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.

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