= 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 $i$. Then
$$
\boxed{P(x,dx')=\frac1p\sum_{i=1}^p\pi_i(x_i'\mid x_{-i})\,dx_i'\prod_{j\ne i}\delta_{x_j}(dx_j').}
$$
Each <Dirac measure> fixes an unupdated coordinate. Equivalently, $P(x,A)=p^{-1}\sum_i\int\mathbf1_A(x_{-i},z)\pi_i(z\mid x_{-i})\,dz$, with $z$ placed in coordinate $i$. Replacing this kernel by an ordinary density on the whole product space would omit those fixed-coordinate constraints.
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