A natural motivation for the notion of measure arises from the problem of determining the area, volume, or hyper-volume of some subset of Euclidean space. For let capture this āintuitiveā notion of size. Then intuitively, we would like to satisfy these following properties:
where are all disjoint from one another, and implies that can be transformed into through some combination of rotations, reflections or translations.
However, such a map cannot exist. In fact, it cannot exist for any choice of . For example, let consist of one element between and each from the set of equivalence classes generated by . Such an exists given the Axiom of Choice. Because takes one element from each equivalence class, for any in , is , we can assert .
Consider the half closed interval . Because of how we defined , . Furthermore, if , we define a map defined by
then is in and and is disjoint if . Furthermore if an element then as consists of elements from every equivalence class there is such that i.e. or . Thus we must have
thus we expect
Clearly, this cannot hold as are all equal to one another because of two. If , the sum would be equal to 0 and if the sum would not converge.
Even if were relaxed to hold only when is finite, such a measure clearly does not exist as should be evident when considering the Banach-Tarsiki paradox.
Thus such a āvolumeā map cannot exist on all elements of . The natural solution is to consider a volume map which is only defined on some elements of . This naturally extends to the notion of a measurable set.