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20.3: Definition of area

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Area is defined as a function Parea P that returns a nonnegative real number area P for any polygonal set P and satisfying the following conditions:

  1. area K1=1 where K1 a solid square with unit side;
  2. the conditions PQarea P=area Q;PQarea Parea Q;area P+area Q=area (PQ)+area (PQ) hold for any two polygonal sets P and 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 area P to be Lebesgue measure of P, then the function Parea 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.


This page titled 20.3: Definition of area is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Anton Petrunin via source content that was edited to the style and standards of the LibreTexts platform.

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