Irrational rotations motivate measurable sets

In 1914 Constantin Carathéodory gave his definition of a measurable set, a definition that is crucial in the general theory of integration. This is because a “primitive” notion of area or measure, on a smaller family of sets, can be extended to the larger family of measurable sets, and it is this larger family that has the necessary properties for a natural and complete theory of integration. This more general theory of integration is of enormous practical importance, for it leads to quite broad conditions under which basic operations on integrals are valid. However, Carathéodory’s definition itself remains mysterious, and has long been the subject of comment to this effect, because of the gap between the definition and the consequences which it has. For example, the existence of this gap was the basis upon which Imre Lakatos, in his work “Proofs and Refutations”, was critical of the type of didactic approach often taken in mathematical exposition. This paper examines critically the argument of Lakatos and responds to it by discussing a specific “problem situation” which leads to the Carathéodory definition. The problem is: calculate the outer measure of a subset of the unit circle that is invariant under an irrational rotation of the circle.