Let $f:\mathbb{R}^n\to\mathbb{R}^n$ be a linear map represented by matrix A.
In this case the determinant of A can be interpreted as a measure of change in orientation and volume of a measurable set in n-dimensional space.
I understand how this is derived when looking at how the volume of a n-dimensional cube is transformed under the linear map, however, I am interesed in the generalization of this property to all measurable sets in n-dimensional space.
Simply put, I am looking for some kind of verification, that the determinant represents this measure of change for any shape or more generally a measurable set. Its easy for me to see, when the shape is a square or cube with vectors as its sides, however, in a more complicated case, its not very clear to me.