TITLE: Conditional probabilities are measurable functions - when are they continuous?
QUESTION [15 upvotes]: Let $\Omega$ be a Banach space; for the sake of this post, we will take $\Omega = {\mathbb R}^2$, but I am more interested in the infinite dimensional setting.  Take $\mathcal F$ to be the Borel $\sigma$-algebra, and let $\mathbb P$ be a probability measure on $(\Omega, \mathcal F)$.
Denote by $\vec x = (x,y)$ a point in $\Omega$, and let $\mathcal F_1$ be the $\sigma$-algebra generated by the first coordinate.  Fix $y_0 \in \mathbb R$ and 
$\eta > 0$, and consider $$f(\vec x) = \mathbb P( ~|y - y_0| \le \eta~ |\mathcal F_1).$$  (More generally, one can consider $f(\vec x) = \mathbb E( \varphi(\vec x) | \mathcal F_1)$ for some suitable $\varphi : {\mathbb R}^2 \to \mathbb R$.)
The function $f$ is measurable; that comes from the definition of conditional expectations.  I would like to find some reasonable sufficient conditions such that $f$ is continuous and positive.  
I feel like this should be relatively elementary material, but unfortunately I'm having trouble finding any references.  How should I approach this?
I've included the [fa.functional-analysis] tag because in general I want to consider $\Omega$ to be a space of smooth functions.  I'm guessing that to give some additional structure, I'll need to assume that $\mathbb P$ is absolutely continuous with respect to a Gaussian measure, because I don't know any other reasonable measures on function space.

REPLY [3 votes]: Tue Tjur studied the existence of continuous disintegrations in a 1975 preprint "A Constructive Definition of Conditional Distributions," Issue 13, Copenhagen Universitet.  He gives necessary and sufficient conditions for their existence.  He also discusses sufficient structure, and there the question of the existence of joint densities arises.  The article is a bit hard to track down, so let me know if you need help finding it.  The existence of continuous disintegrations arises also in the study of the computability of conditional probability, which is my interest.