20.3: Definition of area

Area is defined as a function \(\mathcal{P} \mapsto \text{area } \mathcal{P}\) that returns a nonnegative real number \(\text{area }\mathcal{P}\) for any polygonal set \(\mathcal{P}\) and satisfying the following conditions:

1. \(\text{area }\mathcal{K}_1=1\) where \(\mathcal{K}_1\) a solid square with unit side;
2. the conditions \[\begin{array} {ccc} {\mathcal{P} \cong \mathcal{Q}} & \Rightarrow & {\text{area } \mathcal{P} = \text{area } \mathcal{Q};} \\ {\mathcal{P} \subset \mathcal{Q}} & \Rightarrow & {\text{area } \mathcal{P} \le \text{area } \mathcal{Q};} \\ {\text{area } \mathcal{P} + \text{area } \mathcal{Q}} & = & {\text{area } (\mathcal{P} \cup \mathcal{Q}) + \text{area } (\mathcal{P} \cap \mathcal{Q})} \end{array}\] hold for any two polygonal sets \(\mathcal{P}\) and \(\mathcal{Q}\).

The first condition is called normalization; essentially it says that a solid unit square is used as a unit to measure area. The three conditions in (b) are called invariance, monotonicity, and additivity.

Lebesgue measure, provides an example of area function; namely if one takes \(\text{area }\mathcal{P}\) to be Lebesgue measure of \(\mathcal{P}\), then the function \(\mathcal{P}\mapsto\text{area }\mathcal{P}\) satisfies the above conditions.

The construction of Lebesgue measure can be found in any textbook on real analysis. We do not discuss it here.

If the reader is not familiar with Lebesgue measure, then he should take existence of area function as granted; it might be considered as an additional axiom altho it follows from the axioms I-V.