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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