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Given a regular conditional probability $P(X\in B | T(X) = t)$, where $T$ is a continuous mapping from $\mathcal{X}$ (on which $X$ is defined) to $\mathcal{T}$. Do we know any sufficient condition for such conditional probability to be continuous with respect to $t$? For simplicity one may assume extra structures, such as $\mathcal{T} = \mathbb{R}^d$ (I would like to keep $\mathcal{X}$ to be at least Hilbertian), $T$ is surjective on $\mathcal{T}$, etc.

Related question:Disintegrations are measurable measures - when are they continuous?

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  • $\begingroup$ Does the answer to this question mathoverflow.net/questions/13292 answer your question? $\endgroup$ – R Hahn Nov 26 '16 at 4:32
  • $\begingroup$ @RHahn no, I'm looking into the continuity with respect to the conditioning, namely if the probability changes continuous with the change of the level set of conditioning. $\endgroup$ – newbie Nov 26 '16 at 12:01
  • $\begingroup$ I see you found the actual question I meant to link to, which seems much closer to what you are asking. I don't have anything to add beyond the answers given there, in particular the Tjur article may be helpful. I hope you can get an answer. $\endgroup$ – R Hahn Nov 27 '16 at 1:29

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