Pushforward of Lebesgue Measure under General Linear Group

Theorem
Let $M \in \operatorname O \left({n, \R}\right)$ be an invertible matrix.

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

Then the pushforward measure $M_* \lambda^n$ satisfies:


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