 Solution

Integrating the joint probability density over the auxiliary coordinate, as in (i), gives the marginal distributionIn particular, augmenting with $U$ preserves the target probability distribution of $X$. The other marginal distribution, though not needed to identify $X$, is ${\pi }_U(u)=M{\int }_{\Omega }q(x)1_{\{w(x)\ge Mu\}}dx$ for $0<u<1$. It need not be a uniform distribution on $(0,1)$.