Measurability of the density of a transition kernel

Question

Let $P_\theta$ be a family of probability distributions on $(\mathcal{X},\mathcal{F})$ such that $\theta \rightarrow P_\theta(A)$ is measurable $(\Theta,\mathcal{G})$ for all $A \in \mathcal{F}$. Let $P_\theta$ have densities $p_\theta(x)$ with respect to a $\sigma$-finite dominating measure $\mu$ on $\mathcal{F}$. My question is, can we conclude that $p_\theta(x) = p(\theta,x)$ is measurable $\mathcal{G} \otimes \mathcal{F}$? If so, what is the argument that shows this?