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Let $X$ and $Y$ be Radon spaces, $\mu$ a Borel probability measure on $X$, $F\colon X\to Y$ measurable. Then the disintegration theorem gives us a disintegration $\{\mu^y\}_{y\in Y}$ of $\mu$ with respect to $F$. This disintegration is measurable, in the sense that the map $y\mapsto \mu^y(A)$ is measurable for all $A\subset X$ (Borel) measurable.

I've come across conditions for the disintegration maps $y\mapsto \mu^y(A)$ or $y\mapsto \mu^y(f)$ for bounded continuous $f$ to be continuous (Tue Tjur, A Constructive Definition of Conditional Distributions (1975), Renata Possobon and Christian S. Rodrigues, Geometric Properties of Disintegration of Measures (2023)). However these references assume $X$ and $Y$ to be locally compact. This is limiting for me as I am interested in measures on infinite-dimensional spaces, such as Banach spaces.

My question is whether conditions for continuity exist for non-locally compact spaces, or alternatively for Banach spaces.

Any help or pointers are greatly appreciated.

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    $\begingroup$ If you allow the maps $y\mapsto\mu^y(A)$ to be universally measurable, there might be results involving strong liftings that can allow you to have a disintegration $y\mapsto\mu^y$ such that $y\mapsto\mu^y(f)$ is continuous when $f$ is bounded and continuous. $\endgroup$
    – P. P. Tuong
    Commented May 14 at 21:38

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