Let $\mu$ be a measure over the product space $X\times Y$, $X$ is any topological space and $Y$ is either Lindelöf or compact, and let $\mu_y$ and $\mu_x$ denote the marginals of $\mu$ on $X$ and $Y$, respectively. Suppose that there is a regular conditional distribution $\mu(\cdot|y)$ such that $$ d\mu(x,y)=d\mu(x|y)\times d\mu_y(y). $$
If $\mu(\cdot|y)$ is continuous in the total variation norm with respect to $y$, is it true that $\mu$ is absolutely continuous with respect to the product measure in $X\times Y$?
