*This question was asked and bountied on MSE without answer, so I'm porting it here:*

There's an easy way to change the measure of a set of reals by moving to a larger universe: simply make $\mathbb{R}$ of the ground model be null. Conversely, it's not hard to show that if $M\subseteq N$ are transitive models of ZFC and $A\in\mathcal{P}(\mathbb{R})^M$ is measurable in $M$ and has positive measure in $N$, then $m(A)^M=m(A)^N$, so at least for "nice" sets that's the only way measure can change via this process. *The answer there is stated for forcing extensions only, but that's unnecessary.*

For non-measurable sets things are more complicated. However, the construction given there still somewhat fits the "nullify-or-leave-unchanged" pattern: the anomolous set is built from two pieces such that we can make one null while not changing the outer measure of the other. I'm interested in whether this is optimal:

Question 1: Can there be a pair $M\subseteq N$ of transitive models of ZFC and a set of reals $A\in M$ such that $(i)$ $\mu^*(A)^M>\mu^*(A)^N$ but $(ii)$ there is no partition $A=B\sqcup C$ with $B,C\in M$ such that $\mu^*(B)^N=0$ and $\mu^*(C)^N=\mu^*(C)^M$?

I suspect the answer is no. Annoyingly, I haven't been able to make any progress on this, and in particular I can't even rule out the following extreme (and "obviously" ridiculous) possibility:

Question 2: Can there be a pair $M\subseteq N$ of transitive models of ZFC such that no non-null $A$ in $M$ is null in $N$ but some $A$ in $M$ has $\mu^*(A)^M\not=\mu^*(A)^N$?

notvery intereted in going below ZFC and I'm not interested at all (at the moment) in dropping below ZF + DC, since the question is less exciting if measure is nastier. $\endgroup$1more comment