Suppose I have two forcing notions $\mathbb{P}$ and $\mathbb{Q}$, and a $\mathbb{Q}$-name $\nu$ for a real. Let $G$ be $\mathbb{P}$-generic, and suppose $r$ is a real in $M[G]$ such that for some $H$ which is $\mathbb{Q}$-generic over $M$ we have $\nu[H]=r$.
Technically I'm being a bit sloppy here in saying things like "for some $H$," but we can express the setup above in $M$ appropriately, as usual.
We may then ask whether "$\nu[H]=r$" can be "recovered" in $M[G]$:
- Is there a generic extension $M[G][J]$ of $M[G]$ containing some $J'\subseteq\mathbb{Q}$ which is $\mathbb{Q}$-generic over $M$ and satisfies $\nu[J']=r$?
The answer to this is "yes," and the most natural way to produce such a $J$ is given by Solovay at page 21 of his classic paper on the Solovay model (see also the description in this question; also, see Kanovei's interesting generalization). But I'm not restricting attention to that specific approach.
What I want to know is whether the process of building this $J$ can be "nice," assuming that $\mathbb{P}$ and $\mathbb{Q}$ are each appropriately "nice." For instance:
Suppose $\mathbb{P}$ is proper and $\mathbb{Q}$ is proper in $M[G]$; can we produce $J$ via a proper forcing extension of $M[G]$?
The answer for the analogous question for c.c.c. is almost certainly "no," but the "obvious" counterexample I had doesn't actually work, so I'm also interested in that. And more generally:
What is known about the properties preserved by Solovay's construction?
