 Solution

For the Random-scan Gibbs sampler, the correct transition kernel is a measure, since a one-coordinate update is singular with respect to full-dimensional Lebesgue measure when $p>1$. Write $x_{-i}$ for all coordinates except $i$. ThenEach Dirac measure fixes an unupdated coordinate. Equivalently, $P(x,A)=p^{-1}{\sum }_i\int 1_A(x_{-i},z){\pi }_i(z\mid x_{-i})dz$, with $z$ placed in coordinate $i$. Replacing this kernel by an ordinary density on the whole product space would omit those fixed-coordinate constraints.