Let $\pi$ be the target distribution on $(\mathbb{R}^d,\mathcal{B}(\mathbb{R^d}))$ which is absolutely continuously wrt to the $d$-dimensional Lebesgue measure, i.e :
$\pi$ admits a density $\pi(x_1,...,x_d)$ wrt to $\lambda^d$ with $$\lambda^d(dx_1,...,dx_d) = \lambda(dx_1) \cdot \cdot \cdot \lambda (dx_d)$$
Let us assume that the full conditionals $\pi_i(x_i|x_{-i})$ from $\pi$ are known. So the transition kernel of the Gibbs-Sampler is clearly the product of the full conditionals from $\pi$.
Is the transition kernel absolutely continuously wrt to the $d$-dimensional Lebesgue measure too ?