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 .

Note that

So

This implies .

So we prove the our theorem. Since is a hyperplane of , there is and such that

Define

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

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