Theorem. Let be a hyperplane of . Then has Lebesgue measure zero.
Proof. To show it, we first show
has measure zero.
Let be given. Then set
By archimedian property, we get .
This implies .
So we prove the our theorem. Since is a hyperplane of , there is and such that
Note that by translation invariant of Lebesgue measure, we have
Set and extend to the orthonormal basis for . From this, there is a orthogonal transform such that . So