My question is similar to Radon-Nikodym derivative as a measurable function in a product space but I am interested in the case where no quasi-invariance is assumed.
Suppose $G$ is a Polish group and $P$ is a probability measure on $G$, and $P_x(A) := P(xA)$. Does there exist a measurable function $\rho:G\times G \to \mathbb{R}$ such that for all $x \in G$, $\rho(x,\cdot)$ is a version of $\frac{dP_x}{dP}$? Since $P$ is not assumed to have the same measure zero sets as $P_x$, we interpret this derivative to mean the Radon Nikodym derivative of the absolutely continuous part of $P_x$ with respect to $P$.
