Many proofs in elementary geometry use an intuitive but imprecise definition of the area or the volume. For example, Euclid's first proof of the Pythagorean Theorem uses the fact that all triangles of a given base and a given height have the same area, which (in modern terms) say that the area is invariant by shear motions $\begin{pmatrix} 1 & \lambda \\ 0 & 1 \end{pmatrix}$. On the other hand, my favourite proof of the same theorem (apparently quite close to Euclid's second proof and also discovered by a 12-year old Albert Einstein, see this great article by David Mumford) is an transparent corollary of the fact that the area of something is multiplied by $\lambda^2$ when the something is dilated by a factor $\lambda$.
These and other similar arguments convinced me that what I want of a volume is the two following properties:
The volume is additive : if you cut something into two somethings, the volume of the big one is the sum of the volumes of the two little ones (with obvious but quite tedious to write down generalisations if the two little ones aren't disjoint but have a small intersection, for example an intersection included in a hyperplane).
The volume is a relative invariant, meaning that if you apply an affine transformation $A$ to a something, its volume is multiplied by $|\det(A)|$.
Of course, Lebesgue measure does precisely that. To be more specific, the basic tenets of the Lebesgue measure, at least in the standard presentation, are somehow different (one wants a countable additivity and doesn't hard-code the invariance in the definition, but only requires invariance by translation) but the relative invariance in the sense I gave is a classical and easy property of the Lebesgue measure, so hurray!
My question is: if I only want the two properties I've stated (not countable additivity) and only want to measure the volume of very nice things (let's say: polyhedra), how more low-tech can I be? For example, I could probably define the volume by using a simpler, older integration method (Riemann's one, for instance), but can I get significantly more elementary?
For example, a natural method would be to cut my polyhedron into simplices, define the volume of a simplex to be the determinant of its contituent vectors (up to some constant, n!, if I'm correct) and declare the volume of the polyhedron to be the sum of the volumes of the simplices. However, you would have to prove that this volume doesn't depend on your choice of decomposition in simplices, and I fear this would be a nightmare to prove elementarily, even more so in higher dimensions where all kind of bad things may happen, like the failure of the Hauptvermutung.
So: does such an elementary but rigorous definition of the volume exist?
(I think this question may turn out to have two different answers, one for small dimension and one for the general case, so to be clear, I'm interested in both cases).