Is it true that Lebesgue measure is invariant under isometric map? I mean standard measure of $R^n$.
It is certainly true for interval in $R$ (obvious). I've attempted to prove it in general by induction, and it seems that I succeed in the idea, but don't know how to prove it rigorously. (The idea is to prove that the measure only depends on the mutual distances, which are by definition are equal) Can you, please, give some hints or maybe even solution to this problem, if it's true.
Thank you very much.