Lebesgue Measure Invariant under Orthogonal Group

Theorem
Let $M \in \map {\mathrm O} {n, \R}$ be an orthogonal matrix.

Let $\lambda^n$ be $n$-dimensional Lebesgue measure.

Then the pushforward measure $M_* \lambda^n$ equals $\lambda^n$.

Proof
By Orthogonal Group is Subgroup of General Linear Group, $M \in \GL {n, \R}$.

From Pushforward of Lebesgue Measure under General Linear Group, it follows that:


 * $M_* \lambda^n = \size {\det M^{-1} } \lambda^n$

Since $M^{-1} \in \map {\mathrm O} {n, \R}$ by Orthogonal Group is Group, Determinant of Orthogonal Matrix applies to yield:


 * $\size {\det M^{-1} } = 1$

Hence the result.