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 $X\times Y$?

  • $\begingroup$ Well...is it? What are your thoughts? $\endgroup$
    – Math1000
    Oct 11, 2019 at 19:33
  • $\begingroup$ I would like it not to be true if I’m honest but I can’t find an example. What do you think? $\endgroup$
    – Condor5
    Oct 11, 2019 at 20:31
  • $\begingroup$ I'm not sure actually. Cannot find a counterexample either. $\endgroup$
    – Condor5
    Sep 12, 2020 at 15:18


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