Past exam of the Probability, statistics and options II course of the University of Oxford / 2010 / M / M4 / a / i

The inverse change of variables is $x=u$, $y=uv$, whose Jacobian determinant is $u$. Since $X,Y$ are independent, the original joint density is $f(x)g(y)$. ThusThe transformation is nonsingular away from $u=0$; this line has probability zero because $X$ has a density, so no extra mass is omitted.