Sampling the norm and direction separately on R^n

Question

I am using a random variable that is constructed as such :

1. sample a point X uniformly on the unit sphere;
2. scale that point by a norm Y sampled according to some density f.

Intuitively I would compute the density of the product XY using the following change of variable, as I would do for X sampled on the open ball : $$ \begin{align*} \psi\colon S(0,1)&\times \mathbb{R}^{+} \rightarrow \mathbb{R}^n \times \mathbb{R}^{+}\\ (x,y)&\longmapsto (xy, y) \end{align*} $$

Which gives the following inverse :

$$ \begin{align*} \psi^{-1}\colon \mathbb{R}^n \times \mathbb{R}^{+}_{*}& \rightarrow S(0,1) \times \mathbb{R}^{+}\\ (u,v) &\longmapsto (\frac{u}{v}, v) \end{align*} $$

My problem is : the formula for the change of variable does not apply here directly, since the unit sphere is not an open set. How can I bypass this limitation ? Is there a way of making the change of variable work ?

Thank you very much !