Can there be a pair of models $M\subset N$ of ZFC and an $X\in\mathcal{P}(\mathbb{R})^M$ such that $$0<\mu^*(X)^N<\mu^*(X)^M?$$
(Here "$\mu^*$" denotes Lebesgue outer measure.)
That is, can we change the (outer) measure of a set of reals by passing to a larger model without killing it completely (= making it null)? Certainly we have to have the set $\mathbb{R}^N\setminus\mathbb{R}^M$ be "large" in order to do this (for example, $N$ has to contain a real coding a cover of $X$ more efficient than any cover in $M$), but beyond that I can't seem to get any purchase.
I recall seeing a fairly easy proof that the answer is no, but I can't reconstruct it or find a reference for it at the moment (even under additional assumptions - e.g. that $N$ is a generic extension of $M$).
from Hot Weekly Questions - Mathematics Stack Exchange
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