I am currently reading "Products of Random Matrices with Applications to Schrodinger Operators" book and encountered the following problem.
Let $m$ be the normalized Lebesgue measure on $S\subset\mathbb{R}^{d}$ (unit sphere in $\mathbb{R}^{d}$), i.e $m(S)=1$. Consider $\tau:S\rightarrow S$, $x\rightarrow \frac{Yx}{||Yx||}$, where $Y$ is a matrix with $|det(Y)|=1$. Let $Ym$ be the image of $m$ under the map $\tau$. Show that $\frac{d(Ym)}{dm}(x)=||Y^{-1}x||^{-d}$.
I am completely stuck at this problem and need some help. The hint given in the problem is the following: The measure $\rho$ defined for each borel set $A\subset\mathbb{R}^{d}$ by $\rho(A)=\int_{S}\int_{0}^{\infty} 1_{A}(rx)r^{d-1}dr dm$ is a multiple of Lebesgue measure. I can't see how to use this hint.