Lebesgue Measure Invariant under Orthogonal Group

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Theorem

Let $M \in \operatorname O \left({n, \R}\right)$ 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, $M \in \operatorname{GL} \left({n, \R}\right)$.

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

$M_* \lambda^n = \left\vert{\det M^{-1}}\right\vert \lambda^n$

Since $M^{-1} \in \operatorname O \left({n, \R}\right)$ by Orthogonal Group is Group, Determinant of Orthogonal Matrix applies to yield:

$\left\vert{\det M^{-1}}\right\vert = 1$


Hence the result.

$\blacksquare$

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