Generic Metropolis-Hastings kernel in measure theory

Question

I'm facing difficulties in formulating the Metropolis-Hastings kernel for a specific problem where I need to sample from a probability distribution involving both discrete and continuous degrees of freedom (e.g., a state $s=(x,c)$ representing the position $x$ of a particle and its colour $c$).

1. Is it safe to assume that this probability has a density $p$ with respect to some generic measure $\mu$ on the space $(S,\Sigma)$?
2. Can I define the proposal kernel $Q$ to be $Q(s,A)$ for $A\subset S$? E.g. say I draw my proposal $s'=(x',c')$ by sampling $x'$ from a gaussian and $c'$ from a Bernoulli.
3. Can I assume this proposal kernel has density $q$ with respect to the same measure $\mu$? I'd say this measure is neither the Lebesgue nor the counting measure as the measure space is neither discrete nor continuous.
4. Can I then write the Metropolis-hastings ratio as $$ r(x,x')=\frac{p(x')q(x',x)}{p(x)q(x,x')}, $$ that is, using the densities with respect to $\mu$ and not the measures?