Let $X$ be a set, and let $\mathcal G \subset \mathcal F$ be $\sigma$-fields over $X$. Let $\Delta_\mathcal G$ (resp. $\Delta_\mathcal F$) be the set of probability measures on $\mathcal G$ (resp. $\mathcal F$). Let $\mathcal D_{\mathcal G}$ (resp. $\mathcal D_{\mathcal F}$) be the $\sigma$-field over $\Delta_\mathcal G$ (resp. $\Delta_\mathcal F$) generated by the functions of the form $P \mapsto P(A)$, $A \in \mathcal G$ (resp. $A \in \mathcal F$). If $A$ is a measurable subset of $\Delta_\mathcal G$, let $\mathcal D_{\mathcal G}|_A$ denote the trace $\sigma$-field over $A$.
A measurable extension from $A \in \Delta_{\mathcal G}$ is a measurable function $f:(A, \mathcal D_{\mathcal G}|_A) \to (\Delta_{\mathcal F}, \mathcal D_{\mathcal F})$ such that the restriction of $f(P)$ to $\mathcal G$ is identical to $P$.
I am wondering if measurable extensions have been studied in any detail, and, in particular, what can be said about when they exist. I am also wondering whether there are any interesting applications of this concept.
The motivation for the definition comes from a well-known paper by Blackwell and Ryll-Nardzewski ("Non-existence of everywhere proper conditional distributions,'' 1962). They define a proper conditional distribution on $(X, \mathcal F)$ given $\mathcal G$ to be a $\mathcal G$-measurable function $Q$ from $X$ into $\Delta_{\mathcal F}$ with the property that, for all $x \in X$, the restriction of $Q(x)$ to $\mathcal G$ is point mass at $x$, i.e. $\delta_x$. They then show that when $(X, \mathcal F)$ is a standard space a proper conditional distribution exists if and only if a $\mathcal G$-measurable selection function exists, that is, a $\mathcal G$-measurable function $g: X \to X$ such that $g(x) \in G$ for all $x \in G \in \mathcal G$.
Notice that a proper conditional distribution exists if and only if there is a measurable extension from the set of point masses in $\Delta_{\mathcal G}$. Indeed, if the measurable extension $f$ exists, then (because $d: (X, \mathcal G) \to (\Delta_{\mathcal G}, \mathcal D_{\mathcal G}); x \mapsto \delta_x$ is measurable), $Q(x) = f(d(x))$ defines a proper conditional distribution. Conversely, if a proper conditional distribution $Q$ exists, then $f(\delta_x) = Q(x)$ defines a measurable extension.
So, essentially, I am wondering whether this notion of measurable extension is a potentially interesting generalisation of proper conditional distributions.
