KL divergence of models with both continuous and discrete variables

When $x$ is discrete, KL divergence is $D_{KL}(P||Q)=\sum\limits_{x}P(x)\log \frac{P(x)}{Q(x)}$, when $x$ is continuous, $D_{KL}(P||Q)=\int\limits_{x}p(x)\log \frac{p(x)}{q(x)}dx$. However, when the space of the random variable $x$ is defined on mixed continuous and discrete space, what would be the KL divergence? For example, $x=(r,a)$, where $r$ is a continuous variable that follows Gaussian dis