# Gently changing measure

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 $$\mu^*(A)^M\not=\mu^*(A)^N$$?

• The "forcing" and "inner-models" tags are because those are the primary ways we know how to build transitive models "to order." By "can there be" I mean "is it consistent with ZFC + large cardinals." I'm also interested in the same question nontransitive models (and demanding that $N$ be an end-extension of $M$), but I'm primarily interested in the transitive case. Meanwhile, I'm not very 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. Jan 9, 2020 at 16:47
• Do you have the measure inequality backwards in Question 1? Jan 9, 2020 at 17:53
• If $N$ is a generic extension of $M$ obtained by forcing with a weakly homogeneous forcing, then the asnwer to Question 2 is negative (Lemma 6.3.10 in the Bartoszynski-Judah book). I don't know if we can drop weakly homogeneous here. Jan 13, 2020 at 6:11
• @Ashutosh Ah neat, I didn't know that! But the general situation (esp. non-generic extensions) is still unclear, right? Jan 13, 2020 at 6:12
• Yes. Some related questions are discussed in Kellner, Shelah, Preserving preservation, JSL, Vol. 70 No. 5, 2005 (See section 3). But I didn't see your question being addressed there. Jan 13, 2020 at 6:32