ is inscribed in $\mathcal{Q}_n$ and circumscribed around $\mathcal{P}_n$. Clearly, $\mathcal{P}_n \subset \mathcal{D} \subset \mathcal{Q}_n$.

Show that $\dfrac{\text{area } \mathcal{P}_n}{\text{area } \mathcal{Q}_n} = (\cos \dfrac{\pi}{n})^2$; in particular,

$\dfrac{\text{area } \mathcal{P}_n}{\text{area } \mathcal{Q}_n} \to 1$ as $n \to \infty$.

Next show that area $\mathcal{Q}_n < 100$, say for all $n \ge 100$.

These two statements imply that $(\text{area } \mathcal{Q}_n - \text{area } \mathcal{P}_n) \to 0$. Hence the result.

Since $\mathcal{D}$ is quadrable, the expression $\text{area }\mathcal{D}$ makes sense and the constant $\pi$ can be defined as $\pi=\text{area }\mathcal{D}$.

It turns out that the class of quadrable sets is the largest class for which the area function satisfying the conditions on page is uniquely defined.

If you do not require uniqueness, then there are ways to extend area function to all bounded sets. (A set in the plane is called bounded if it lies inside of a circle.) In the hyperbolic plane and in the sphere there is no similar construction. If you wonder why, read about doubling the ball paradox of Felix Hausdorff, Stefan Banach, and Alfred Tarski.