>>14096745No. I will try to give you a explanation because the subject is actually quite subtle.
The ideas of "length", "area" and "volume" are all derivatives of a single concept: measure, but not any measure, Haar measures.
In essence a Haar measure is a function that assigns positive numbers to sets (including 0 are infinity) such that is invariant by isometries (translations, rotations and reflections). But clearly there are infinite Haar measures, so, what you prove is that given a particular Lebesgue-measurable set E, there is only one Haar measure such that E has measure 1.
Traditionally for lengths, such set is a segment, for areas it is the unit square, and for volumes the unit cube. Finally by proving that an "area function" considers segments as null, you realize that scaling a rectangle depends on the scale of both of its sides, therefore the "base * height" formula.
The Lebesgue measure is a Haar measure, and you can prove that if B is a matrix and E is Lebesgue-measurable, then ?( B*E ) = |det B| * ?(E) which also gives rise to the "base x height" formula and many others.
