Can we "slightly" change the measure of a set? 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 $M$).
 A: The following answer is due to hot_queen in the comments above; I'm posting it here and accepting it in order to move this question off the "unanswered" queue. If hot_queen posts their argument as an answer I'll upvote/accept it and delete this one; meanwhile, I've marked this CW so I don't get reputation for their work.
The answer is yes. Briefly, we build a model $M$ with a set of reals $X$ in $M$ such that $(i)$ in $M$, both $X\cap [0,1]$ and $X\cap (1,2]$ have positive outer measure but $(ii)$ we can make the former null without making the latter null.
In a bit more detail:

Over a ground model $S$, add $\omega_1$-many random reals $A=(a_\eta)_{\eta<\omega_1}$ in $[0,1]$, then a Cohen real $c$, then $\omega_1$-many random reals $B=(b_\eta)_{\eta<\omega_1}$ in $(1,2]$. Let $N$ be the whole generic extension $S[A][c][B]$, and $M$ the submodel $S[A][B]$. Then going from $M$ to $N$, $c$ nullifies $A$ but not $B$.

And that basic recipe is highly tweakable.
Of coures this raises the question of whether a mild measure change can happen "irreducibly," which I've asked here.
