We have a linear map $L : \mathbb R^n \rightarrow \mathbb R^m $ ($m\geq n$)
I have two questions:
- How does one prove that $\mathcal H^n (L(B(x,r)))=\mathcal L^n (O^* \circ L(B(x,r)))$?
(We have the polar decomposition of L as $L=O\circ S$ where $O$ is an orthogonal matrix and $S$ is a symmetric matrix).
- Why is the push-forward of $\mathcal H^n$ under the map L, i.e. $\nu (A)= \mathcal H^n (L(A))$ $A \subset \mathbb R^n$ a Radon Measure?